Sajmohan Mohandas Moolayil

Antonio Da Costa

Jean-François Tahon

Vincent Bouad

Ahmed Hamieh

Freddy Ponchel

Vincent Ladmiral

Denis Rémiens

Jean-Marc Lefebvre

Rachel Desfeux

Sophie Barrau

Anthony Ferri

New insight into nanoscale identification of the polar axis direction in organic ferroelectric films

Keywords: Ferroelectric organic, Poly(vinylidene fluoride-co-trifluoroethylene), Polar axis, Edge-on lamella, Nanoscale, Piezoresponse Force Microscopy

Ferroelectric poly(vinylidene fluoride-co-trifluoroethylene) [P(VDF-co-TrFE)] thin films have been deposited by spin-coating onto Bi0.5Na0.5TiO3(BNT)/LNO/SiO2/Si heterostructure.

Copolymer microstructure investigated by using Grazing-Incidence Wide-Angle X-ray Diffraction (GIWAXD) and deduced from the (200)/(110) reflections demonstrates that the baxis in P(VDF-co-TrFE) orthorhombic unit cell is either in-plane or out-of-plane, depending on the face-on or on the two types of edge-on (called I and II) lamellar structures locally identified by Atomic Force Microscopy (AFM). For edge-on I lamellae regions, the electroactivity is found to be twice as high (dzz eff ~ -50.3 pm/V) that measured for both edge-on II or face-on crystalline domains, as probed by Piezoresponse Force Microscopy (PFM). This result is directly correlated to the direction of the ferroelectric polarization vector in the P(VDF-co-TrFE) orthorhombic cell: larger nanoscale piezoactivity is related to the b-axis which lies along the normal to the substrate plane in the case of the edge-on I domains. Here, the ability to thoroughly gain access to the asgrown polar axis direction within the edge-on crystal lamellae of the ferroelectric organic layers is evidenced by combining the nanometric resolution of the PFM technique with a statistical approach based on its spectroscopic tool. By gathering information at the nanoscale, two orientations for the polar b-axis are identified in edge-on lamellar structures. These findings contribute to a better understanding of the structure-property relationships in P(VDF-co-TrFE) films, which is a key issue for the design of future advanced organic electronic devices.

INTRODUCTION

The growing emphasis on understanding the nanoscale properties, in correlation with the structure and morphology of electroactive polymer materials, has become major scientific concern in recent years. Electroactive organic materials are extensively studied and manipulated to enhance their properties and induce new functionalities for the design of electronic devices such as field effect transistor (FET), transducers and data storage devices. [START_REF] Asadi | Crossbar Memory Array of Organic Bistable Rectifying Diodes for Nonvolatile Data Storage[END_REF][START_REF] Asadi | Organic Non-Volatile Memories from Ferroelectric Phase-Separated Blends[END_REF][START_REF] Naber | Organic Nonvolatile Memory Devices Based on Ferroelectricity[END_REF][START_REF] Naber | High-Performance Solution-Processed Polymer Ferroelectric Field-Effect Transistors[END_REF] One major candidate among ferroelectric organic materials is poly(vinylidene difluoride) (PVDF) and its derivative copolymer, poly(vinylidene-co-trifluoroethylene) P(VDF-co-TrFE). Regarding the latter, the low temperature ferroelectric phase presents orthorhombic lattice Cm2m similar to PVDF with extended lattice parameters. [START_REF] Calleja | Structure and Properties of Ferroelectric Copolymers of Poly(vinylidene fluoride)[END_REF][START_REF] Tashiro | Structural Study on Ferroelectric Phase Transition of Vinylidene Fluoride-Trifluoroethylene Copolymers (III) Dependence of Transitional Behavior on VDF Molar Content[END_REF][START_REF] Kim | High-Performance Needle-Shaped Crystals in Thin and Ultrathin P(VDF-TrFE) Films Formed by Melt Recrystallization[END_REF][START_REF] Chang | Direct-Write Piezoelectric Polymeric Nanogenerator with High Energy Conversion Efficiency[END_REF][START_REF] Liu | Rapid Nanoimprinting and Excellent Piezoresponse of Polymeric Ferroelectric Nanostructures[END_REF] The copolymer displays all-trans chain conformations, denoted as FE (ferroelectric) phase or sometimes β-phase in reference to β-phase of PVDF, as shown in Figure 1a. Indeed, this FE-phase has the planar zigzag all-trans (TTTT) conformation, resulting in the CH2-CF2 and CHF-CF2 dipoles lying along the normal direction to the polymer chains backbone and thus generating at the macroscopic scale a maximum and observable electric dipole moment (remnant polarization Pr ~6 µC/cm) along the b-axis of the unit cell. [START_REF] Martins | Electroactive Phases of Poly(vinylidene fluoride): Determination, Processing and Applications[END_REF][START_REF] Ruan | Properties and Applications of the β Phase Poly(vinylidene fluoride)[END_REF][START_REF] Park | Irreversible Extinction of Ferroelectric Polarization in P(VDF-TrFE) Thin Films Upon Melting and Recrystallization[END_REF][START_REF] Katsouras | The Negative Piezoelectric Effect of the Ferroelectric Polymer Poly(vinylidene fluoride)[END_REF][START_REF] Hafner | Multi-Scale Characterisation of a Ferroelectric Polymer Reveals the Emergence of a Morphological Phase Transition Driven by Temperature[END_REF] Such polar fluoropolymers exhibit a negative longitudinal piezoelectric coefficient (dzz ~ -30 pm/V) [START_REF] Katsouras | The Negative Piezoelectric Effect of the Ferroelectric Polymer Poly(vinylidene fluoride)[END_REF][START_REF] Hafner | Multi-Scale Characterisation of a Ferroelectric Polymer Reveals the Emergence of a Morphological Phase Transition Driven by Temperature[END_REF] , i.e. they contract in the direction of an applied electric field, unlike more conventional inorganic ferroelectrics. The physical properties of electroactive P(VDF-co-TrFE) polymer generally depend on the structure and organization of crystalline lamellae, which depend in turn on the processing conditions, e.g. annealing temperature, applied stress or nature of the solvent used for casting. [START_REF] Asadi | Organic Non-Volatile Memories from Ferroelectric Phase-Separated Blends[END_REF][START_REF] Naber | High-Performance Solution-Processed Polymer Ferroelectric Field-Effect Transistors[END_REF][START_REF] Xia | Removable Polytetrafluoroethylene Template Based Epitaxy of Ferroelectric Copolymer Thin Films[END_REF] Crystallinity improvement of the P(VDF-co-TrFE) thin films may be achieved by annealing or heating the film above its Curie transition (around 135°C). Rod-like topographical features are one of the signatures of optimized crystallinity and ferroelectric response of the films. [START_REF] Ohigashi | Lamellar and Bulk Single Crystals Grown in Annealed Films of Vinylidene Fluoride and Trifluoroethylene Copolymers[END_REF][START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF] The crystalline lamellae of P(VDF-co-TrFE) films grown by different methods have usually an isotropic orientation, generating a lower and non-uniform piezoresponse region. [START_REF] Wu | Orientation of Lamellar Crystals and its Correlation with Switching Behavior in Ferroelectric P(VDF-TrFE ) Ultra-Thin Films[END_REF] While the negative longitudinal piezoelectric response in the crystalline domains was recently described by Liu et al. [START_REF] Liu | Ferroelectric Polymers Exhibiting Negative Longitudinal Piezoelectric Coefficient: Progress and Prospects[END_REF] to be independent of the amorphous regions and of the amorphous-crystalline interface, there is still a debate on the contribution to the piezoelectric response on each component. [START_REF] Zhang | Recent Progress on Structure Manipulation of Poly(vinylidene fluoride)-Based Ferroelectric Polymers for Enhanced Piezoelectricity and Applications[END_REF] The nanoscale piezoelectric/ferroelectric properties of P(VDF-co-TrFE) samples have already been studied using the piezoresponse force microscopy (PFM) technique. [START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF][START_REF] Choi | Enhancement of Local Piezoresponse in Polymer Ferroelectrics via Nanoscale Control of Microstructure[END_REF][START_REF] Qian | Unveiling the Piezoelectric Nature of Polar α-Phase P(VDF-TrFE) at Quasi-Two-Dimensional Limit[END_REF] In particular, the domain switching dynamic was shown by Zhang et al. [START_REF] Zhang | Microimprinting and Ferroelectric Properties of Poly(vinylidene fluoride-trifluoroethylene) Copolymer Films[END_REF] and Gaynutdinov et al. [START_REF] Gaynutdinov | Polarization Switching at the Nanoscale in Ferroelectric Copolymer Thin Films[END_REF] An in-depth study of PFM domain imaging was reported by Sharma et al. [START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF], while the enhancement of ferroelectric properties in P(VDF-co-TrFE) by nanoconfinement was revealed by Liew et al. [START_REF] Liew | Nanoconfinement Induced Crystal Orientation and Large Piezoelectric Coefficient in Vertically Aligned P(VDF-TrFE) Nanotube Array[END_REF] In the present study, we focus on the local identification of the copolymer macromolecular chain orientations and the related polarization by using the spectroscopic tool of the PFM technique, i.e. the switching spectroscopy PFM (SS-PFM) mode. To the best of our knowledge, although it is well-known that the orientation of the ferroelectric polarization is crucial to induce electroactive properties in such organic fluoropolymers (along the normal direction to the macromolecular chains for optimized properties), no detailed investigation about the accurate polar b-axis direction in the orthorhombic P(VDF-co-TrFE) unit cell has been conducted at the nanometer scale when considering the specific vertical crystal lamellae. Indeed, while it is known that these vertical domains lead to a significant enhancement of the piezoelectric response [START_REF] Wu | Orientation of Lamellar Crystals and its Correlation with Switching Behavior in Ferroelectric P(VDF-TrFE ) Ultra-Thin Films[END_REF] , there are two distinct types of out-of-plane oriented crystalline lamellae within the P(VDF-co-TrFE) copolymer, depending on the orientation of the polarization axis in these domains. It thus becomes crucial to be able to identify both types of edge-on crystal lamellae.

EXPERIMENTAL SECTION

Materials.

The commercial (VDF/TrFE 80/20 mol%) P(VDF-co-TrFE) copolymer was provided by Piezotech. (Bi0.5Na0.5)TiO3(BNT)/LaNiO3(LNO)/SiO2/Si heterostructure used as ferroelectric substrate was prepared as described below.

Growth of inorganic underlying layers. The LNO film was sputter deposited at room temperature on top of a SiO2/Si substrate using an RF magnetron sputtering system. It was then annealed at 700 o C in air to form a (00l)-oriented film acting as a conducting substrate and to improve the electrical conductivity of the film. Details on the manufacturing process and properties of LNO are given elsewhere. [START_REF] Hamieh | Synthesis of Lead-Free (Bi0.5Na0.5)TiO3 Thin Film by RF Magnetron Sputtering: Impact of Na on the Properties of Film[END_REF] BNT layer was also sputtered deposited at 250 o C on top of the LNO/SiO2/Si heterostructure previously fabricated using similar RF magnetron sputtering system. The film is then annealed at 650 o C in air to lead to the (00l)-oriented ferroelectric film. A detailed description of these BNT films can be found elsewhere. [START_REF] Detalle | Chemical and Physical Characterization of LaNiO3 Thin Films Deposited by Sputtering for Top and Bottom Electrodes in Ferroelectric Structure[END_REF] The deposited thickness of LNO and BNT layers are 200 and 350 nm, respectively. Fabrication of P(VDF-co-TrFE) thin films. The P(VDF-co-TrFE) films were prepared using the following procedure. 0.5 g of copolymer was dissolved in 10 g of acetone and stirred at ambient temperature until complete solubilization of the copolymer. 400 μL of this solution was then dropped on top of the substrate and spin coated at 2500 rpm for 45 s. The residual solvent was removed by drying the film at 70°C in a vacuum oven for at least 5 h. To finally achieve a complete smooth layer of the copolymer and to lead to self-arranged features on the substrates, we have considered the melt-crystallization process. The films were melted at 180 o C for 3 hours under air, then slowly cooled down with a cooling rate of around 1.25°C/min to room temperature under ambient conditions. A 300-nm-thick copolymer layer was grown.

Characterizations. The structure of the P(VDF-co-TrFE) thin films was characterized by the Grazing-Incidence Wide-Angle X-ray Diffraction (GIWAXD). The experiments were carried out using a Xeuss 2.0 (XENOCS) with a GeniX3D micro-source (λCu = 1.54 Å). The sample was put on the specific GIWAXD sample holder and the sample-to-detector distance was 100 mm.

The surface morphology and the roughness of the copolymer films were characterized by the Tapping mode of the Atomic Force Microscopy (TM-AFM) (MultiMode 8-HR, Bruker, USA).

To assess the nanoscale piezoelectric properties, the dual AC resonance tracking (DART) method of the PFM mode provided by a commercial AFM microscope (MFP-3D, Asylum Research/Oxford Instruments, USA) was used under environmental conditions. The DART-PFM mode allows for amplifying the detected electromechanical signal by scanning at the contact resonance frequency of the cantilever. [START_REF] Rodriguez | Dual-Frequency Resonance-Tracking Atomic Force Microscopy[END_REF] By using SS-PFM, piezoresponse hysteresis loops were recorded over the free surface of the polymer films giving access simultaneously to polarization reversal and piezoelectric activity. The loops were measured in remnant mode, where rectangular voltages pulses were applied and the PFM signal was collected at zero field, promoting electromechanical response while minimizing electrostatic contribution. All local piezoelectric measurements were carried out by using Pt/Ir-coated tip and cantilever with a nominal ∼3.8 N/m spring constant (PPP-EFM, Nanosensors) as the nanometric conductive probe.

RESULTS AND DISCUSSIONS

The structure of deposited P(VDF-co-TrFE) copolymer thin films were finely characterized by WAXD measurements at grazing incidence. The GIWAXD pattern in Figure 2a shows the presence of one main discontinuous ring indicative of a non-isotropic distribution of the P(VDFco-TrFE) macromolecular chains on the BNT substrate. Figure 2b represents the integrated intensity profiles of P(VDF-co-TrFE) layer. The diffractogram exhibits one main peak at 2 = 19.95° corresponding to the (200)/(110) planes and attributed to the polar P(VDF-co-TrFE) crystal phase. [START_REF] Choi | Phase Transition in the Surface Structure in Copolymer Films of Vinylidene Fluoride (70%) with Trifluoroethylene (30%)[END_REF] The calculated interplanar distances are related to an orthorhombic unit cell with the following parameters: a = 0.890 nm, b = 0.505 nm and c = 0.255 nm, [START_REF] Bellet-Amalric | Crystalline Structures and Phase Transition of the Ferroelectric P(VDF-TrFE) Copolymers, a Neutron Diffraction Study[END_REF][START_REF] Hu | Regular Arrays of Highly Ordered Ferroelectric Polymer Nanostructures for Non-Volatile Low-Voltage Memories[END_REF] the macromolecular chains being along the c-axis and the polarization direction oriented along the b-axis. [START_REF] Roth | Aligning In-Plane Polarization Multiplies Piezoresponse in P(VDF-TrFE) Films on Graphite[END_REF] It should be noted that the two (200) and (110) orientations cannot be differentiated by GIWAXD measurements due to similar interplanar distances (see Figure 1b).

As reported by Bargain et al., [START_REF] Bargain | Tencé-Girault S. From Solvent-Cast to Annealed and Poled Poly(VDF-co-TrFE) Films: New Insights on the Defective Ferroelectric Phase[END_REF] a fraction of defective ferroelectric (DFE) phase comprising conformational defects including gauche conformations contributes to a shoulder on the copolymer main diffraction peak. The GIWAXD spectra were then fitted by a pseudo-Voigt function in order to estimate the degree of crystallinity () and the fraction (F) of the ferroelectric (FE) and DFE crystal phases from the ratio of their respective peak area contribution to the total crystal peak area. The peak associated to the DFE is observed at 2 = 18.6°. The crystallinity degree calculated for the copolymer is around  = 70% with a fraction FFE = 55% and FDFE = 14% for the copolymer. Depending upon the different defects present in the P(VDF-co-TrFE) the d-spacing along a-and b-axis of the orthorhombic unit cell increases and the peak position shifts to the lower 2 values. In addition, Spampinato et al. [START_REF] Spampinato | Enhancing the Ferroelectric Performance of P(VDF-co-TrFE) Through Modulation of Crystallinity and Polymorphism[END_REF] reported three different DFE phases existing in the copolymer with respect to the amount of defects.

Figure 3a represents the evolution of the main peak intensity at 2θ = 19.95 o with the azimuthal angle () from 0° to 180°. The intensity is maximum at  = 25° and 160° suggesting the tilt of the d-spacings d200/d110 at the same angle. Considering the (200) reflections in the orthorhombic unit cell, the a-crystallographic axis presents an angle of ±20-25° and the b-axis lays parallel to the substrate plane, respectively. In the case of the (110) reflections, the a-crystallographic axis presents an angle slightly lower than ±20-25° while the b-axis lays quasi-parallel to the substrate plane.

A maximum with a lower intensity is observed at  = 90°. In this case, for the (200) reflection in the orthorhombic unit cell, the a and b crystallographic axes lay perpendicular and parallel to the substrate plane, respectively, while for the (110) reflection, the a and b axes present an angle of 30 and 60° with the substrate plane, respectively. [START_REF] Gutiérrez-Fernández | Morphology and Ferroelectric Properties of Semiconducting/Ferroelectric Polymer Bilayers[END_REF] These oriented orthorhombic unit cells with respect to the plane of the BNT substrate are shown on Figure 3b. The macromolecular chains are then laid along to the substrate implying a fraction of the crystalline lamellae oriented in a direction perpendicular to the substrate (so-called edge-on lamellar structure). These various orientations for the crystal lamellae are illustrated in Figure 3c.

AFM surface morphology

Figure 4 represents the TM-AFM topography of the P(VDF-co-TrFE) thin film. These AFM images give access to the direct visualization of the microstructure of the melt-crystallized copolymer correlated to the macromolecular chain arrangement. As seen in Figure 4a, the P(VDF-co-TrFE) layer deposited onto BNT film displays a complex morphology, leading to a high RMS roughness of ~ 41.0 nm. Horizontal crystalline domains appearing as irregular flakes coexist with perpendicular arranged lamellae showing particular needle-shaped, leading to the generation of many interfaces. Very similar AFM morphology was reported for P(VDF-co-TrFE) film deposited onto silicon substrate. [START_REF] Xia | Epitaxy of Ferroelectric P(VDF-TrFE) Films via Removable PTFE Templates and Its Application in Semiconducting/Ferroelectric Blend Resistive Memory[END_REF] This result is also in correlation with the previously described GIWAXD analyses (Figure 2), where the crystalline lamellae were found to be in-plane (IP) and out-of-plane (OP) oriented, related to well-known face-on and edge-on crystals in such semi-crystalline polymers (see Figure 3c). By further analyzing the surface morphology over the whole surface of the sample, two well-defined kinds of microstructures are clearly evidenced, corresponding to horizontal (face-on) and perpendicular (edge-on) lamellae regions.

Figure 4b highlights the specific face-on zone, where a horizontal growth of the crystalline lamellae leads to the IP stacking of lamellae, forming steps. The curve profile of the AFM topography associated to the green dotted line and shown in Figure 4e reveals a height of around 20 nm for the stacking layers. This value corresponds to the domain periodicity, including crystalline lamella and amorphous region existing in-between two lamellae, and called the long period. The obtained values are in agreement with the value of the long period typically found in semi-crystalline polymers, i.e. of the order of a few tens of nm. [START_REF] Bargain | Tencé-Girault S. From Solvent-Cast to Annealed and Poled Poly(VDF-co-TrFE) Films: New Insights on the Defective Ferroelectric Phase[END_REF][START_REF] Li | Direct Observation of the Growth of Lamellae and Spherulites by AFM[END_REF] Figure 4c represents the detailed morphology of domains formed by perpendicular lamellae structure, called edge-on I lamellar structure hereafter. The lamellae are periodically stacked together along the c-axis of the orthorhombic cell of the P(VDF-co-TrFE) and separated by the amorphous phase, leading to perpendicular arranged structures. The lamellae thickness ranges from 140 nm up to 180 nm, as depicted in the height profile in Figure 4f. Such large values of lamellar thickness (>100 nm) have already been reported for fluorinated polymers. [START_REF] Cabarcos | Toward Chain Extension in Crystals of Fluorinated Copolymers As Revealed by Real Time Ultra-Small-Angle X-ray Scattering[END_REF] Considering that in the present case nonisothermal crystallization occurs with a fairly large residence time between the melt and the Curie transition temperatures, chain extended crystallization process may occur as postulated in [START_REF] Ohigashi | Lamellar and Bulk Single Crystals Grown in Annealed Films of Vinylidene Fluoride and Trifluoroethylene Copolymers[END_REF][START_REF] Hikosaka | Unified Theory of Nucleation of Folded-Chain Crystals and Extended-Chain Crystals of Linear-Chain Polymers[END_REF][START_REF] Hikosaka | Unified Theory of Nucleation of Folded-Chain Crystals (FCCs) and Extended-Chain Crystals (ECCs) of Linear-Chain Polymers: 2. Origin of FCC and ECC[END_REF] . In addition, we observe another type of morphology for edge-on lamellae, called edge-on II in this study, as seen on Figure 4d. In this case, the lamellae are periodically stacked together but perpendicularly to the longitudinal axis seen in the AFM image. The thickness of these stacking arrangements (including both crystalline structures and amorphous regions) ranges from 40 nm up to 120 nm, as shown in the associated height profile in Figure 4g. Such inhomogeneous surface morphology was already observed in the literature for P(VDFco-TrFE) films, depending on various elaboration parameters (thin film fabrication processes, pressure, annealing temperature, thickness or substrate nature, …) [START_REF] Kim | High-Performance Needle-Shaped Crystals in Thin and Ultrathin P(VDF-TrFE) Films Formed by Melt Recrystallization[END_REF][START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF][START_REF] Choi | Enhancement of Local Piezoresponse in Polymer Ferroelectrics via Nanoscale Control of Microstructure[END_REF]2521,3636,[START_REF] Martínez-Tong | Improving Information Density in Ferroelectric Polymer Films by using Nanoimprinted Gratings[END_REF][START_REF] Zhu | Crystalline Morphologies of P(VDF-TrFE) (70/30) Copolymer Films Above Melting Point[END_REF][START_REF] Lee | Annealing Effect Upon Chain Orientation, Crystalline Morphology, and Polarizability of Ultra-Thin P(VDF-TrFE) Film for Nonvolatile Polymer Memory Device[END_REF][START_REF] Guo | Thermally Induced Cooperative Molecular Reorientation and Nanoscale Polarization Switching Behaviors of Ultrathin Poly(vinylidene fluoridetrifluoroethylene) Films[END_REF][START_REF] Hahm | Crystallization and Microstructure-Dependent Elastic Moduli of Ferroelectric P(VDF-TrFE) Thin Films[END_REF] , and is in perfect agreement with the GIWAXD experiments described above. From application perspectives such as nanodevices, the high roughness associated to the heterogeneity of the crystalline morphology is an important limitation. Indeed, the control of the polarization is difficult due to the non-uniform local fields [START_REF] Martínez-Tong | Improving Information Density in Ferroelectric Polymer Films by using Nanoimprinted Gratings[END_REF][START_REF] Sharma | High-Resolution Studies of Domain Switching Behavior in Nanostructured Ferroelectric Polymers[END_REF] as we will show afterwards in the case of the domain manipulation by PFM lithography experiments. Nevertheless, in the context of the present study, such a complex surface morphology is essential.

Poling experiments

The electroactive properties of the PVDF-based copolymer deposited were then investigated at the sub-micrometer scale by locally manipulating the ferroelectric domains via the conductive probing tip of the AFM instrument. Poling experiments were carried out using PFM lithography with DC voltages of ±70 V applied to the AFM tip when scanning square areas at the free surface of the films (the DC voltages were intentionally set at these values in order to overcome the polarizing voltage coercivities deduced from local piezoresponse loops shown afterwards, usually allowing for efficient polarization reversal in both samples). The recorded OP-PFM phase and amplitude responses are shown on Figure 5b,c as well as the corresponding AFM topographic image in Figure 5a. Besides, the sequences of the applied DC voltages are specified on the PFM phase pattern (Figure 5b). Clear inverted contrasts are observed from the phase signal (Figure 5b) which are associated to the domains with upward and downward polarization for bright and dark regions, respectively. These signals suggest the effective switching of the ferroelectric polarization within the polar phase of the grown copolymers originating from the rotation of the permanent dipoles around the macromolecular chains and the movement of domain walls. This result is further confirmed by the profile curve of the PFM phase signal on 5e where the profile curve associated to the dashed green line seen in Figure 5c is shown (blue arrows indicate the amplitude sharply dropping at the created domain walls). In addition, no change in the morphological properties is noticed after the polarizing process, as revealed on the simultaneously recorded AFM image in Figure 5a, attesting that the detected PFM contrasts are exclusively due to the electroactive properties of the PVDF-based copolymer. It is worth noting that uniform OP-PFM phase and amplitude contrasts observed for the copolymer may also agree with the existence of edge-on structure lamellae in which the polarization reversal is more effective owing to the rotation of the dipoles about the caxis upon normal electric field and for which the ferroelectric retention properties are superior. [START_REF] Guo | Flexible Robust and High-Density FeRAM from Array of Organic Ferroelectric Nano-Lamellae by Self-Assembly[END_REF] It should be noted that although the thermal conditions employed in the present study are not ideal for achieving optimized piezoelectric performances in P(VDF-co-TrFE), the quality of domain switching related to the -CF2dipole moments rotation along the poling direction is definitively shown. Indeed, a relatively high degree of polarization alignment is obtained in the manipulated areas.

In addition, we notice that some artificially switched regions were grown beyond the AFM tip writing areas, as marked by the white arrows on the Figure 5b, which can be explained by the domain flow process under the high tip electric fields [START_REF] Hu | Universal Ferroelectric Switching Dynamics of Vinylidene Fluoride-trifluoroethylene Copolymer Films[END_REF] and can further provide an insight in the degree of non-uniformity in the local switching potential previously mentioned. [START_REF] Sharma | High-Resolution Studies of Domain Switching Behavior in Nanostructured Ferroelectric Polymers[END_REF] Furthermore, a particular behavior is observed for the edge-on lamellae structures. The electrical bias-induced polarization orientation within these vertical lamellae during PFM lithography process propagates beyond the scanned area by several micrometers. This phenomenon is particularly highlighted by the white dotted squares and the corresponding insets in Figure 5a,b,c. Bright contrasts for vibration amplitude response as well as for phase signal related to the upwards polarization are clearly visible compared to as-grown (not manipulated) dark regions. This direction of the polar vector well agrees with the surrounded positively polarized regions. The extension of the imposed orientation for the edge-on crystals may again be mainly explained by the domain flow process already reported by Hu et al. 48Erreur ! Signet non défini. This phenomenon has already been observed in PVDF copolymer films, [START_REF] Martínez-Tong | Improving Information Density in Ferroelectric Polymer Films by using Nanoimprinted Gratings[END_REF] where the authors showed that this large extension of ferroelectric signal could be controlled by nanoconfinement. Besides, higher PFM response as well as better ferroelectric retention properties were reported for similar edge-on oriented crystals. [START_REF] Kim | High-Performance Needle-Shaped Crystals in Thin and Ultrathin P(VDF-TrFE) Films Formed by Melt Recrystallization[END_REF]

Piezoloop measurements

To get more precise information about electroactive behavior of the P(VDF-co-TrFE) crystalline domains, the local polarization reversal and electromechanical activity were then probed over the free surface of the sample by using the SS-PFM tool. The conductive AFM tip acting as a nanometric top electrode allows access to the coercive voltage (Vc) and the effective piezoelectric coefficient (dzz eff ) associated with the OP component of polarization of the single probed domain. A statistical study was conducted by recording fifty piezoloops over each the different morphologies previously described, i.e. on the face-on or edge-on (I and II) lamellar crystal domains observed for the polymer layer. Figure 6 displays the histograms showing the distribution of the Vc and dzz eff values deduced from the simultaneously measured phase and amplitude piezoloops, respectively. Regardless the type of surface morphology, Gaussian distributions of coercive voltages and amplitude responses are obtained, the average values of which are summarized in Table 1. These PFM signals mainly arise from the crystalline part of the semi-crystalline P(VDF-co-TrFE) copolymer. However, knowing the flipping behavior highly depends on the crystallinity degree of the film, we cannot completely exclude the influence of the presence of amorphous regions when probing the domains beneath the AFM tip. Indeed, we previously shown that about 30% of amorphous phase exists in the sample. The Vc values were determined by measuring the half-width of the phase piezoloops, while the dzz eff values were estimated by using the following equation: A = dzz eff ×Vac×Q, where A is the PFM amplitude, Vac is the driving voltage (1.5 V for all measurements) and Q is the quality factor (Q was found to be ranging between 20 and 30 in the experiments, in concordance with the polymeric nature of the material).

Relatively wide distributions for Vc and dzz eff values are obtained. They are centered around 33.8 V and -30.0 pm/V, respectively, for the face-on lamellae regions. For the vertically stacked lamellae, average values of 36.4 V and -50.3 pm/V for edge-on I on the one hand, and 37.4 V and -27.1 pm/V for edge-on II are determined. No significant difference in switching properties is observed for the three kinds of morphologies observed at the surface of this sample, as shown by the very similar values of the polarizing voltage coercivity summarized in Table 1. This result is rather unexpected considering the heterogeneity of the probed crystalline morphology.

Although 180° for phase difference is systematically detected in accordance with the switching process of the dipole moments, the face-on or edge-on nature does not seem to significantly impact the average coercive voltages measured, unlike the study reported by Martínez-Tong et al. [START_REF] Martínez-Tong | Improving Information Density in Ferroelectric Polymer Films by using Nanoimprinted Gratings[END_REF] From the Vc values and based on the 300-nm-thick P(VDF-co-TrFE) layers, the coercive fields can be estimated (even if we have to keep in mind that the electric field seen by the film is highly inhomogeneous due to the particular contact between the film surface and the nanoscale geometry of the AFM tip acting as top electrode). They are ranging from 112.7 MV.m -1 (Vc = 33.8 V) up to 124.7 MV.m -1 (Vc = 37.4 V), higher than usual values reported for bulk PVDF or copolymers materials (50 MV.m -1 ). [START_REF] Furukawa | Ferroelectric Properties of Vinylidene Fluoride Copolymers[END_REF][START_REF] Fridkin | General Features of the Intrinsic Ferroelectric Coercive Field[END_REF] These higher energy barriers required for dipole rotation leading to efficiently flipping of the -CF2groups around the macromolecular chains (c-axis) are mainly related to the non-optimized crystal quality of the films which are crystallized upon cooling from melting. Now, when considering the trend of the piezoelectric response observed over the polymer surface, it is worth noting that a notably strong piezo-activity is locally detected when probing the specific edge-on I lamellae regions (~ -50.3 pm/V), much higher than for the similarly vertically-grown lamellae (denoted edge-on II) or for the face-on structures. Indeed, in the latter case the electromechanical responses measured by the AFM tip as well as over the edge-on II structures (-27.1 pm/V) are nearly similar. Consequently, the particular morphological arrangement of the crystalline lamellae described in detail above (Figure 4a,b,c,d) does not seem to be the main parameter to explain such difference in the measured deformation amplitude, even if it should be still pointed out that the observed vertical structures reveal two distinct morphological arrangement of stacked lamellae. In an attempt to explain such a difference in the nanoscale mechanical deformation, the specific orientation for the polymer chains in the P(VDFco-TrFE) cell has to be considered. The face-on crystal lamellae observed for the organic layer possess their a-and b-axis of the orthorhombic cell in the IP direction, as previously mentioned and confirmed by GIWAXD analysis. In the case of the edge-on I and II lamellae, the c crystallographic axis lies definitively along the substrate plane, but there is no certainty about the exact orientation of the b-axis, i.e. it can be either parallel or oriented at 60° with respect the substrate plane (see Figure 3c). As already mentioned, in the orthorhombic cell of P(VDF-co-TrFE), it is established that the polarization vector is aligned parallel to the b-axis where the remnant polarization is maximized. [START_REF] Liew | Nanoconfinement Induced Crystal Orientation and Large Piezoelectric Coefficient in Vertically Aligned P(VDF-TrFE) Nanotube Array[END_REF][START_REF] Hu | Regular Arrays of Highly Ordered Ferroelectric Polymer Nanostructures for Non-Volatile Low-Voltage Memories[END_REF] This specific direction allows for a more effective switching of polarization since the ferroelectric dipole moment can easily rotate around the horizontal chain c-axis (i.e. the chain backbone) under application of OP electric field. [START_REF] Wu | Orientation of Lamellar Crystals and its Correlation with Switching Behavior in Ferroelectric P(VDF-TrFE ) Ultra-Thin Films[END_REF][START_REF] Choi | Enhancement of Local Piezoresponse in Polymer Ferroelectrics via Nanoscale Control of Microstructure[END_REF][START_REF] Guo | Flexible Robust and High-Density FeRAM from Array of Organic Ferroelectric Nano-Lamellae by Self-Assembly[END_REF] In addition, the piezoelectric activity mostly comes from the global dimensional change of the material, which induces modification in the density of dipole moments. This results in ferroelectric polarization change inducing the piezoelectric effect. [START_REF] Katsouras | The Negative Piezoelectric Effect of the Ferroelectric Polymer Poly(vinylidene fluoride)[END_REF][START_REF] Hafner | Multi-Scale Characterisation of a Ferroelectric Polymer Reveals the Emergence of a Morphological Phase Transition Driven by Temperature[END_REF][START_REF] Liew | Nanoconfinement Induced Crystal Orientation and Large Piezoelectric Coefficient in Vertically Aligned P(VDF-TrFE) Nanotube Array[END_REF] As a result, the preferential polar axis orientation directly causes the enhancement of the piezoelectric constant.

By assuming the local deformation mainly occurs in the thickness direction, the much higher dzz eff piezoelectric coefficient detected in this work over the edge-on I crystal lamellae can be directly related to the direction of the b crystallographic axis in the P(VDF-co-TrFE) orthorhombic cell, which would have in this case a large component along the normal direction to the substrate plane, same as the ferroelectric polar vector direction. On applying vertical electrical voltage during poling process via the AFM tip, the CF2 molecular dipoles preferentially align along this direction, perpendicular to polymer chain (c-axis). In this case, we can state we locally probe the (110)-oriented edge-on lamellae, previously identified as edge-on I structures, in which the ferroelectric polarization is inclined at an angle of 60° to the substrate plane (Figure 3c) leading to the strong OP-PFM signal. On the other hand, it is well-established that the local polarization reversal upon OP electric field in such edge-on crystal domains with (110) orientation can occur according three different angles: 180°, 120° and 60° depending on the strength of the applied electric field. [START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF] On the contrary, for the lower dzz eff , we can reasonably establish that the b-axis in edge-on II lamellae would be aligned along the substrate plane, i.e. when the polarization vector alignment is not favorable for leading to highest deformation under applied OP driving voltage. [START_REF] Park | Irreversible Extinction of Ferroelectric Polarization in P(VDF-TrFE) Thin Films Upon Melting and Recrystallization[END_REF][START_REF] Wu | Orientation of Lamellar Crystals and its Correlation with Switching Behavior in Ferroelectric P(VDF-TrFE ) Ultra-Thin Films[END_REF] In this case, the rotation of dipole moments in the direction of applied electrical bias would require to change the alignment of the chain axis. A mechanism where the switching would be possible thanks to the torque on the dipole moment that eventually would enable its rotation was suggested but in any case this would be less effective. [START_REF] Gutiérrez-Fernández | Morphology and Ferroelectric Properties of Semiconducting/Ferroelectric Polymer Bilayers[END_REF] This is further supported by the nearly similar polarization reversal and piezoelectric deformations of the face-on structures where the baxis lies definitively along the substrate plane. These cases are attributed to the (200)-oriented edge-on lamellae, previously so-called as edge-on II. As a result, whatever the face-on or edgeon orientation, the key parameter to consider in order to fully understand the particular nanoscale electromechanical behavior in such P(VDF-co-TrFE) films is the alignment of dipoles, i.e. the specific polar b-axis direction. The schematic representations of the different molecular chain orientations for crystalline P(VDF-co-TrFE) lamellae with the related polarization axis, with respect to the local piezoelectric deformation detected beneath the AFM tip, are presented in Figure 7. The evidence of the impact of the polar axis orientation in P(VDF-co-TrFE) layers on their electrical properties was already extensively reported in the literature [START_REF] Park | Irreversible Extinction of Ferroelectric Polarization in P(VDF-TrFE) Thin Films Upon Melting and Recrystallization[END_REF][START_REF] Gaynutdinov | Polarization Switching at the Nanoscale in Ferroelectric Copolymer Thin Films[END_REF][START_REF] Ge | Preparation of Perovskite Conductive LaNiO3 Films by Metalorganic Decomposition[END_REF] , but rarely at such local scale. Lau et al. have reported the recording of piezoresponse loops on both face-on and edge-on PVDF copolymer structures, where the coercive voltage was found to be similar and the PFM signal higher in the case of the edge-on crystals. [START_REF] Lau | Effect of Annealing Temperature on the Morphology and Piezoresponse Characterisation of Poly(vinylidene fluoride-trifluoroethylene) Films via Scanning Probe Microscopy[END_REF] Kim et al. and Martínez-Tong et al.

have also described better local ferroelectric performances of needle crystals by means of PFM technique in the case P(VDF-co-TrFE) layers. [START_REF] Kim | High-Performance Needle-Shaped Crystals in Thin and Ultrathin P(VDF-TrFE) Films Formed by Melt Recrystallization[END_REF] However, they did not thoroughly discriminate between the two types of edge-on crystalline lamellae since no information on the specific location probed over the film surface is given, thus they did not provide insight on the precise polarization direction. More interesting is the absence of PFM hysteresis loop detected when probing flat-on oriented P(VDF-co-TrFE) ultra-thin films and the well-defined PFM loops for the edge-on oriented counterpart, as reported by Wu et al. [START_REF] Wu | Orientation of Lamellar Crystals and its Correlation with Switching Behavior in Ferroelectric P(VDF-TrFE ) Ultra-Thin Films[END_REF] Nevertheless, this latter result reflects only two extreme cases, not taking into account more complex situations in the edge-on oriented crystals. Gutiérrez-Fernández et al. have recently highlighted the homogeneous OP polarization reversal attributed to the (110) orientation of the edge-on P(VDF-co-TrFE) lamellae when artificially manipulated by PFM lithography, while inhomogeneous OP-PFM phase contrasts were observed for the (100) reflection of the edge-on structures. [START_REF] Gutiérrez-Fernández | Morphology and Ferroelectric Properties of Semiconducting/Ferroelectric Polymer Bilayers[END_REF] These latter results clearly show the influence of the polarization vector on the local electromechanical performances of the PVDF-based copolymer. However, they are carried out by manipulating the domains and then by imaging the poled regions, i.e. in this case the signal is recorded at the micrometer scale and without any information about the associated surface morphology. Finally, Sharma et al. [START_REF] Sharma | Orientational Imaging in Polar Polymers by Piezoresponse Force Microscopy[END_REF] have reported on the visualization of the polarization distribution in the P(VDF-TrFE) compound using OP-and IP-PFM imaging. In that case, they focused on copolymer nanostructures (i.e., isolated highly crystalline nanomesas with lateral dimensions ranging from 100 to 500 nm and of thickness 3 and 12 monolayers, prepared via Langmuir-Blodgett deposition), rather than thick films as in the present study. While their results are of very high quality, our investigation aims to explore both types of edge-on structures, which do not exist in the case of nanomesas.

Therefore, the reported approach enables to gain access to the as-grown polar axis direction within the crystalline structures of the organic layer by taking advantage of the nanometric resolution as well as the very high sensitivity of the spectroscopic PFM tool. More specifically the polar crystallographic axis direction is identified within the well-defined local morphology.

In particular, clear assessment of the crystallographic orientation of the edge-on crystal lamellae allows for a better understanding of the structure-property relationship in such electroactive P(VDF-co-TrFE) polymers.

CONCLUSIONS

In the present work, we carefully investigated the polar b-axis direction in P(VDF-co-TrFE) orthorhombic unit cell with respect to the surface of the substrate, when deposited in thin films.

Taking advantage of GIWAXD

Figure

  Figure5dcorresponding to the dashed blue line seen in Figure5bwhere 180° for phase

FIGURESFigure 1 .

 1 FIGURES

Figure 2 .

 2 Figure 2. (a) GIWAXD pattern and (b) integrated intensity profiles for P(VDF-co-TrFE) film

Figure 3 .

 3 Figure 3. (a) Intensity evolution of (200)/(110) main peak with azimuthal angle for P(VDF-co-

Figure 4 .

 4 Figure 4. AFM surface morphology of the grown P(VDF-co-TrFE) layer. (a) Large (60×60 μm²)

Figure 5 .

 5 Figure 5. Local manipulation of the polarization direction in the P(VDF-co-TrFE) layer. (a)

Figure 6 .

 6 Figure 6. Statistical study of the local switching and electromechanical behaviors in P(VDF-co-

Figure 7 .

 7 Figure 7. Schematic illustrations of P(VDF-co-TrFE) crystalline lamellae associated to PFM

TABLES Table 1 .

 TABLES1 Summarized of the crystalline lamellar structures, average coercive voltages (Vc) and average effective piezoelectric coefficients (dzz eff ) determined on the P(VDF-co-TrFE) film.

		Face-on	Edge-on I	Edge-on II
	Vc (V)	33.8	36.4	37.4
	dzz eff (pm/V)	-30.0	-50.3	-27.1

This work was supported through grant ANR-16-CE08-0025, ANR NanoPiC from French National Research Agency. It was carried out on the X-Ray Diffraction and Diffusion Facility of the Advanced Characterization Platform of the Chevreul Institute. Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation, Hauts-de-France Region, Fonds Européen de Développement Régional (FEDER) and Major Domain of Interest (DIM) "Eco-Energy Efficiency" of Artois University are acknowledged for supporting and funding partially this work. Région Hauts-de-France and FEDER are gratefully acknowledged for funding the MFP-3D microscope under Program "Chemistry and Materials for a Sustainable Growth". The authors gratefully thank Dr. S. Menuel from the University of Artois for his assistance to 3D schematic illustrations.

AUTHOR INFORMATION

Corresponding Author

*Anthony Ferri, E-mail: anthony.ferri@univ-artois.fr *Sophie Barrau, E-mail: sophie.barrau@univ-lille.fr ACKNOWLEDGMENT


Jérôme Rech

Tobias Micklitz

K A Matveev

Conductance of fully equilibrated quantum wires

Keywords: numbers: 71, 10, Pm

We study the conductance of a quantum wire in the presence of weak electron-electron scattering. In a sufficiently long wire the scattering leads to full equilibration of the electron distribution function in the frame moving with the electric current. At non-zero temperature this equilibrium distribution differs from the one supplied by the leads. As a result the contact resistance increases, and the quantized conductance of the wire acquires a quadratic in temperature correction. The magnitude of the correction is found by analysis of the conservation laws of the system and does not depend on the details of the interaction mechanism responsible for equilibration.

Experimental studies on the dc transport of short quantum wires have shown the quantization of their conductance in units of 2e 2 /h [1]. This phenomenon is well understood within a model of non-interacting electrons [2], even though the interactions in the wire are usually not weak, i.e., e 2 /hv F > ∼ 1, where v F is the Fermi velocity in the wire. The absence of any effect of electron interactions on the conductance is usually attributed to the fact that the quantum wires are always connected to two-dimensional leads, where interactions between electrons do not play a significant role. Indeed, it has been shown that within the so-called Luttinger-liquid theory, the interactions inside the wire do not affect conductance [3,4,5].

A number of recent experiments revealed deviations from perfect quantization in low-density wires [6,7,8,9,10,11,12,13,14,15]. These deviations often take the form of a shoulder-like feature, which develops at finite temperature just below the first quantized plateau, around 0.7 × 2e 2 /h . At the moment, there is no consensus on the theoretical interpretation of this so-called "0.7 structure." It is generally accepted, however, that electron-electron interactions are involved in this feature, thus generating a lot of interest in understanding the effect of interactions on the transport properties of onedimensional conductors. Here we study one of the most fundamental aspects of interactions, when they are so weak that their only effect is to equilibrate inside the wire the electrons coming from the two leads.

In the absence of interactions, the electrons propagate through the wire ballistically. Therefore the distribution functions of the right-and left-moving electrons are controlled by the left and right leads, respectively,

f p = θ(p) e (ǫp-µ l )/T + 1 + θ(-p) e (ǫp-µr)/T + 1 . (1)
Here ǫ p is the energy of an electron with momentum p, θ(p) is the unit step function, and we assume that the left and right leads have the same temperature T , but different chemical potentials µ l = µ + eV and µ r = µ. It is important to note that even weak processes of electronelectron scattering will modify the distribution (1). Indeed, such processes will force some left-moving electrons to change their direction of motion and become rightmovers. Thus the basic assumption of Eq. ( 1) that all the right-movers originate from the left lead and are in equilibium with it, will be violated.

The exact shape of the true steady state distribution of electrons can be understood easily if the wire is very long and the electron system is Galilean invariant, ǫ p = p 2 /2m. In this case it is convenient to view the electron system in the reference frame moving with the drift velocity v d = I/ne, where I is the electric current in the system and n is the electron density. In this frame the electron system is at rest, and must be described by the equilibrium Fermi-Dirac distribution characterized by a single chemical potential. In the stationary reference frame this distribution takes the form

f p = 1 exp ǫp-v d p-µeq Teq + 1 , (2)
where µ eq and T eq approach µ and T at V → 0. The distribution functions (1) and (2) coincide only in the limit of zero temperature. (In this case, to linear order in the drift velocity, µ l,r = µ eq ± v d p F , where p F is the Fermi momentum.) At nonzero temperature, one can expect the full equilibration of the electron system in the wire to significantly affect its transport properties.

Recently, the equilibration of the electrons in the moving frame was shown to have a strong effect on the Coulomb drag between two parallel quantum wires [16] and to give rise to a finite resistivity of long inhomogeneous wires [17]. On the other hand, because the equilibration processes that relax the distribution (1) to the form (2) involve converting right-moving electrons into left-moving ones, it is natural to expect that the interactions will affect conductance of the wires even in the absence of inhomogeneities, Fig. 1. Such an effect was re-j R (l)

j L (l) j R (r) j L (r)
T, µ l T, µ r FIG. 1: Quantum wire in the regime of small applied bias, µ lµr = eV . The electric current is given by total currents of the right-and left-moving electrons, I = e(j R + j L ).

The equilibration processes convert some of the right-moving electrons into the left-moving ones, thereby reducing the conductance of the wire.

cently discussed by Lunde, Flensberg, and Glazman [18] in the case of short wires, where the effect of the equilibration processes is weak, and the distribution function is still close to the unperturbed form (1). They found a negative correction to the quantized conductance of the wire, which grows linearly with its length L.

In this paper we explore the opposite limit of a long wire, L → ∞, in which the electron distribution function in the wire does assume the limiting form (2), and the correction to the conductance saturates at a value independent of L. The crossover between this regime and that of short wires will be discussed elsewhere [19].

Throughout this paper we assume that the interactions between electrons are very weak, and their only effect is to provide a mechanism of relaxation of the distribution function to the form (2). The exact nature of the scattering processes is unimportant, as long as these processes conserve the number of electrons, the energy of the system, and its momentum. Below we obtain the conductance of the wire by detailed analysis of these conservation laws.

The conservation of the number of particles means that in the steady state regime the total particle current j(x) is constant along the wire. It is convenient to present the total current as the sum j = j R + j L of currents of the right-and left-moving electrons,

j R,L (x) = 2 ∞ -∞ dp h θ(±p)v p f p (x), (3)
where the factor of 2 accounts for the spins, v p = p/m is the electron velocity, positive sign in the step function corresponds to j R , while the negative one to j L . It is important to realize that the distribution function f p in Eq. (3) depends on the position in the wire. Inside a long wire, the relaxation processes ensure that f p has the universal form (2), but near the ends of the wire f p is affected by the leads. For example, at the left lead the distribution of the right-moving electrons is controlled by the lead and takes the form of the first term in Eq. ( 1). This enables one to easily evaluate the current j R (l) of the right-movers at the left lead.

Unlike the total current j, the current j R (x) is not uniform along the wire, as the equilibration processes allow electrons to change direction. The rate Ṅ R of the change of the number of right-movers due to the electronelectron collisions is given by the difference of the values of j R at the two ends of the wire, Ṅ R = j R (r) -j R (l).

Although the current j R (r) of the outgoing rightmovers is not known, it can be expressed in terms of the total current j and the current j L (r) of incoming leftmovers, j R (r) = j-j L (r). It follows then that the change in the number of right-movers per unit time Ṅ R now depends on the electric current I = ej flowing through the wire, as well as the sum of incoming particle currents from both leads

j R (l) + j L (r) = I e -Ṅ R . (4)
In analogy with j R (l), the current j L (r) is controlled by the right lead and can be found by using the distribution function f p given by the second term in Eq. ( 1). Since both terms in the left-hand side of Eq. ( 4) are determined by the distribution functions in the non-interacting leads, the result of the routine evaluation of the two currents is given by the Landauer formula j R (l)+j L (r) = 2eV /h, up to corrections exponentially small in µ/T . Thus we find the following relation between the applied bias, electric current, and Ṅ R ,

2e 2 h V = I -e Ṅ R . (5)
An equivalent relation was obtained earlier in Ref. [18] using the Boltzmann equation formalism. It formally expresses the idea that the processes changing the number of right-movers in the wire will result in a correction to the quantized conductance. The conservation of energy in electron-electron collisions implies that the total energy current j E (x) is uniform along the wire. It is instructive to express the energy current as the sum j E = j R E +j L E of the contributions of the right-and left-moving particles,

j R,L E (x) = 2 ∞ -∞ dp h θ(±p)v p ǫ p f p (x). (6)
In the same fashion that we could relate the particle current j to how the number of right-moving electrons changes over time, one can find a relation between the energy current j E flowing through the wire and the rate of change ĖR of the energy of right-movers due to the electron collisions. Indeed, the reasoning that led to Eq. ( 4) can be readily extended to the case of energy currents rather than particle ones, leading to 2: Illustration of the relaxation processes converting a right-moving electron into a left-moving one. The parabolas represent the quadratic dispersion ǫp = p 2 /2m of the electrons in the wire. (a) A three-particle process studied in Ref. [18]. Momentum and energy conservation ensure that as a right-mover changes direction, another right-mover increases its energy and a left-mover decreases it. (b) Transfer of an electron from the right to the left Fermi point is accompanied by multiple right-movers increasing their energies and multiple left-movers decreasing energies.

j R E (l) + j L E (r) = j E -ĖR . ( 7
) (a) (b) µ µ T T FIG.
The energy currents in the left-hand side of Eq. ( 7) are again controlled by the leads and can be easily computed using the distribution function (1) of noninteracting electrons. At low temperatures T ≪ µ one finds j R E (l) + j L E (r) = (2eV /h)µ, up to corrections small as e -µ/T .

Since the energy current j E does not depend on the position in the wire, one can calculate it in a region away from the leads, where the distribution function is given by Eq. ( 2). The formal calculation gives j E = µ(1 + π 2 T 2 eq /6µ 2 eq )j, where we discarded terms of order O (T eq /µ eq ) 4 and higher. This result can be compared with the above-mentioned calculation for the unperturbed distribution (1), which can be summarized as j E = µj. The difference can be traced back to the energy dependence of the term v d p in the Fermi function (2). Expressing the particle current as j = I/e we then find

2e 2 h V = 1 + π 2 6 T µ 2 I - e µ ĖR . ( 8
)
This result is obtained in the linear order in the applied bias, which enabled us to substitute T eq = T and µ eq = µ.

In the absence of the scattering processes changing the number of the right-and left-moving electrons, not only Ṅ R , but also ĖR would vanish. Indeed, in this case the two branches of excitations would represent two electron systems with no particle exchange allowed. Then the distribution function (1) would describe the systems of right-and left-movers in thermal equilibrium with each other, and net heat transfer between them would vanish. Thus both Ṅ R and ĖR arise as a consequence of the same relaxation mechanism inside the wire, and we will now show that there is a simple relation between these rates.

In the case of a short wire it was shown [18] that the dominant process changing the number of right-movers involves three electrons, with a right-mover near the bottom of the band reducing its momentum and thus the direction of motion, Fig. 2(a). The conservation of momentum then requires that the other two electrons increase their momenta. Finally, conservation of energy requires one of these two electrons to be near the right Fermi point, and the other near the left one. The typical momentum change is controlled by the temperature, |δp| ∼ T /v F . As a result of such scattering events the distribution function (1) shows only a small modification whereby the exponentially small discontinuity near p = 0 is smeared.

A much more significant change occurs in longer wires, where the relaxation processes bring the distribution function to the form (2). A comparison of the distribution functions (1) and (2) shows that the main difference between them is at the values of momentum p near the Fermi points ±p F . Thus the dominant relaxation processes contributing to Ṅ R take electrons with p ≈ p F and move them to p ≈ -p F . Such processes are realized in many small steps of |δp| ∼ T /v F and are accompanied by multiple electrons scattering near the two Fermi points, see Fig. 2(b). The total momentum transferred to these electrons is 2p F . Energy conservation requires that it is distributed evenly between the right-and left-movers, so that the resulting energy increase δE R = v F p F is compensated by the decrease δE L = -v F p F .

In the end, the energy balance for the right-moving electrons consists of a loss of ǫ F , which was the energy of the electron changing direction, and a gain of v F p F = 2ǫ F due to the redistribution of momentum. As a result, for every right-moving electron that changes direction, ∆N R = -1, the right-movers' energy increases by an amount ∆E R = ǫ F . We thus conclude that

ĖR = -µ Ṅ R , (9)
where we replaced ǫ F with µ, as the small difference µ-ǫ F ∼ T 2 /µ turns out to be irrelevant for our purposes.

It is important to note that the result ( 9) is not sensitive to the specific details of the electron relaxation mechanism. Indeed, the two key ingredients of this derivation are the conservation laws that control the redistribution of momentum 2p F between the right-and left-movers, and the quadratic dispersion that governs how the energies of the two subsystems change as a result of that redistribution.

By analyzing the conservation laws we have so far been able to establish three linear relations (5), (8), and (9) between four quantities, V , I, Ṅ R , and ĖR . Assuming that the applied bias V is known, we can now express the remaining three quantities in terms of V . Most importantly, we find I = GV , with the conductance

G = 2e 2 h 1 - π 2 12 T µ 2 , (10)
where we restricted ourselves to the leading order term in (T /µ) 2 . The quadratic in temperature correction to the quantized conductance 2e 2 /h of the wire is our main result. Unlike the correction to the conductance of a short wire δG ∝ e -µ/T [18], our correction shows power-law dependence on the temperature. Earlier papers [3,4,5] on the conductance of long quantum wires did not find any correction to the conductance, as the Luttinger-liquid theory used there does not account for the relaxation processes leading to our result (10).

Experimentally, small temperature-dependent corrections to quantized conductance have been observed in quantum point contacts [6,7,8,9]. The latter are essentially short quantum wires, with only a few electrons in the one-dimensional part of the device. In order for our result (10) to be fully applicable the length of the system should be sufficient to ensure full equilibration of the electron distribution function. Comparison of our correction δG ∼ (T /µ) 2 with the result [18] for short wires, δG ∝ Le -µ/T implies that our result is applicable at L ≫ l eq ∝ e µ/T , in agreement with the more detailed calculation [19]. Although some experiments with longer quantum wires have been reported [10,12], a careful study of the temperature dependent corrections to the conductance is not yet available.

In summary, we have shown that in a long quantum wire, the full equilibration of the electron distribution function leads to a finite correction to the conductance, which at T ≪ µ is quadratic in temperature, Eq. (10). Our derivation relied uniquely on an analysis of the conservation laws for energy, momentum, and particle number, without making specific assumptions regarding the process of equilibration.

We are grateful to A. V. Andreev and L. I. Glazman for helpful discussions. This work was supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357.


Jérôme Charles

Sébastien Descotes-Genon

Zoltan Ligeti

Stéphane Monteil

Michele Papucci

Karim Trabelsi

Future sensitivity to new physics in B d , B s and K mixings

for the CKMfitter Group

We estimate, in a large class of scenarios, the sensitivity to new physics in B d and Bs mixings achievable with 50 ab -1 of Belle II and 50 fb -1 of LHCb data. We find that current limits on new physics contributions in both B d,s systems can be improved by a factor of ∼ 5 for all values of the CP violating phases, corresponding to over a factor of 2 increase in the scale of new physics probed. Assuming the same suppressions by CKM matrix elements as those of the standard model box diagrams, the scale probed will be about 20 TeV for tree-level new physics contributions, and about 2 TeV for new physics arising at one loop. We also explore the future sensitivity to new physics in K mixing. Implications for generic new physics and for various specific scenarios, such as minimal flavor violation, light third-generation dominated flavor violation, or U (2) flavor models are studied.

I. INTRODUCTION

Before the impressive results from the B factory experiments, BaBar and Belle, the simple picture of Kobayashi and Maskawa for the origin of the CP violation [1] observed in K decays was not confirmed experimentally. The BaBar and Belle results showed that the SM description of the flavor sector is correct at the order one level. However, in most flavor-changing neutral-current processes, new physics (NP) can still contribute at least at the level of 20-30% compared to the SM.

Many extensions of the SM receive stringent constraints from data on flavor changing processes and CP violation, and may give observable effects as the sensitivity improves. The mixings of the four neutral mesons, K, D, B d , and B s , provide particularly strong bounds. For each neutral-meson system, contributions generated by new heavy degrees of freedom can be described by two real parameters. For example, in low-energy supersymmetry B mixing receives contributions (besides the SM box diagrams with W bosons and top quarks) from box diagrams with winos and stops or gluinos and sbottoms. The magnitudes and phases of such contributions depend crucially on the mechanism of supersymmetry breaking and the origin of flavor symmetry breaking.

However, the extraction of NP contribution to meson mixing is entangled with the determination of the SM parameters, in particular the CKM elements. It is not enough to measure the mixing amplitude itself, only the combination of many measurements can reveal a deviation from the SM. In this paper we perform such a fit, taking into account the latest expectations for future LHCb and Belle II measurements, and anticipated progress in lattice QCD, in order to investigate the sensitivity to NP in neutral-meson mixing in the near future.

In most of this paper, we consider the well-defined scenario where no deviations from the SM predictions are observed. This allows us to explore the expected progress in constraining NP in the mixings of neutral mesons in an unambiguous way. An illustration of the prospects to reveal a possible NP signal is given in the last section.

II. NEW PHYSICS IN MESON MIXING

In a large class of NP models the unitarity of the CKM matrix is maintained, and the most significant NP effects occur in observables that vanish at tree level in the SM. In the SM CKM fit, the constraints come from (i) ∆F = 1 processes dominated by tree-level charged current interactions, and (ii) ∆F = 2 meson mixing processes, which only arise at loop level. Therefore, it is simple to modify the CKM fit to constrain new physics in ∆F = 2 processes, under the assumption that it does not significantly affect the SM tree-level charged-current interactions [2]. Within this framework (for a review, see [3]), we can parameterize the NP contributions to the B d,s mixing amplitudes as

M d,s 12 = (M d,s 12 ) SM × 1 + h d,s e 2iσ d,s . (1)
Until the first measurements of α and γ around 2003, it was not known if the SM gives the leading contribution to B d -B d mixing [4,5] (similarly, for B s -B s mixing, the LHCb constraint on sin 2β s was needed).

The motivation for the above parameterization is that any NP contribution to M 12 is additive, and using Eq. ( 1) one can easily read off both the magnitude and the CP violating phase of the total NP contribution. In particular, for a NP contribution to the mixing of a meson with q i qj flavor quantum numbers due to the operator

C 2 ij Λ 2 (q i,L γ µ q j,L ) 2 , (2)
one finds that

h ≃ 1.5 |C ij | 2 |λ t ij | 2 (4π) 2 G F Λ 2 ≃ |C ij | 2 |λ t ij | 2 4.5 TeV Λ 2 , σ = arg C ij λ t * ij , (3)
where λ t ij = V * ti V tj and V is the CKM matrix. We used NLO expressions for the SM and LO for NP, and neglected running for NP above the top mass. Operators of different chiralities have conversion factors differing by O(1) factors [6]. Minimal flavor violation (MFV), where the NP contributions are aligned with the SM ones, correspond to σ = 0 (mod π/2).

Analogously, in K mixing, we choose to parameterize NP via an additive term to the so-called tt contribution to M K 12 in the SM. This is justified by the short distance nature of NP, by the fact that in many NP models the largest contribution to M K 12 arise mostly via effects involving the third generation ("23-31" mixing), and more practically, since this allows one to maintain a consistent normalization for NP across the three down-type neutral meson systems. In this paper, D-meson mixing is not considered, due to the large uncertainties related to long-distance contributions.

Comments are in order concerning our assumption of neglecting NP in charged current b → u, c transitions. If a NP contamination is present and has a different chiral structure than the SM, it will manifest itself by modifying decay distributions, such as the lepton spectrum in semileptonic B decays. On the contrary, if NP has the same chiral structure as the SM, it cannot be physically separated in the determination of ρ and η. In such a case, the extracted values of these parameters will not correspond to their SM values. This discrepancy will propagate to the NP fit, and will manifest itself as a nonzero value for h d,s [7], with a specific pattern for h d,s and σ d,s .

III. GENERIC FIT FOR Bd AND Bs MIXINGS

Table I shows all inputs and their uncertainties used in our fit, performed using the CKMfitter package [4,8,9] with its extension to NP in ∆F = 2 [10] (for other studies of such NP, see Refs. [5,[11][12][13][14][15][16]). We use standard SM notation for the inputs, even for quantities affected by NP in ∆F = 2 whose measurements should be reinterpreted to include NP contributions (e.g. α, β, β s ). We consider 2003 (before the first measurements of α and γ) and 2013 (as of the FPCP 2013 conference), and two future epochs, keeping in mind that any estimate of future progress involves uncertainties on both experimen- Belle II data, and probably corresponds to the middle of the 2020s, at the earliest. Estimates of future experimental uncertainties are taken from Refs. [17,18,[START_REF] Meadows | [END_REF][START_REF] Bona | [END_REF].

(Note that we display the units as given in the LHCb and Belle II projections, even if it makes some comparisons less straightforward; e.g., the uncertainties of both β and β s will be ∼ 0.2 • by Stage II.) For the entries in Table I where two uncertainties are given, the first one is statistical (treated as Gaussian) and the second one is systematic (treated through the Rfit model [8]). Considering the difficulty to ascertain the breakdown between statistical and systematic uncertainties in lattice QCD inputs for the future projections, for simplicity, we treat all such future uncertainties as Gaussian.

The fits include the constraints from the measurements of A d,s SL [10,11], but not their linear combination [START_REF] Abazov | [END_REF], nor from ∆Γ s , whose effects on the future constraints on NP studied in this paper are small. While ∆Γ s is in agreement with the CKM fit [10], there are tensions for A SL [START_REF] Abazov | [END_REF]. The large values of h s allowed until recently, corresponding to (M s 12 ) NP ∼ -2(M s 12 ) SM , are excluded by the LHCb measurement of the sign of ∆Γ s [24]. We do not consider K mixing for the fits shown in this Section, as it may receive NP contributions unrelated to B d and B s mixings in the general case considered in this section.

Figure 1 shows the evolution of the constraints on (ρ, η) in the presence of NP in both B d and B s meson mixings, for 2003, 2013, Stage I, and Stage II. 1 The main constraints on ρ and η come from the tree-level inputs γ and |V ub |, and also from the combination γ(α) = π -β -α which is not affected by NP in ∆F = 2 [2]. This constraint is more precise than γ itself until Stage I, but of similar precision by Stage II. The γ and γ(α) measurement constraints are known modulo π, leading to a sign 1 Considering anticipated results from only one experiment, plots similar to Fig. 1, and with a different parameterization, Fig. 2, appear in Refs. [17,[START_REF] Bona | [END_REF].

ambiguity in the determination of ρ and η. 2 The intersection of the γ, γ(α) and |V ub | constraints yields two 95.5% CL regions in Fig. 1 (yellow for positive ρ and η, mauve for negative ρ and η) symmetric with respect to the origin. This degeneracy is lifted by the addition of the other experimental inputs, in particular A d SL , leading to a single and small 95.5% CL region (in yellow) for ρ and η. (In 2013, the degeneracy is only partially lifted: the ρ < 0, η < 0 solution is excluded at 68.2% CL, but it is allowed at 95.5% CL.)

Figures 2 and3 show the corresponding evolutions of the constraints on (h, σ) in the B d and B s meson systems. Each plot is obtained by considering all the inputs in Table I and treating ρ, η, and the other physics parameters not shown as nuisance parameters. This corresponds to the case of generic NP, ignoring possible correlations between different ∆F = 2 transitions. Since we are interested in the future sensitivity of LHCb and Belle II to NP, for Stage I and Stage II, we chose the central values of future measurements to coincide with their SM predictions using the current best-fit values of ρ and η. Thus, the future best fit corresponds to h = 0. Figure 4 shows the projection on the (h d , h s ) plane.

Future lattice QCD uncertainties for Stage I are taken from Refs. [START_REF] Blum | Lattice QCD at the Intensity Frontier[END_REF][START_REF] We | Water for helpful correspondance about future lattice QCD expectations[END_REF] (where they are given as expectations by 2018). These predicted lattice QCD improvements will be very important, mainly for the determination of |V ub | and for the mixing matrix elements,

B q |( bL γ µ q L ) 2 |B q = (2/3) m 2
Bq f 2 Bq B Bq . The current expectation is that the uncertainties of f Bq will get below 1%, and may be significantly smaller than those of B Bq . The reduction of the uncertainty of the latter to a similar level would be important. Up to now, due to the chiral extrapolations to light quark masses, more accurate results were obtained for matrix elements involving the B s meson or for ratios between B d and B s hadronic inputs, compared to the results for B d matrix elements. This leads us to use the former quantities as our lattice inputs for decay constants and bag parameters in Table I. This choice might not be the most suitable one This may not reflect the way lattice results will improve in the future, and correlations will affect the shape of the allowed regions in those plots.

From the discussion in the introduction, one may think that ρ and η are determined mostly by SM tree-level processes (|V ub /V cb | and γ from B → DK decays), while the additional loop-level observables in the standard CKM fit constrain the NP. In particular, ∆M d,s , sin 2β d,s , and α would constrain h d,s and σ d,s , while ǫ K constrains h K and σ K . This simple separation of SM and NP has not been possible yet, given the large uncertainty of γ compared to the combination, γ(α) ≡ π -β -α, which is independent of NP in the classes of models under con-sideration [2]. (Note that in the determination of α from B → ρρ, ρπ, ππ, an isospin analysis is used to remove the penguin contribution. To use this measurement to constrain new physics in mixing, one has to assume that NP conserves isospin, which holds in most scenarios, and is strongly supported by data.) As one can clearly see from Table I and Fig. 1, when the direct measurement of γ becomes as precise as π -β -α in the future, the separation of the two sectors will be simpler to understand, even in a combined SM + NP fit.

For our analysis, precise determination of CKM parameters from tree-level measurements is essential, as illustrated in Fig. 1. Depending on future experimental results, the tension between inclusive and exclusive |V ub | (and |V cb |) determinations might remain a cause for concern [START_REF] See | Determination of V cb and V ub[END_REF]. The CKM part of our analysis also relies on the expectation that the determination of γ will indeed reach the 1 • level. (For α, a comparison of the ρρ, ππ, ρπ results may help to constrain the effects of isospin breaking and to reach the expected accuracy.)

One can see from Figs. 2 and 3 that recent LHCb measurements have imposed comparable constraints on NP in B s mixing to those in the B d system. This qualitative picture will continue to hold for Stage I and Stage II. At Stage I, we will have h d,s < ∼ 0.1 for generic NP phases, with an improvement by an additional factor of more than two at Stage II. This is not surprising, as the uncertainties on β and β s will be comparable, and improvements in the determination of ρ and η from γ and |V ub /V cb | will affect the constraints on the two systems in a similar way. It is also interesting to see that the MFV regions (σ d,s = 0 mod π/2) will be less constrained also in the future. Figure 4 provides a different view of these results, by showing the magnitudes of NP allowed in the B d vs. B s systems.

The better than factor-of-four improvement in the sensitivity to h d,s from the current constraints to Stage II, more than doubles the energy range probed by these observables, and parallels the improvements in the high energy reach of the LHC, going from LHC7 to LHC14. If NP contains the same CKM suppressions of ∆F = 2 transitions as those present for the SM contributions, typical for models with nontrivial flavor structure in the LHC energy range, the scales probed by the mixing constraints are

Λ ∼ 17 TeV (B d ) , Λ ∼ 19 TeV (B s ) . (4)
Here we used Eq. ( 3) with |C ij | = |λ t ij |, and the 95% CL bounds, h d < 0.07 and h s < 0.06 from Figs. 2 and3 Equation ( 4) implies that LHCb and Belle II will probe new particles with CKM-like couplings with masses, M , in the 10-20 TeV range if they contribute at tree level (i.e., Λ ∼ M ), and in the 1-2 TeV range if they enter with a loop suppression (i.e., Λ ∼ 4πM ). Considering color factors, RGE effects, etc., which can differ for other operators, one sees that these constraints are in the ballpark of gluino masses explored at LHC14 [26].

IV. INCLUDING NEW PHYSICS IN K MIXING

Next, we consider the neutral kaon system, in addition to B d and B s . We only include the constraint from ǫ K , since there are large uncertainties in the long-distance contribution to ∆m K (for the same reason, we do not study D-meson mixing). Figure 5 shows the evolution of the constraints on NP in K mixing. Larger values of h K for certain values of the CP -violating phase will still be allowed, even at Stage II. Due to the presence of only one observable, ǫ K , constraining two parameters, h K and σ K , such "throats" cannot be eliminated. They correspond to the values for which the imaginary part of the NP contribution vanishes, that is σ K ∼ π-β SM or π/2-β SM , where β SM is the value of the true CKM β angle shown in Figure 1. In the 2013 plot, the two additional branches with low p-values correspond to the less favored second solution for the CKM parameters ρ < 0, η < 0.

NP contributions as large as 30% of the SM tt contribution will be allowed in the future, even in the MFV case, as can be seen by considering the σ K = 0, π/2 values in the Stage II plot in Fig. 5. Note that the improvement from Stage I to Stage II is much less significant than the one from the current status to Stage I. Indeed, despite the almost factor-of-two improvement on the uncertainty on B K and the improvements on ρ and η, other parameters entering ǫ K are not expected to have similar improvements, as shown in Table I. This includes the uncertainty associated with higher-order terms in the OPE emphasized in Ref. [27], and higher-order QCD corrections discussed in Refs. [28,29] (in particular for cc contribution).

In many scenarios with TeV-scale NP, the constraints from kaon mixing provide the strongest constraints to date, especially for the case of chirality-flipping left-right (LR) operators, due to chiral enhancements in the matrix elements and stronger QCD running. This situation will be maintained in the Stage II era as well, with comparable constraining power for non-LR NP, and a significant advantage of the kaon system over the B d,s systems in constraining chirality-flipping operators. Furthermore, if NP is decoupled from the weak scale and carries unsuppressed flavor violation (e.g., intermediate-scale split supersymmetric scenarios [30]), the kaon system will provide the most stringent probe (or the first place where a deviation can be observed), since it carries the strongest CKM suppression in the SM.

We next consider more specific NP scenarios, where the contributions to the different neutral-meson systems are correlated. In the MFV case mentioned in Sec. II, Additional particularly interesting scenarios are those in which the dominant effects are mediated by the third generation, motivated by the natural stabilization of the electroweak scale, and those in which the approximate horizontal U (2) 3 symmetry of the SM, induced by m u,c /m t ≪ 1 and m d,s /m b ≪ 1, also applies to the NP contributions [31,32]. In the first case, the NP contribution to kaon mixing is attained via mixing with the third generation, and is therefore related to those in B d and B s mixings. In a fundamental theory representing this scenario the mixing parameters C ij in the kaon sector, similar to Eq. (2), will be the product of those entering the B d -and B s -mixing expressions, up to small correc-tions. Therefore, there is a correlation among the phases,

h d = h s = h K , 0 = σ d = σ s = σ K (mod π/2) . (6)
σ K = σ d -σ s . (7)
On the other hand, the magnitudes of the NP contributions, h K,d,s also depend on the typical mass scale, coupling constants, and kinematic function, represented by Λ in Eq. ( 2). Thus, in general, 3rd generation mediation in the kaon system does not imply a relation between h K and h d,s . The constraint on such models is shown in Fig. 7, for the future Stage II scenario. Mild correlations between the limits on the magnitudes of NP in B d,s and K mixings arise due to the relations on the CP phases σ i described above, and to a lesser extent via ρ and η. The plot is easily understood: the largest NP contribution in B d,s mixing is allowed for σ d,s ∼ 0 (mod π/2), which is allowed for sufficiently small h K < ∼ 0.6. The presence of the "throats" in Fig. 5 allow larger values for h K for the non-U (2) 3 case, but at the price of not allowing σ s,d ∼ 0 (mod π/2), hence the (small) reduction in the allowed magnitude of NP in the B d,s sector. This effect can be more pronounced, if the actual future data agrees less well with the SM, than assumed in this paper.

In the case of the minimal U (2) 3 class of models [33], the NP contributions to B d and B s should be equal. Furthermore, minimality implies that the bulk of the NP contribution in the kaon sector is controlled by the same spurions as in the B d,s sectors via 3rd generation media-tion, "23-31". Therefore, one has

h B ≡ h d = h s , σ B ≡ σ d = σ s , σ K = 0 . (8)
The constraints on such scenarios are shown in Figs. 8 and9. In Fig. 8 the minimal U (2) 3 scenario is shown in the h B -σ B plane. While the 2003 and 2013 fits show interesting patterns arising from the combination of the B d,s constraints, the future projections for the U (2) 3 models look very similar to Figs. 2 and3.

The correlation between the limits on the magnitudes of NP in B d,s and K mixings in the minimal U (2) 3 case is shown in Fig. 9 in the h K -h B plane. Similar considerations as in Fig. 7 apply here. As can be seen in Fig. 5, the constraint σ K = 0 limits the size of h K < ∼ 0.25. In the case of generic U (2) 3 models, which allow additional NP contributions in the kaon system unrelated to those in the B d,s systems, h d = h s and σ d = σ s are maintained, but the correlation with the K systems is lost. Therefore, the constraints in Fig. 9 no longer apply, while those in Fig. 8 are still valid.

Constraints on NP in K will improve if lattice QCD gives a precise SM calculation of ∆m K [START_REF] Christ | ANL Intensity Frontier Workshop[END_REF]. For Re(M K 12 ) in the SM, the ratio of the tt and cc contributions is about 0.5%, so a 1% calculation of ∆m K could exclude h K > ∼ 2. Lattice QCD progress may also reduce the uncertainty in the higher order terms in ǫ K discussed in Ref. [27], improving the overall constraints. Due to its unpredictability, we do not include possible improvements in this term (κ ǫ ) in our Stage I and II fits. Even assuming a much reduced uncertainty of η cc , ±0.2 instead of ±0.76 at NNLO now (see Ref. [START_REF] Buras | [END_REF]), would only improve the bounds on h K shown in Fig. 5 slightly; e.g., at Stage II for σ K = 0, from K < 0.31 to h K < 0.24.

In certain classes of models, improvement in sensitivity compared to Fig. 5 can also arise from future measurements of K + → π + ν ν and K L → π 0 ν ν [START_REF] Grossman | [END_REF]. These decays are also sensitive to NP in s → d penguins, which can be parameterized by another magnitude h ; thus the difference of the number of observables vs. NP parameters will not change. However, in certain well-motivated scenarios, σ

(∆S=1) K = σ K [12], or h (∆S=1) K
∼ 0, and in such cases including future data on these rare decays will improve the sensitivity to NP.

V. SUMMARY AND OUTLOOK

We studied the anticipated future improvements in the constraints on NP in B d , B s , and K mixings. We found that if no NP signal is seen, the bounds on h d and h s will improve by about a factor of 5. This corresponds to probing NP at scales more than a factor of two higher than currently (for a fixed set of couplings). Interestingly, compared to the allowed regions to date, we expect the MFV-like regions, σ = 0 (mod π/2), to be nearly as strongly constrained as those with generic NP phase in the future. Our results for the future sensitivity to a NP contribution given by Eq. ( 2) in B d and B s mixings at Stage II are summarized in Table II. For K mixing, the large h K regions in Fig. 5 complicate the interpretation in terms of NP scales. If we assume that lattice QCD will exclude h K > 2 as discussed in Sec. IV, we get sensitivity up to 3 TeV (0.3 TeV) at tree level (one loop) for CKMlike couplings, and up to 9 × 10 3 TeV (7 × 10 2 TeV) at tree level (one loop) for non-hierarchical couplings.

So far in this paper we assumed that future measurements agree with the SM predictions. However, future data can not only set better bounds on NP, they may also reveal deviations from the SM. This is illustrated in Fig. 10, where we set ρ, η, h d,s and σ d,s to their current best-fit values (allowing for NP in ∆F = 2), and performed a fit assuming for all future measurements the corresponding central values, but uncertainties as given in Table I ness, Fig. 10 may give an impression of the sensitivity to reveal a deviation from the SM.

Similar predictions could be made for many other higher dimension flavor-changing operators. The ∆F = 1 observables dominated by one-loop contributions in the SM probe different NP contributions. Such analyses have been performed for b → sγ, b → sℓ + ℓ -, etc. [37]. The progress for the constraints imposed by some of these observables, especially those corresponding to not yet observed processes, will be greater than those for B d and B s mixings studied in this paper. This example is particularly interesting, as many NP models do predict an effect which may be observable in the coming decade. Furthermore, ∆F = 2 generically provides the strongest constraints for high-scale models with unsuppressed flavor violation, while still providing competitive constraints for lower scale NP (where flavor transitions are parametrically suppressed as in the SM). Finally, the significant improvements on the bounds in the h -σ planes for B d and B s mixings give an impressive yet conservative illustration of the anticipated future progress coming from the LHCb upgrade and the Belle II experiment.

  tal and theoretical sides. Our Stage I projection refers to a time around or soon after the end of LHCb Phase I, corresponding to an anticipated 7 fb -1 LHCb data and 5 ab -1 Belle II data, towards the end of this decade. The Stage II projection assumes 50 fb -1 LHCb and 50 ab-1

FIG. 2 .

 2 FIG. 2. The past (2003, top left) and present (top right) constraints on h d -σ d in B d mixing. The lower plots show future sensitivities for Stage I and Stage II described in the text, assuming measurements consistent with the SM. The dotted curves show the 99.7% CL contours.

FIG. 3 .

 3 FIG. 3. The past (2003, top left) and present (top right) constraints on hs -σs in Bs mixing. The lower plots show future sensitivities for the Stage I and Stage II described in the text, assuming measurements consistent with the SM. The dotted curves show the 99.7% CL contours.

FIG. 4 .

 4 FIG. 4. The past (2003, top left) and present (top right) constraints on h d -hs in B d and Bs mixings. The lower plots show future sensitivities for the Stage I and Stage II scenarios described in the text, assuming measurements consistent with the SM. The dotted curves show the 99.7% CL contours. For Stage I and Stage II, the white dashed curves indicate the 95% CL contours obtained by setting theoretical uncertainties to zero.

  . If, instead, we use |C ij | = 1 (corresponding to nonhierarchical NP contributions), the probed scales are Λ ∼ 2 × 10 3 TeV (B d ) , Λ ∼ 5 × 10 2 TeV (B s ) . (5)

FIG. 5 .

 5 FIG. 5. The past (2003, top left) and present (top right) constraints on hK -σK in K mixing. The lower plots show future sensitivities for the Stage I and Stage II scenarios described in the text, assuming measurements consistent with the SM. The dotted curves show the 99.7% CL contours.

Figure 6

 6 Figure 6 shows the p-values for the real (positive or negative) h ≡ h d exp(2iσ d ) in 2003, 2013, Stage I, and Stage II.

FIG. 6 .FIG. 7 .

 67 FIG. 6. Constraints on h ≡ h d e 2iσ d = hs e 2iσs = hK e 2iσ K in MFV scenarios, in which σ d = σs = σK = 0 (mod π/2), for the different epochs considered.

FIG. 8 .

 8 FIG. 8. The past (2003, top left) and present (top right) constraints on U (2) 3 scenarios, where hB ≡ h d = hs, σB ≡ σ d = σs. The lower plots show future sensitivities for the Stage I and Stage II scenarios described in the text, assuming measurements consistent with the SM. The dotted curves show the 99.7% CL contours.

FIG. 9 .

 9 FIG.9. Correlations between limits on NP in K and B d,s mixing, at "Stage II", in minimal U (2) 3 models. This fit corresponds to h d = hs ≡ hB, σB ≡ σ d = σs, and σK = 0. The dotted curve shows the 99.7% CL contour.

TABLE I

 I

. Central values and uncertainties used in our analysis (see definitions in Ref.

[10]

). The entries "id" refer to the value in the same row in the previous column. The 2003 and 2013 values correspond to Lepton-Photon 2003 and FPCP 2013 conferences

[4]

. The assumptions entering the Stage I and Stage II estimates are described in the text.

TABLE II .

 II for Stage II. While any assumption about possible future NP signals includes a high degree of arbitrari-The scale of the operator in Eq. (2) probed by B d and Bs mixings at Stage II (if the NP contributions to them are unrelated). The impact of CKM-like hierarchy of couplings and/or loop suppression is indicated.

	Couplings	NP loop Scales (in TeV) probed by order B d mixing Bs mixing
	|Cij | = |VtiV * tj | tree level	17	19
	(CKM-like) one loop	1.4	1.5
	|Cij | = 1	tree level 2 × 10 3	5 × 10 2
	(no hierarchy) one loop 2 × 10 2	40

  FIG. 10. Hypothetical Stage II fits for NP, assuming that all future experimental results correspond to the current best-fit values of ρ, η, h d,s and σ d,s (with measurement uncertainties as given in Table I, but different central values).

	excluded area has CL > 0.95	CKM f i t t e r
		Stage II (NP)

In 2013, the combined constraint from the ππ, πρ and ρρ data allows a second solution for α near 0, with a lower significance than the SM solution in Table I[4]. This second solution is shown as the negative-slope γ(α) wedge in Fig.1, and is ruled out once combined with the γ constraint. We assume that this lowsignificance solution will disappear with more data by Stage I.

ACKNOWLEDGMENTS

We thank Riccardo Barbieri, Filippo Sala, and Stéphane T'Jampens for helpful comments. ZL and MP were supported in part by the Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under contract DE-AC02-05CH11231. We would like to thank all members of the CKMfitter group for suggestions on various aspects of this article.