X Ribeyre
email: ribeyre@u-bordeaux.fr

S E D'humières

V T Jequier

Tikhonchuk

Effect of differential cross section in Breit-Wheeler pair beaming

Keywords: gamma-ray sources, Breit-Wheeler process, pair electron-positron, Active Galactic Nuclei

The pair beaming in the Breit-Wheeler (BW) process is investigated. We examine the effect of the BW differential cross section on pair angular and energy distributions. Although, this study is relevant for laser induced intense gamma-ray source collisions experiments, we apply the pair beaming in astrophysical context, in particular for Active Galactic Nuclei (AGN).

Introduction

Electron-positron pair production in the collision of two photons is a fundamental physical process predicted by the quantum electrodynamics (QED) [START_REF] Beresteskii | Quantum Electrodynamics[END_REF]. The electronpositron pair annihilation in two photons: e + + e -→ γ + γ was theoretically predicted by Dirac [START_REF] Dirac | [END_REF] and has obtained extremely accurate experimental verification [3]. But the reverse process: γ + γ → e + + e -, theoretically predicted by Breit-Wheeler (BW) [4], although conceptually simple, has been by far more difficult to verify experimentally.

The photon-photon collision physics was first studied in 1970's in Novosibirsk [5] and Frascati [6]. In these experiments the pairs were created in the collision of electron and positron beams: e + +e -→ e + +e -+e + +e -in the so-called Landau-Lifshitz process. The photons in this process are virtual, they present an intermediate step of the pair creation: γ * + γ * → e + + e -. Another similar QED process verified experimentally is the Bethe-Heitler process [7], γ + γ * → e + + e -, where a real photon collides with a virtual photon produced in the Coulomb electric field of an ion. Concerning the collision of real photons only the non-linear or multi-photon BW process [8] has been observed [9], where a real photon is coupled to several (4-5) low energy laser photons. The two photons collision process is still awaiting a direct observational verification. Only recently, thanks to the high power laser technology, first scheme for its direct observation have been proposed [10,11,12].

In the astrophysical context, the BW process plays an important role in high energy phenomena [13]. According to Nikishov [14], the background photons put a stringent limit on the maximum energy of photons coming from hard X-ray sources. The BW process also takes place in Active Galactic Nuclei (AGN) [15,16], where the gamma rays near the central black hole are converted into electron-positron pairs, which are further accelerated by the radiation pressure of the X-ray flux emission from the disk.

Electron-positron pair production by two gamma photon beams was studied in Ref. [17] by considering only the pairs kinematics in the BW process. The pair angular distribution in the center of mass (CM) frame was assumed to be uniform, i.e. the pairs are emitted isotropically. The effect of pair beaming in the laboratory frame was explained by the directed motion of the CM frame. This effect may facilitate the experimental observation of the BW process.

In this paper, we account for the effect of the BW differential cross section anisotropy on the pair beaming. This effect changes the angular and energy distribution of pairs in the laboratory frame. In Sect. 2, we recall the main features of the pair beaming assuming an anisotropic pairs distribution in the CM frame. Section 3 is devoted to the analysis of the effect of BW differential cross section anisotropy on pair beaming. The BW differential cross section is derived in Sec. 3.1, and the pair angular and energy distributions are studied in Sect. 3.2. Section 4, presents an analysis of the BW process in the astrophysical context, in particularly for AGN. Section 5 contains our conclusions.

Kinematics of e + , e -pairs and pair beaming

In this section we recall the main results obtained in Ref. [17], were we demonstrated the pair beaming effect in the collision of two photon beams. We use a unit system where the speed of light c = 1 and the electron mass m e = 1. We consider two colliding photons with momenta p γ 1 , p γ 2 and energies E γ 1 , E γ 2 , respectively and θ p is the angle between them. The scalar product in Sec. 3.1, of the four-momenta of two photons is a Lorentz invariant: 2E

γ 1 E γ 2 (1 -cos θ p ) = E 2 cm
, where E cm is the total energy in the CM frame. The pair creation threshold is defined by the condition E cm ≥ 2, electrons and positions are created at rest if E cm = 2. In the general case this condition writes:

E γ 1 E γ 2 (1 -cos θ p ) ≥ 2.
For a given energy of the two photons, the threshold angle θ th is defined by: θ p ≥ θ th where θ th = arccos(1-1/E γ 1 E γ 2 ). From the Lorentz transformation of the e -, e + pair momenta and energy, one finds a condition in which all pair momenta are aligned in the direction of the CM frame velocity. This condition is referred to as the: pair beaming condition. This condition writes: θ p ≤ θ p,beam , where:

θ p,beam = arccos(1 -[E γ 1 + E γ 2 ]/E γ 1 E γ 2 ).
Moreover, in the case where (E

γ 1 + E γ 2 )/2E γ 1 E γ 2 > 1,
there is beaming for θ p ≥ θ th . This latter beaming condition is relevant for collisions between very high and very low energy photons. For θ p = π and equal photon beam energies, there is no beaming because the laboratory frame and the CM frame are identical, and the pairs e -, e + are emitted in opposite directions. When the pair beaming conditions apply, the emission cone aperture θ e,beam writes:

tan θ e,beam = ± E cm E 2 cm -4 4(E γ 1 + E γ 2 ) 2 -E 4 cm .
However, the results in Ref. [17] are obtained assuming that in the CM frame, the pair emission probability is isotropic, that is, the differential cross section dσ/dΩ cm = 1 is constant. The pair beaming is then possible for θ th < θ p < θ p,beam . An example of the beaming effect, is shown in Figure 1 for the case of two photon beams with the same energy: 4 MeV crossing at an angle θ p = 40 • which satisfy the beaming condition: θ p,beam ≃ 42 • , θ th ≃ 15 • . Five thousands of pairs were emitted isotropically in the CM frame and their distribution in the laboratory frame is shown in Figure 1. Figure 1a shows the angular pair distribution versus the polar and azimuthal angles (θ e , φ e ). The pairs are emitted along the CM frame velocity. The pairs are distributed inside a cone angle θ e,beam = ±64.7 • (see Fig. 1b). Figures 1c,d present the angular distribution in the photon incidence plane and perpendicular to this plane, respectively. The pair distribution has an azimuthally symmetric bell shape.

According to the Lorentz transformation the minimum and maximum pair energies are [17]: The pair energy distribution is presented in Figure 2a. The spectrum is uniform between the two energy values given before. The effect of anisotropy of pair emission is discussed in the next section.

E max,min = (E γ 1 + E γ 2 ) 2 ± (E γ 1 + E γ 2 ) 2 -E 2 cm E 2 cm -4 2E cm .
3. Effect of the cross section anisotropy on pair beaming

BW differential cross section

The Feynman diagram technique for the differential cross section calculation is described in Ref. [START_REF] Beresteskii | Quantum Electrodynamics[END_REF]. Figure 3 shows the two first order Feynman diagrams of the BW process: the production of an e -, e + pair from the collision of two real photons. The total scattering matrix M is a sum of two scattering matrices M 1 and M 2 corresponding to the processes shown in Figures 3. The module of M averaged over all spin and polarization configurations writes:

|M| 2 = 1 4 state [M 1 M * 1 + M 2 M * 2 + 2Re(M 2 M 1 )].
The pairs are characterized by there four-momenta P and P ′ and photons by their four-momenta K and K ′ . The kinematic invariants called the Mandelstam variables are then given by: s

= (P + P ′ ) 2 = (K + K ′ ) 2 , t = (P -K) 2 = (P ′ -K ′ ) 2 and u = (P -K ′ ) 2 = (P ′ -K) 2 .
In particular √ s = E cm , is the total energy in the CM frame.

|M| 2 = 2e 4 u -1 t 2 -1 + t -1 u -1 -4 1 t -1 + 1 u -1 -4 1 t -1 + 1 u -1 2 ,
where e the electron electric charge. The BW differential cross section in the CM frame writes [18]: where, p is the momentum of the positron or electron. Expressing the Mandelstam variables through the CM frame velocity β cm and the angle between the photon direction and electron emission direction θ x , one finds:

dσ dΩ cm = 1 64π 2 s 2| p| √ s |M| 2 ,
dσ dΩ cm = r 2 e β cm s -1 + 3 -β 4 cm 2 1 1 -β cm x + 1 1 + β cm x - 2 s 2 1 [1 -β cm x] 2 + 1 [1 + β cm x] 2 , (1)
where, r e is the electron classical radius, x = cos θ x and dΩ cm = dxdφ. Figure 4a shows the BW differential cross section in CM frame for the collision of photons with equal energies of 4 MeV and θ p = 40 • . The energy dependence of the differential cross section is shown in panel (b). The probability of pair emission is symmetric along the photons direction axis and achieves a maximum in the photon beam direction, see Fig. 4a. The anisotropy increases with the CM energy. It appears clearly for √ s > 3, very close to the the pair creation threshold √ s = 2.

Pairs kinematics and energy distributions

By using the Lorentz transformations the pair emission distribution is obtained in the laboratory frame. The pair distribution has been calculated by a random generation of a five thousand pairs in the CM frame with the probability given by Eq. (1). Figure 5 shows the pair emission characteristics for the same set of parameters as in Fig. 1, with E cm = √ s = 5.3. The pair angular distribution is much more affected by the anisotropy of the differential cross section. This is expected as according to Fig. 4, the pairs emission is much more anisotropic for √ s > 3. This indeed can be seen in Figs. 5c,d where the pair beam in the collision plane is larger than in the perpendicular direction. In particular, the angular pair distribution shows two peaks in the photon beam direction. The pair energy distribution is shown in in Fig 2b . Compared to Fig. 2a, the maximum of the energy distribution is shifted to higher energies.

To investigate further the effect of the BW differential cross section on pair beaming, we varied the incoming photon energies. Two different cases are considered: 4 MeV-1 MeV, 4 MeV-7 MeV, while keeping the same crossing angle θ p = 40 • . For the two cases the values of √ s are: ∼3 and ∼7 corresponding to an decreasing and increasing anisotropy respectively according to the Fig. 4b. Figure 6 shows the pair emission characteristics for 4 MeV and 1 MeV photon beams collision. The pair angular distribution (see Fig. 6a) shows a stronger beaming than in Fig. 5, indeed, the pair beaming condition gives: θ e,beam = ±14.7 • . Moreover, because of the unbalanced photon energy beams, the CM frame velocity is almost aligned along the high energy photon beam direction. Then most of the pairs are emitted in the direction of the highest photon beam energy. The pairs energy distribution is shown in in Fig 7a . Compared to Fig. 2b, the maximum of the energy distribution is shifted to higher energies and uniform as in Fig. 2a.

For 4 MeV and 7 MeV photon energies, the pair emission characteristics are show in Fig. 8. The angular distribution presented in Fig. 8a, shows that the CM frame velocity direction is aligned toward the high energy photon direction. Because, the beam angle θ p,beam = 37 • , the beaming condition is not totally satisfied, some pairs are emitted backward compared to the CM frame velocity direction. Figs. 8b,c,b show that the pair beam is split in two parts as in Fig. 5c. However, the pair beam presents an asymmetry in the photon incidence plane and more pairs are emitted in the direction of the highest photon energy beam. The pair energy distribution is shown in Fig 7b . The shape of the energy distribution shown in Fig. 2b is close to the energy distribution plotted in Fig. 2b, but shifted to higher energies.

In summary, to investigate the BW differential cross section effect on pair beaming, three cases are considered in terms of photon beam energies: 4-4 MeV, 4-1 MeV, 4-7 MeV, all cases with θ p = 40 • . In the case 4-4 MeV, the influences of the differential cross section are important: on pair angular distribution as well as on the energy distribution. The pairs are emitted mainly in the photon beam directions. Then, less pairs are emitted in the bisector between the photon beam directions compared to the case of the isotropic differential cross section. The pair energy distribution achieved a maximum at the mean energy of the two photon beams. Concerning the two last cases: 4-1 MeV; 4-7 MeV, the pair beam is emitted mainly along the CM frame velocity direction. Because of the photon beams energy difference, less pairs are emitted on the bisector between the two photon beams. However, for of 4-1 MeV photon beams collision, the effect of BW differential cross section on the angular pair distribution and energy distributions are weak. In the case 4-7 MeV photon beams, the effect becomes important and the pair distribution is beamed in the direction of the most energetic photon beam. The energy distribution is shifted to higher energies and achieves a maximum for the mean energy of the two photon beams.

Application on pair production in Active Galactic nuclei

In this section we present an analysis of the BW process in the astrophysical context, in particular in AGN. An AGN produces jets of relativistic particles [19]. The relativistic flows are characterized by their bulk Lorentz factor Γ b (equivalent to E on the pair energy distribution see the previous section). The observations show that in AGN the Lorentz factor for jet flows is in range: Γ b = 2 -10. It is mainly assumed that the relativistic flow is dominated by a e -, e + pair plasma. Moreover, the idea of two flow structures was supposed. The jet is composed by a mildly relativistic sheath composed of e -/p + and driven by magnetohydrodynamical (MHD) forces and an ultra-relativistic flow composed of e -, e + pairs, which is responsible for most of the emission (see Fig. 9). We assume that the AGN is composed of a central rotating black hole (BH), with two different accretion disks: around the central BH for radius R < 10r g an advectiondominated accretion flow (ADAF) (r g is the Schwarzschild radius, r g = 2MG/2c 2 , M is the BH mass, G the gravitational constant and c the speed of light) and an external standard accretion disk (SAD). The SAD emits photons mainly in ultra-violet and X rays and the ADAF emission is between the radio and gamma-ray range [21].

The mechanisms of e -, e + pair plasma production and acceleration are not well understood, however we will assume that the pair plasma is produced from the Breit-Wheeler process. As it is shown in Fig. 9, the gamma-rays emitted from the ADAF collide along the BH rotation axis. Once the pair plasma is created by the BW process, the anisotropy of the high energy photons emitted from the accretion disk can transfer a strong momentum to the pair plasma via the inverse Compton process. This mechanism is know as Compton rocket [20]. We propose to study direct pair plasma creation and acceleration by the BW process, and show that this flow can be produce with high Lorentz factor Γ b ∼ 10 without the requirement of the Compton rocket mechanism.

The characteristics of the photon energy distribution emitted from the ADAF was calculated in Ref. [21], the photons are emitted from the radio to gamma-ray range, with an energy cuttoff at 4 MeV (see Fig. 2 in Ref. [21]). From the pair threshold production definition, only photons between 0.065 -4 MeV energies can participate to the BW pair production (see Sect. 2). We propose to study the photon beams collision for different angles θ p , in two different cases: for 4 MeV-4 MeV and 4-0.5 MeV photon beam energies. For the second situation, 0.5 MeV has been chosen to highlight the beaming effect and the consequence of energy difference between the two photon beams.

Figure 10 shows for θ p = 180 • , 90 • , 40 • , 16 • the pair momenta distributions for 4 MeV-4 MeV photon beams collision. From the pair beaming criteria, only the pairs corresponding to θ p ∈ [14.7 • , 41.9 • ] are beamed. For θ p = 180 • the pairs are emitted toward the photon beam directions and the energy distribution is mono-energetic (see Fig. 10c), all the pairs are emitted with Lorentz factor Γ b = 8. For θ p = 90 • most of the pairs are emitted in CM frame velocity direction, however, a small fraction are emitted toward the BH. The pair energy distribution is peaked around Γ b ∼ 8, with a maximum Γ b ∼ 13 (see Fig. 10d). For θ p = 40 • the pairs are beamed, all the pairs propagate along the CM frame velocity direction. The pair angular distribution is shown in Fig. 10a and the inner beam pair structure is due to the BW differential cross section (see previous section). For the beaming angle, the pair beam achieves the maximum energy: Γ b ≃ 15. Between θ p 40 • -16 • the pair beaming increases (see Fig. 10b) and at the same time the pair energy distribution range decreases (see Fig. 10f).

Figure 11 shows the pair distribution characteristics for 4 MeV-0.5 MeV photon beams collision. We represent the momenta distribution for θ p = 180 • , 90 • , 45 • . Because of the different photon energies (see the previous section), the pair beam is emitted mainly along the higher photon energy direction. For θ p = 180 • the pair momenta distribution are preferentially emitted in the direction of the CM frame velocity, that is the direction of the maximum photon energy beam. The corresponding energy distribution in Fig. 11c shows that the energy distribution is no more a Dirac distribution. As in the previous case, the distribution is enlarged between Γ b ∼ 1 -7.5 and the pair beam is emitted preferentially at low and high energies. The beaming condition is satisfied for θ p ∈ [42.4 • , 98.6 • ], then all the pairs are emitted in the CM frame velocity direction. However, the pair beam axis is no longer on the AGN symmetry axis, but with an angle of 45 • from the symmetry axis. On Fig. 11a the pair angular distribution is peaked due to BW differential cross section effect. The energy distribution is more uniform and achieves the maximum extent, as we observe in the previous case, with Γ b 1 -8 (see Fig. 11d). Finally, for θ p = 45 • the pair momenta distribution is pinched along the CM frame velocity direction (see Fig. 11b). The pair beam is emitted with an average angle of ∼ 20 • from the AGN symmetry axis and pairs are beamed inside ±5 • . The corresponding energy distribution varies between Γ b ∼ 3 -6 (see Fig. 11e).

In summary, the two different photon collision situations allow to observe that for 4 -4 MeV photon beam energies, the pair beam is aligned along the AGN symmetry axis, with the maximum Lorentz factor Γ b ∼ 15 when the beaming criteria is reached. The pair beam is more collimated due to the BW differential cross section effect. While in the 4 -0.5 MeV case the pair beam is off axis with a lower Lorentz factor Γ b < 8 and is less collimated.

Conclusions

The pair beaming condition in BW process has been previously studied in Ref. [17]. In this paper, we go further to investigate in details the effect of the BW differential cross section on the pair beaming. We show that, for equal photon beam energies, this effect is weak, i.e. for √ s < 3, the pairs are emitted mainly on the bisector of the initial photon beam directions. However, for different photon beam energies, the pairs are beamed in the CM frame velocity direction. For √ s > 3, the effect the BW differential cross section becomes important. Two peaks in the pair angular distribution appear in the photon incident plane along the photon beam directions. In the case of equal photon energy beams, the two peaks are symmetric and for unbalanced energies more pairs are emitted toward the highest photon energy direction. The energy distributions are modified in consequence.

An application of the BW process to an astrophysical situation, in particular to AGN, shows that the pair beam achieved the maximum Lorentz factor when the beaming conditions are satisfied. For the 4 MeV-4 MeV photon collisions situation, Lorentz factors Γ b greater than 10 can be reached. In this case Γ b values obtained are close to the AGN jets observations and the BW differential cross section effect is important. For the 4 MeV-0.5 MeV photon collision situation, the pair beam is off-axis and the pair beam energy distribution is shifted to lower energies. Then, this study suggests that the BW process could be taken into account in initial conditions or in addition of the Compton rocket pair acceleration process.

Figure 1 .

 1 Figure 1. Pair emission in the laboratory frame for E γ1 = E γ2 = 4 MeV and θ p = 40 • . (a) Angular distribution of pair emission, θ e and φ e are the polar and azimuthal angles. Red and blue arrows delimit the pair emission directions, black arrow the direction of CM frame velocity. (b) Pair angular distribution in the direction of the CM frame velocity, blue and black points show the positrons and electrons, respectively. (c) Angular histogram (θ ex ) in the plane of incidence of photons. (d) Angular histogram (θ ey ) in the direction perpendicular to the incidence plane.

Figure 2 .

 2 Figure 2. Pair energy distribution for E γ1 = E γ2 = 4 MeV and θ p = 40 • . (a) For isotropic (b) for BW differential cross section.

Figure 3 .

 3 Figure 3. Electron-positron pair production from a pair of photons represented to the first order by Feynman diagrams, corresponding to the BW process: γ + γ → e + + e -.

Figure 4 .

 4 Figure 4. (a) BW differential cross section dσ/dxdφ (in units of r 2 e ) blue and red arrows give the photon direction and black arrow the CM frame velocity direction for photon beams with equal energy of 4 MeV and θ p = 40 • . (b) Dependence of the differential cross section on the photon energy and polar angle.

Figure 5 .

 5 Figure 5. Pair emission in the laboratory frame for E γ1 = E γ2 = 4 MeV and θ p = 40 • with anisotropic differential cross section. (a) Angular distribution of pair emission, θ e and φ e are the polar and azimuthal angles. Red and blue arrows delimit the pair emission directions, black arrow the direction of CM frame velocity. (b) Pair angular distribution in the direction of the CM frame velocity, blue and black points show the positrons and electrons, respectively. (c) Angular histogram (θ ex ) in the plane of incidence of photons. (d) Angular histogram (θ ey ) in the direction perpendicular to the incidence plane.

Figure 6 .

 6 Figure 6. Pair emission in the laboratory frame for E γ1 = 4 MeV, = E γ2 = 1 MeV and θ p = 40 • with anisotropic differential cross section. (a) Angular distribution of pair emission, θ e and φ e are the polar and azimuthal angles. Red and blue arrows delimit the pair emission directions, black arrow the direction of CM frame velocity. (b) Pair angular distribution in the direction of the CM frame velocity, blue and black points show the positrons and electrons, respectively. (c) Angular histogram (θ ex ) in the plane of incidence of photons. (d) Angular histogram (θ ey ) in the direction perpendicular to the incidence plane.

Figure 7 .

 7 Figure 7. Pair energy distribution for θ p = 40 • . (a) For E γ1 = 4 MeV,E γ2 = 1 MeV and (b) for E γ1 = 4 MeV, E γ2 = 7 MeV.

Figure 8 .

 8 Figure 8. Pair emission in the laboratory frame for E γ1 = 4 MeV, = E γ2 = 7 MeV and θ p = 40 • with anisotropic differential cross section. (a) Angular distribution of pair emission, θ e and φ e are the polar and azimuthal angles. Red and blue arrows delimit the pair emission directions, black arrow the direction of CM frame velocity direction. (b) Pair angular distribution in the direction of the CM frame velocity, blue and black points show the positrons and electrons, respectively. (c) Angular histogram (θ ex ) in the plane of incidence of photons. (d) Angular histogram (θ ey ) in the direction perpendicular to the incidence plane.

Figure 9 .

 9 Figure 9. Schematic picture of the rotating BH accretion disk axis along with the emission from the inner ADAF and outer SAD. The ADAF emits in the gamma-ray range and the SAD in the UV and X-ray range (r g is the Schwarzschild radius). The two source flows are represented: a non relativistic outflow from the ADAF driven by the opened magnetic field, and plasma pairs that escape with relativistic speed along the inner flux tubes.

Figure 10 .

 10 Figure 10. Schematic picture of the AGN, two beam sources of equal energy 4 MeV are emitted symmetrically by the ADAF and collide on the axis. For θ p = 180 • ; 90 • , 40 • , 16 • , the momenta for each photon beam collisions are plotted. (a)-(b) show angular pairs distribution when beaming condition is satisfied. (c)-(f) show the energy distribution for the different θ p values.

Figure 11 .

 11 Figure 11. Same situation than as in Figure 10, but with the two photon beam sources of energies 4 MeV-0.5 MeV colliding on the symmetry axis for θ p = 180 • ; 90 • , 45 • .

Acknowledgments

We acknowledge the financial support from the French National Research Agency (ANR-17-CE30-0033-01) -TULIMA Project. This work is partly supported by the Aquitaine Regional Council.


Ilkser Erdem Kiper

Paulette Bloomer

Philippe Borsa

Thierry Bernard Hoareau
email: thoareau@gmail.com

Characterization of genome-wide microsatellite markers in rabbitfishes, an important resource for artisanal fisheries in the Indo-West Pacific

Keywords: Microsatellite resolution, Short-tandem repeat, Cross-species amplification, Siganus spp

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

Introduction

Rabbitfishes (genus Siganus) are Indo-West Pacific reef-associated fishes currently comprising 30 valid species [1,[START_REF] Eschmeyer | Catalog of fishes: genera, species, references, electronic version[END_REF]. These fishes are economically important as a food source in coastal areas across the Indo-West Pacific where they support local fisheries and aquaculture [START_REF] Lam | Siganids: their biology and mariculture potential[END_REF][START_REF] Woodland | Revision of the fish family Siganidae with description of two new species and comments on distribution and biology[END_REF][START_REF] Kuiter | Coastal fishes of south-eastern Australia[END_REF][START_REF] Kuriiwa | Phylogenetic relationships and natural hybridization in rabbitfishes (Teleostei: Siganidae) inferred from mitochondrial and nuclear DNA analyses[END_REF][START_REF] Lemer | Cytochrome b barcoding, molecular systematics, and geographic differentiation in rabbitfishes (Siganidae)[END_REF]. Overfishing threatens this resource, as in the case of the cordonnier Siganus sutor (Valenciennes 1835), along the coasts of eastern Africa [START_REF] Wambiji | Reproductive biology of three siganid species: Siganus canaliculatus (Park 1797), Siganus sutor (Valenciennes 1835) and Siganus stellatus (Forsskal 1775) in family Siganidae along the Kenyan coast[END_REF][START_REF] Samoilys | Artisanal fisheries on Kenya's coral reefs: decadal trends reveal management needs[END_REF]. Appropriate resource management requires the identification of demographically independent populations, which is often achieved using microsatellite markers [START_REF] Selkoe | Microsatellites for ecologists: a practical guide to using and evaluating microsatellite markers[END_REF]. The aim of the present study was to develop microsatellite markers for siganid fishes. To this end, we identified polymorphic microsatellite loci in a target species (S. sutor) and evaluated their utility as genetic markers in twelve other congeneric species.

Materials and methods

Siganus spp. fin tissue samples were obtained from several locations across the Indo-West Pacific (Online Resource 1). The rabbitfish species included in this study were S. argenteus, S. canaliculatus, S. corallinus, S. doliatus, S. fuscescens, S. laqueus, S. luridus, S. puellus, S. punctatus, S. rivulatus, S. spinus, S. sutor and S. vulpinus (Online Resource 1). Genomic DNA was extracted from fin tissue by using the DNeasy kit (Qiagen, Valencia CA, USA) following the manufacturer's protocol.

The microsatellite library was developed for S. sutor using the protocol of fast isolation by amplifiedfragment length polymorphism of sequences containing repeats, which involves a protocol based on digestion and ligation reactions [START_REF] Zane | Strategies for microsatellite isolation: a review[END_REF] combined to 454 sequencing [START_REF] Abdelkrim | Fast, cost-effective development of speciesspecific microsatellite markers by genomic sequencing[END_REF]. We followed the same procedures as [START_REF] Reid | High-throughput microsatellite marker development in two sparid species and verification of their transferability in the family Sparidae[END_REF]. Genomic DNA was fragmented by enzyme digestion using the restriction enzyme MseI (New England Biolabs, Ipswich, MA). This was followed by the ligation of the small fragments to an adaptor (5′-TACTCAGGACTCAT-3′/5′-GACGATGAGGTCCTGAG-3′) [START_REF] Zane | Strategies for microsatellite isolation: a review[END_REF][START_REF] Reid | High-throughput microsatellite marker development in two sparid species and verification of their transferability in the family Sparidae[END_REF]. The ligation products were then hybridized to the repeat containing biotinylated probes (CAAA 6 , GAAA 6, and GATA 6 ) [START_REF] Tóth | Microsatellites in different eukaryotic genomes: survey and analysis[END_REF] and the hybridized regions were captured on streptavidin-coated beads (Roche, Basel, Switzerland). After hybridization, non-specific binding elements were removed by washing. The eluted DNA, which consists of chemically and heat-separated DNA fragments containing selected repeats removed from the probes, was amplified according to [START_REF] Zane | Strategies for microsatellite isolation: a review[END_REF] and the amplification product was pyrosequenced using GS-FLX titanium reagents by the 454 Genome Sequencer FLX™ (Inqaba Biotech, Pretoria, South Africa).

The repeat-enriched sequence library was mined for potential microsatellites, i.e., nucleotide sequences containing ≥ 8 short tandem repeats for all the repeat types including mono-, di-, tri-, tetra-, penta-and hexanucleotides, using MSATCOMMANDER v. 0.8.2 [START_REF] Faircloth | MSATCOMMANDER: detection of microsatellite repeat arrays and automated, locusspecific primer design[END_REF]. Sequences > 100-bp long were selected to ensure adequate flanking regions for primer design. Duplications were verified using the online version of the alignment program MAFFT (http://mafft.cbrc.jp/alignment/server/), selecting the default settings, but choosing the option for adjusting the direction of nucleotide sequences. A single copy of identical sequences was kept. The program PRIMER DESIGNER v. 4.20 (SECentral, Scientific & Educational Software, Durham NC, USA) was used to generate 59 primer sets based on standard criteria (Online Resources 2-4). A total of 19 of these primer sets were selected for further tests. Additionally, we included 14 microsatellite markers that universally amplify in acanthopterygian fishes [START_REF] Gotoh | Characterization of comparative genome-derived simple sequence repeats for acanthopterygian fishes[END_REF]. All 33 markers were used for cross-species amplification in S. sutor and the 12 other siganid species. A total of 17 markers found to be polymorphic in S. sutor were further genotyped in three other species, S. spinus, S. luridus and S. rivulatus. Amplification reactions were done in 10 µL volumes and the reaction mixture included ~ 40 ng of template DNA, 1× PCR buffer, 1.5 mM MgCl 2 , 0.2 mM dNTPs (Promega, Madison WI, USA), 0.2 pM primers (Whitehead Scientific, Johannesburg, South Africa), and 0.04 U Supertherm Taq polymerase (Southern Cross Biotechnologies, Cape Town, South Africa). The PCR cycling conditions were: 94 °C for 5 min, followed by 30 cycles of 94 °C for 30 s, 1 min at annealing temperature (details in Online Resource 3), 72 °C for 1 min and by a final elongation at 72 °C for 7 min. Amplification products were subjected to electrophoresis on 3% agarose gels (1.5 h, 100 V) and visualised with GelRed™ stain (Biotium, Hayward CA, USA). The amplification reactions were repeated using fluorescently-labelled ChromaTide® Alexa Fluor™ 488-5-dUTPs (Invitrogen, Carlsbad CA, USA) to confirm the presence of multiple alleles at each locus. Forward primers of Siganus-specific loci that were polymorphic were labelled by the fluorescent G5 dye set (Applied Biosystems, Foster City CA, USA). Genotyping was done using the Quantitect Multiplex PCR kit (Qiagen), following the manufacturer's recommendations. The fragments were analysed on an ABI 3500xl genetic analyzer (Applied Biosystems) with the GeneScan Liz™ 500 Size Standard (Applied Biosystems). Scoring of genotypes was done using the GENEMARKER v. 1.5 software (SoftGenetics, State Collage, Pennsylvania, USA).

To investigate the transferability of the 19 new Siganus-specific markers across the family, we evaluated the relationship between genetic distance to the target species S. sutor and cross-species amplification success. To estimate genetic distance, nucleotide sequences of the mitochondrial cytochrome b (cytb) gene were used. This included sequences available from GenBank (Online Resource 5) and new sequences generated in the present study (Online Resource 6). We followed the procedure described in [START_REF] Hoareau | Design of phylum-specific hybrid primers for DNA barcoding: addressing the need for efficient COI amplification in the Echinodermata[END_REF] to design specific primers for the family Siganidae. Cycling conditions included initial denaturation at 94°C for 3 min; 35 cycles of denaturation at 94°C for 30 s, annealing at 55°C for 30 s, elongation at 72°C for 1 min; and final elongation at 72°C for 7 min. The amplified products were precipitated using 2.8 volumes of absolute ethanol, 0.4 volumes of Sabax water and 0.08 volumes of 3M sodium acetate, followed by a washing step using 90 µL 70% ethanol. The products were eluted in 15 µL Sabax water. Cycle sequencing reactions contained approximately 30 ng purified DNA, 1 µL ABI PRISM BigDye Terminator Cycle Sequencing Ready Reaction Kit v. 3.1 (Applied Biosystems), 5 x sequencing buffer and 3.2 pM primer. Sequencing was done on an ABI 3500xl sequencer (Applied Biosystems).

Genetic distance between species was estimated as the net mean nucleotide divergence (Kimura-2parameter model; MEGA6) [START_REF] Tamura | MEGA6: molecular evolutionary genetics analysis version 6.0[END_REF] at the cytb locus between S. sutor and the other species considered. The distances were used to reconstruct a Neighbor-Joining tree in MEGA6, to illustrate the diversity of species used in the present study and their phylogenetic relationships (Fig. 1a). The GenBank sequences that represented possibly misidentified specimens, possible mitochondrial-DNA introgression or cryptic species were removed from the analysis (Online Resource 5).

Summary statistics on the microsatellite genotype data, and their significance, were estimated using GENETIX v.4.05 [START_REF] Belkhir | GENETIX 4.05, logiciel sous WINDOWS™ pour la génétique des populations[END_REF]. The possible presence of null alleles was assessed using MICROCHECKER v. 2.2. 3 [20]. Linkage disequilibrium (LD) among markers was estimated and tested using permutation tests in ARLEQUIN v. 3.0 [START_REF] Excoffier | ARLEQUIN (ver. 3.0): an integrated software package for population genetics data analysis[END_REF]. BAYESCAN v. 2.1 [START_REF] Foll | A genome-scan method to identify selected loci appropriate for both dominant and codominant markers: a Bayesian perspective[END_REF] and LOSITAN [START_REF] Antao | LOSITAN: a workbench to detect molecular adaptation based on a F st -outlier method[END_REF] were used to detect potential outlier loci based on allelic differences estimated by F ST , and comparison between the observed F ST and expected heterozygosity among the populations, respectively. We used the program POWSIM v. 4.1 [START_REF] Ryman | POWSIM: A computer program for assessing statistical power when testing for genetic differentiation[END_REF] to evaluate the statistical power of the markers to detect subtle genetic structure. This program identifies the threshold above which F ST values become significant, given the distribution of allele frequencies in a set of markers. The specific test that we performed relied on simulated pairs of populations (100 replicates) with a set sample size (50 individuals) and effective population size (N e = 5000), but a varying number of generations since isolation (t = 5-300) to generate various levels of divergence (i.e. F ST values). To further evaluate the utility of the markers with regards to potential spatial genetic structuring, an analysis of molecular variance (AMOVA) [START_REF] Excoffier | Analysis of molecular variance inferred from metric distances among DNA haplotypes: application to human mitochondrial DNA restriction data[END_REF] was done among samples of S. spinus and S. luridus, the two species from the study with multiple samples from different locations. Each sampling location was defined as a population (see Online Resource 1). The significance of the variance components among populations and within populations was determined using 1000 permutations.

Results and discussion

The repeat-enriched library contained 6112 sequences and the mining step uncovered 310, 61, and 421 sequences containing di-, tri-and tetra-nucleotide repeats, respectively (Online Resource 7). Only a subset of these sequences was longer than 100 bp and contained sufficiently long flanking regions at both ends. These included 28, 12, and 38 sequences containing di-, tri-, and tetra-nucleotide repeats, respectively. Among these, 59 sequences were used for primer design (Online Resource 2). Due to budget limitations, only 19 of these potential markers were tested for amplification in 13 Siganus species, including S. sutor.

Cross-species amplification success of the 33 markers (of which 19 were Siganus-specific and14 were universal in ray-finned fishes) across 12 siganids ranged from 36% (S. laqueus) to 76% (S. corallinus, S. doliatus and S. fuscescens). The genetic distance to S. sutor, estimated from nucleotide sequences at the cytb locus, ranged from 0.035 (S. rivulatus) to 0.165 (S. argenteus) (Online Resource 8).The amplification success of the 19 Siganusspecific markers ranged from 32% (in S. vulpinus) to 79% (in S. fuscescens) (Fig. 1a) and showed a weak yet significant decrease with the genetic distance to S. sutor (Fig. 1b). The new microsatellite markers therefore amplify in species other than S. sutor but phylogenetic distance affects the amplification efficiency. Alternative primers might need to be designed for the few markers that do not, or poorly amplify in some species. Similar negative correlation has been reported previously [START_REF] Reid | High-throughput microsatellite marker development in two sparid species and verification of their transferability in the family Sparidae[END_REF][START_REF] Carreras-Carbonell | Utility of pairwise mtDNA genetic distance for predicting cross-species microsatellite amplification and polymorphism success in fishes[END_REF]. This trend is likely a general one across taxa and markers, and should be accounted for in the evaluation of cross-species amplification.

Twelve of the 19 Siganus-specific markers and five of the universal acanthopterygian fish markers were polymorphic in S. sutor (Online Resource 3). These were tested for polymorphism in three other siganid species (Table 1). The expected heterozygosity averaged across all loci was 0.822 for S. luridus, 0.753 for S. rivulatus, 0.682 for S. spinus, and 0.823 for S. sutor, while the mean number of alleles across loci was 8.78, 5.73, 6.75, and 11.69, respectively. Significant deviation from Hardy-Weinberg equilibrium was detected at 3/9 loci in S. luridus, 1/11 loci in S. rivulatus, 2/8 loci in S. spinus, and 7/17 loci in S. sutor, but null alleles were detected in only a smaller proportion of the loci: SIG006 in S. spinus, and SIG008 and SIG013 in S. sutor. Significant linkage disequilibrium values were detected for 2/28 locus-pairwise comparisons in S. spinus (Online Resource 9) and for 5/120 comparisons in S. sutor, after Bonferroni correction [27] (Online Resource 10). Considering the low occurrence of significant values, both null alleles and linkage disequilibrium are expected to have a marginal effect on the analysis of genetic differentiation between populations in S. spinus and S. sutor. BAYESCAN [START_REF] Foll | A genome-scan method to identify selected loci appropriate for both dominant and codominant markers: a Bayesian perspective[END_REF] and LOSITAN [START_REF] Antao | LOSITAN: a workbench to detect molecular adaptation based on a F st -outlier method[END_REF] detected a single outlier locus (SIG017), relative to the level of genetic differentiation between populations. The occurrence at this locus of an F ST value above the 99% confidence interval suggests directional selection, but this can also be due to non-overlapping allele sizes between species.

By assessing the resolution power of the microsatellite markers, we were able to evaluate whether the sets of microsatellites in S. rivulatus, S. spinus and S. sutor could potentially identify genetically distinct populations. A previous simulation study showed that the threshold F ST values for the detection of genetic subdivision among populations ranged from as low as 0⋅005-0⋅007 when using the appropriate set of markers [START_REF] Waples | What is a population? An empirical evaluation of some genetic methods for identifying the number of gene pools and their degree of connectivity[END_REF]. In the present study, the set of microsatellite markers was able to detect subtle levels of genetic differentiation through the power analysis. These values were as low as F ST = 0.0135 in S. luridus, F ST = 0.015 in S. spinus, and F ST = 0.011 in S. sutor (Fig. 2); S. rivulatus was excluded from this analysis due to low sample sizes. These results based on simulations were confirmed by the AMOVA analyses, which showed significant genetic differentiation among locations in both S. luridus (F ST = 0.152, P = 0.001) and S. spinus (F ST = 0.077, P = 0.021) despite low sample sizes per location ranging from one to ten (Table 2). The set of markers proposed in this study therefore proved useful in detecting subtle genetic structure in Siganus spp.

The amplification success, the level of polymorphism, the resolution power, and the pilot study for the detection of population differentiation show that these microsatellites represent a valuable molecular resource for rabbitfish species. These markers will be useful for developing studies aimed at improving resource management in artisanal rabbitfish fisheries across the Indo-West Pacific. Number of sequences with designed primers 21 5 33 a Primers were designed using the program PRIMER DESIGNER v4.20 based on the standard criteria (melting temperature, GC content, presence of dimers, hairpin and false priming, differential stability between the 5' and 3' ends, and presence of adjacent repeats of the same base).

Online Resource 9

Pairwise linkage disequilibrium estimates in S. spinus

Fig. 1

 1 Fig. 1 Amplification success of the new set of 19 microsatellites in 13 Siganus species. a Genetic relationships among 13 Siganus species illustrated by a Neighbour Joining tree based on 184 cytochrome b gene sequences [MEGA6: Kimura-2 parameter (K2P) model; pairwise deletion] [18]. The amplification success of the 19 loci represented by circles whose surface is proportional to the proportion of amplifying loci. Supported nodes (≥ 75% bootstrap score) are indicated with a star. cytb gene sequences of three surgeonfish species (Acanthurus blochii, A. triostegus, Naso unicornis) were chosen as outgroup. b Amplification success plotted against K2P distance to S. sutor. Linear regression model: y = -0.0178x + 0.7172, R = 0.3531. The null hypothesis that the slope is equal to 0 was rejected (F = 5.46; P = 0.042).

Fig. 2 2

 22 Fig. 2 Power analysis for the set of polymorphic microsatellite markers illustrating the proportion of significance (PS) as a function of the F ST value in S. luridus (9 loci; 17 individuals; PS = 2.076 + 0.261. ln(F ST ); F ST95 = 0.0135), S. spinus (8 loci; 20 individuals; PS = 2.055 + 0.265. ln(F ST ); F ST95 = 0.015), and S. sutor (16 loci; 32 individuals; PS = 1.991 + 0.232. ln(F ST ); F ST95 = 0.011). Dotted line indicates 95% significance threshold (F ST95 ). The relatively low F ST values observed at this threshold show that fine-scale population structure can be detected using this set of markers

Table 1

 1 Summary statistics for 17 polymorphic loci tested in four siganids (with sample size in brackets) including the number of observed alleles (N A ), expected heterozygosity (H E ), observed heterozygosity (H O ), and inbreeding coefficient (F IS ) estimates[START_REF] Weir | Estimating F-Statistics for the analysis of population structure[END_REF] P < 0.001. NA: locus with no amplification, MD: locus with missing data (a proportion of the individuals were not amplified)

	Locus	Statistic	Species
			S. luridus (17)	S. rivulatus (7)	S. spinus (20)	S. sutor (32)
	ORLA2-91
		N A	13	6	12	14
		H E	0.927	0.929	0.840	0.905
		H O	0.882	1	0.700	0.931
		F IS	0.050	-0.091	0.191***	-0.029
	ORLA8-113
		N A	8	1	11	4
		H E	0.831	0	0.853	0.696
		H O	0.625	0	0.900	0.594
		F IS	0.254*	0	-0.030	0.149
	ORLA11-71
		N A	5	2	4	5
		H E	0.546	0.533	0.650	0.668
		H O	0.588	0.400	0.700	0.719
		F IS	-0.081	0.273	-0.051	-0.078
	ORLA12-160
		N A	NA	MD	11	11
		H E			0.837	0.883
		H O			0.824	0.906
		F IS			0.047	-0.027
	ORLA14-42
		N A	15	5	5	17
		H E	0.944	0.800	0.555	0.913
		H O	0.857	1	0.650	0.844
		F IS	0.096	-0.290	-0.146	0.077
	SIG006
		N A	NA	7	3	8
		H E		0.890	0.539	0.732
		H O		1	0.300	0.406
		F IS		-0.135	0.464***	0.449***
	SIG007
		N A	7	4	NA	10
		H E	0.836	0.714		0.780
		H O	0.546	0.857		0.677
		F IS	0.358***	-0.220		0.133
	SIG008
		N A	NA	NA	NA	11
		H E				0.882
		H O				0.308
		F IS				0.655***
	SIG009
		N A	MD	11	MD	23
		H E		0.956		0.953
		H O		1		0.739
		F IS		-0.050		0.228***
	SIG010
		N a	MD	MD	NA	16
		H E				0.886
		H O				0.636
		F IS				0.286***
	SIG011
		N A	9	7	5	16
		H E	0.874	0.933	0.709	0.937
		H O	0.769	1	0.600	0.760
		F IS	0.124	-0.081	0.178	0.192***
	SIG013

Table 2

 2 Analysis of molecular variance for Siganus luridus and Siganus spinus. Degrees of freedom (df), percentage contribution to overall genetic variance (GV%) and fixation index (F ST ) with related P-value are presented.

	Species	Source of variation	df	GV%	Fixation index	P-value
	S. luridus	Among populations	2	15.19	F ST : 0.152	0.001
		Within populations	31	84.81
	S. spinus	Among populations	4	7.67	F ST : 0.077	0.021
		Within populations	35	92.33

* P < 0.05 (1000 permutations)

Acknowledgements

We would like to thank the following collaborators for facilitating the collection of samples: A. Brito and R. Mutombene from the Instituto Nacional de lnvestiqacao Pesqueira (Mozambique), N. Wambiji from the Marine & Fisheries Research Institute (Kenya), S. Abeare from Reef Doctor (Madagascar), G. Gouws from the South African Institute for Aquatic Biodiversity (South Africa), A. Collet, G. Mou-Tham and D. Lecchini from IRD (New Caledonia), P. Boblin and E. Tardy from SPC (New Caledonia), E. Tessier from Parc Marin (La Réunion), F. Giancarlo from Proyek Biokor (Indonesia and West Papua), M. Hassan from SMEL (France), and M. Low from NUS (Singapore).

Funding

This project was funded by the Western Indian Ocean Marine Science Association (WIOMSA): MARG I/11/2013 and the South African Department of Science and Technology/National Research Foundation African Coelacanth Ecosystem Programme (ACEP) II. Sampling was funded in part by IRD, SPC and SAIAB. IEK benefitted from a PhD scholarship from the University of Pretoria's PhD programme. TBH was supported by the University of Pretoria's research fellowship programme and the University of Pretoria Research Development Programme.

Conflict of interest

We have no conflict of interest regarding the content of the present paper.

Molecular Biology Reports

Supplementary information to:

Characterization of genome wide microsatellite markers for rabbitfishes, an important resource for artisanal fisheries in the