Introduction 1 2 The polarization potential of a non-spherical shell 2 3 The P-wave instability 4 4 The D-wave instability 5 5 Discussion 6

A Explicit expressions for the coefficients of the tachyonic potentials. 9

Introduction

The physics of antibranes placed in backgrounds that contain charges dissolved in the fluxes has been a constant source of surprises. The supergravity solutions that describe these antibranes have certain singularities, that are visible both when the fields sourced by the branes are treated as a perturbation [START_REF] Bena | On the Existence of Meta-stable Vacua in Klebanov-Strassler[END_REF][START_REF] Bena | On Metastable Vacua and the Warped Deformed Conifold: Analytic Results[END_REF][START_REF] Dymarsky | On gravity dual of a metastable vacuum in Klebanov-Strassler theory[END_REF][START_REF] Bena | The backreaction of anti-D3 branes on the Klebanov-Strassler geometry[END_REF][START_REF] Bena | The Backreaction of Anti-M2 Branes on a Warped Stenzel Space[END_REF][START_REF] Giecold | Insane Anti-Membranes?[END_REF] and when the full backreaction of these fields is taken into account [START_REF] Blåbäck | The problematic backreaction of SUSY-breaking branes[END_REF][START_REF] Blåbäck | Anti-)Brane backreaction beyond perturbation theory[END_REF][START_REF] Bena | Anti-D3's -Singular to the Bitter End[END_REF][START_REF] Gautason | Cosmological Constant, Near Brane Behavior and Singularities[END_REF]. The fields diverging at the singularity have the right form and g s dependence to cause the antibranes to polarize into higher-dimensional branes, but in all regimes of parameters where the exact coefficients of these fields could be computed they are such that polarization either does not happen [START_REF] Bena | Persistent anti-brane singularities[END_REF][START_REF] Bena | Polchinski-Strassler does not uplift Klebanov-Strassler[END_REF], or if it happens it is accompanied by a brane-brane-repelling tachyon, which causes the antibranes to run away from each other [START_REF] Bena | Tachyonic Anti-M2 Branes[END_REF][START_REF] Bena | Giant Tachyons in the Landscape[END_REF] 1 .

The generic potential for n anti-D3 branes to polarize into N 5 five-branes wrapping a twosphere of radius R in a three-form field strength proportional to C is of the form [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF]:

V polarization = N 2 5 n R 4 -CN 5 R 3 -m 2 nR 2 . (1.1)
The first term represents the excess mass brought about by the presence of the five-branes, the second term is the polarization force exerted by the six-form potential (Hodge dual to threeform field strength) on the five-branes, and the last term is the potential felt by the n anti-D3 branes. When a tachyon is present (m 2 > 0) the polarization potential has always a minimum, regardless of the sign a C. This implies that anti-D3 branes can polarize into five-branes even when the cubic term in the polarization potential (and hence the six-form potential that induces the branes to polarize) is absent or has the wrong sign [START_REF] Zamora | Nonsupersymmetric SO(3) invariant deformations of N=1* vacua and their dual string theory description[END_REF]. Hence, the presence of a branebrane-repelling tachyon appears to favor brane polarization. One may even go so far as to argue

1 When the worldvolume of the branes is not flat but AdS, this singularity can be resolved by polarization [START_REF] Apruzzi | AdS 6 solutions of type II supergravity[END_REF][START_REF] Junghans | Curvature-induced Resolution of Anti-brane Singularities[END_REF], but this can only happen when the scale of the AdS space is the same as the scale of the polarizing fields [START_REF] Bena | Persistent anti-brane singularities[END_REF]. Hence, this phenomenon is irrelevant for resolving the singularities of the antibranes that are added to AdS flux compactifications to obtain de Sitter spaces with small cosmological constant [START_REF] Kachru | De Sitter vacua in string theory[END_REF].

that the presence of this tachyon should be taken as an indication that antibrane singularities will always be resolved by brane polarization, and that the tachyon simply signals the intention of the unstable unpolarized anti-D3 branes to move to a stable vacuum where the antibranes are polarized into five-branes. It is the purpose of this paper to investigate the stability of the vacua with polarized antibranes in the presence of a brane-brane-repelling tachyon. We will examine polarized branes with a tachyonic term in the polarization potential and examine two possible instability modes, both of which break the spherical symmetry of the polarization shell: the P-mode, which corresponds to shifting the center of the polarization shell, and the D-mode, which corresponds to deforming the spherical polarization shell into a ellipsoidal (cucumber) shape. We will find that the P-mode deformation is always tachyonic, while the D-mode is only tachyonic when the threeform field strength that polarizes the branes is smaller than a certain value (C < 2 3 m). Hence, the presence of a brane-brane-repelling tachyon renders unstable the configurations where these branes are polarized into higher-dimensional branes.

The paper organized as follows. In Section 2 we write down the potential describing the polarization of D3 branes into a five-brane wrapping a (shifted and squashed) non-spherical polarization shell. The next two sections are devoted to the stability analysis of this shell: In Section 3 we study the stability under a P-wave deformation, corresponding to a shift of the center of the sphere, and find that this deformation is tachyonic whenever a tachyon is present in the polarization plane, regardless of the magnitude of the charges or of the polarizing fields. In Section 4 we study the stability under a D-wave (ellipsoidal) deformation and show that this deformation is also tachyonic whenever the tachyon is stronger than a certain value. In the last section we discuss possible extensions and generalizations of our results and their implications for the stability of anti-D3 branes polarized into NS5 branes [START_REF] Kachru | Brane / flux annihilation and the string dual of a nonsupersymmetric field theory[END_REF] at the bottom of the Klebanov-Strassler solution [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF]. Some lengthy formulae are relegated to the Appendix.

The polarization potential of a non-spherical shell

The goal of this section is to derive the polarization potential for D3 branes polarized into fivebranes that wrap a two-sphere perturbed with P-and D-mode deformations inside an R 3 along which the inter-D3 potential is tachyonic. The five-brane worldvolume is spanned by the four Minkowski coordinates of the D3 branes and two compact directions inside the R 3 parameterized as:

x 1 = R 1 sin θ cos ϕ x 2 = R 1 sin θ sin ϕ (2.1)
x 3 = R 2 cos θ + a . (2.2)
When the shift parameter a is non-zero and the radii are equal this embedding describes a P-wave deformation of the spherical shell. The D-wave deformation corresponds to a = 0 and

R 1 = R 2 .
The pull-back of the R 3 metric on the deformed two-sphere wrapped by the five-brane is:

g ⊥ = R 2 1 • ξ 2 1 -ξ 2 + R 2 2 dξ 2 + R 2 1 (1 -ξ 2 )dϕ 2 , (2.3)
where we introduced: ξ ≡ cos θ .

(2.4)

The D3 brane charge of the five-brane gives rise to a nontrivial 2-form flux on its worldvolume:

F 2 = nf (ξ)dξ ∧ dϕ , (2.5)
where n is the total number of D3 branes and f (ξ) is the D3 charge density function normalized as:

1 -1 f (ξ)dξ = 1 . (2.6)
For the spherically symmetric configuration f (ξ) = 1 2 , but when the sphere is deformed f (ξ) will change, and has to be determined anew. To ensure that the total D3 brane charge is the same, one has to add a global (ξ-independent) Lagrange multiplier term to the probe action:

λ • f (ξ) - 1 2 . (2.7)
As explained in [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF], the quartic term in the polarization potential is equal to the mass difference between n unpolarized D3 branes and n D3 branes inside a five-brane shell. When the total mass of the D3 branes is bigger than the five-brane mass this is proportional to:

detg ⊥ 4 √ detF 2 = 1 4nf (ξ) R 2 1 ξ 2 + R 2 2 1 -ξ 2 R 2 1 , (2.8)
where the factor of 4 in the denominator was chosen such that the integral of this term matches the first term in (1.1). The cubic term in the polarization potential (1.1) comes from the six-form potential that forces the five-brane shells to expand. In the near-brane region the field strength corresponding to this potential is proportional to the volume form of the R 3 plane in which the polarization happens:

dC 6 ∼ dx 1 ∧ dx 2 ∧ dx 3 ∧ Vol Mink , (2.9)
Integrating the pullback of C 6 over the worldvolume of the five-brane gives the potential density:

- 3 2 CR 2 1 R 2 ξ 2 , (2.10)
where C is a constant proportional to the strength of the polarizing fields, the factor of 3/2 was chosen for later convenience and the minus sign reflects the fact that the orientation of the five-brane is such that the force exerted by the six-form potential favors polarization. The quadratic term in the polarization potential is given by the D3 brane density multiplied by the value of the tachyonic potential at their location:

-m 2 nf (ξ) x 2 1 + x 2 2 + x 2 3 , (2.11)
where the minus sign reflects the presence of a brane-brane-repelling tachyon and the expression x 2 1 + x 2 2 + x 2 3 has to be pulled back on the worldvolume using the parameterization (2.1). When the number of five-branes is larger than one, the full potential can be obtained from the potential above by simply replacing n by n/N 5 and multiplying by an overall factor of N 5 .

The P-wave instability

In this section we are interested in the polarization potential into a five-brane wrapping a shifted sphere given by (2.1) with R 1,2 = R and non-zero a. With this parametrization we find that x2 1 + x 2 2 + x 2 3 = R 2 + 2Raξ + a 2 and, using equations (2.7), (2.8), (2.10) and (2.11), we obtain the potential density:

V (f (ξ), ξ, R, a, λ) = -m 2 nf (ξ) R 2 + 2Raξ + a 2 - 3 2 CR 3 ξ 2 + R 4 4nf (ξ) + λ • f (ξ) - 1 2 . (3.1)
To find the final potential,

V (R, a) = 1 -1 V (f (ξ), ξ, R, a, λ) dξ one has to solve δV δf (ξ) = 0 , (3.2)
together with the constraint (2.6). Unfortunately, for a non-zero a the solution of (3.2) for the charge distribution function,

f (ξ) = R 2 2 λn -m 2 n 2 R 2 + 2Raξ + a 2 -1/2 , (3.3)
has a complicated dependence on ξ which does not yield an analytic final expression for λ and consequently for the potential V (R, a). 2 Nevertheless, we are interested only in the small-a behavior of the potential, and so it will suffice to solve for the Lagrange multiplier in a series expansion:

λ(a) = λ 0 + λ 1 a + λ 2 a 2 + . . . , (3.4)
where the coefficients depend on m, n and R. Expanding (3.3) in a and integrating term by term in ξ we find that (2.6) implies:

λ 0 = R 4 n 1 + mn R 2 , λ 1 = 0 , λ 2 = m 2 n 1 + mn R 2 (3.5)
and for the purpose of this analysis we will not need higher-order coefficients. Substituting this expansion into (3.3) and (3.1) and integrating the potential density over ξ, we arrive at the final expression for V (R, a):

V (R, a) = -m 2 nR 2 -CR 3 + R 4 n -m 2 n 1 + 1 3 mn R 2 • a 2 + O a 3 . (3.6)
The coefficient of the a 2 term is strictly negative, but this does not yet imply P-wave instability. Indeed, we have to expand the potential around the polarization radius R = R pol which minimizes the full potential including the a 2 term: Substituting R = R pol into (3.6) one arrives at the final form of the potential we are interested in:

R pol = n 8 3C + 9C 2 + 32m 2 - 1024 3
m 4 n C 243C 4 + 1296C 2 m 2 + 1536m 4 + 81C 4 + 288C 2 m 2 + 128m 4 9C 2 + 32m 2 -1 • a 2 + . . . . (3.7)
V on-shell (n, C, m) = n 3 C 4 v 0 (ω) + nC 2 v 2 (ω) • a 2 + O a 3 , where ω ≡ m C , (3.8)
and the full expressions for v 0 (ω) and v 2 (ω) are given in the Appendix. The leading term, v 0 (ω), is the value of the potential for a spherically symmetric (a = 0) configuration, and one can see from equation (A.2) and from Figure 1 that the term quadratic in a is always negative, regardless of the values of m, C and n.

Hence, a small shift of the center of the polarized sphere lowers its potential energy and makes this configuration P-wave unstable.

The D-wave instability

In this section we will study a D-wave (ellipsoidal) deformation of the spherical shell. When a = 0, equation (2.1) implies that

x 2 1 + x 2 2 + x 2 3 = R 2 1 (1 -ξ 2 ) + R 2 2 ξ 2 ,
V (f (ξ), ξ, R 1 , R 2 , λ) = -m 2 nf (ξ) R 2 1 (1 -ξ 2 ) + R 2 2 ξ 2 - 3 2 CR 2 1 R 2 ξ 2 + + R 2 1 4nf (ξ) R 2 1 ξ 2 + R 2 2 (1 -ξ 2 ) + λ • f (ξ) - 1 2 . (4.1)
As in the previous section, we will consider an infinitesimal deviation from a spherical shell:

R 1 = R + σ • r 1 , R 2 = R + σ • r 2 , (4.2)
where σ is a (dimensionless) "squashing" parameter, and R is the radius of the undeformed sphere (given by the leading-order (a = 0) term in (3.7)).

Proceeding as earlier, we solve (3.2) to find the D3 charge density distribution, f (ξ), subject to the constraint (2.6). This allows us to obtain the perturbative σ-expansion for the Lagrange multiplier λ. For our purposes it is enough to determine λ only up to O(σ 2 ), which upon substituting back to the potential gives us the following quadratic term:

nC 2 b 1 (ω)r 2 1 + b 2 (ω)r 2 2 + b 3 (ω)r 1 r 2 • σ 2 , (4.3)
where the expressions for b i (ω) are given in the Appendix. The spherical solution is then stable with respect to D-wave fluctuations if and only if both eigenvalues of the corresponding matrix are positive. The product of the eigenvalues is 4b 1 b 2 -b 2 3 , and this expression is zero when

m = 3 2 C . (4.4)
It is straightforward to check that when the tachyon is larger3 than this critical value, m > 3 2 C, one of the eigenvalues is always negative, and hence the configuration is unstable. In particular, all the polarization channels where the polarizing fields are zero or negative (C ≤ 0) and the polarization only happens because of the tachyon [START_REF] Zamora | Nonsupersymmetric SO(3) invariant deformations of N=1* vacua and their dual string theory description[END_REF], give rise to configurations that are D-wave unstable. However, when the polarizing field is positive and much stronger than the tachyon, the D-wave instability is absent.

Discussion

We have found that brane polarization in the presence of brane-brane-repelling tachyons results in unstable configurations. Our analysis focused of the stability of polarized branes that lay inside an R 3 plane along which there is a brane-brane-repelling tachyon, and the perturbations that we explore do not take the polarization shell outside of this plane. However, in general a D3 brane solution perturbed with transverse three-form fields can have many polarized vacua, corresponding to polarization into arbitrary (p, q) five-branes wrapping two-spheres laying in various R 3 subspaces of the R 6 transverse to the D3 branes [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF]. Hence, in order to show that tachyons are incompatible with metastable vacua with polarized branes one must in principle analyze the stability of all such polarization shells, and consider more general perturbations than the ones we have considered in this paper. While we leave the thorough analysis of the stability of these configurations to a subsequent paper [22], we outline below a few aspects of this analysis that reveal that the instability we found may be fatal for all brane polarization channels, regardless on whether they are along a plane where there is a brane-brane-repelling tachyon.

In order to do this, it is important to recall that in the supersymmetric Polchinski-Strassler dual of the N = 1 * gauge theory [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF], the polarization of the D3 branes into (p, q) five-branes only gave rise to a supersymmetric vacuum when the polarization two-sphere was inside a specific R 3 , where the cubic term in the polarization potential is maximal (since the cubic term comes from the polarization force exerted on the shell, we can call this the "maximum-force" R 3 ). However, nothing stops one from considering D3 branes polarized into (p, q) five-branes wrapping a twosphere inside a different R 3 plane, such as a D5 shell in an oblique plane, or an oblique shell in the D5 or NS5 planes. These polarized shells will have more energy than the supersymmetric vacua, and are unstable under deformations that keep the polarized shell inside the same R 3 plane. However, if we tilt these shells towards the "maximum-force" R 3 , their energy is lowered, and hence these shells are not even sitting on extrema of the polarization potential.

When supersymmetry is broken there is no longer a natural way to associate a given type of (p, q) five-brane to a given polarization R 3 subspace of R 6 . Since both the cubic and the quadratic terms depend on the orientation of the polarization plane, one can also identify, besides the "maximum-force" R 3 , a "maximum-tachyon" R 3 , along which the quadratic term in the potential is the largest. 4 In general these two three-planes will not be the same, and the minimum-energy round shell will lay inside an R 3 plane between them. If there is a branebrane-repelling tachyon in this plane our result implies that this round shell will be unstable. However, if this plane has a positive quadratic term in the polarization potential than one will have to perform a more thorough stability analysis [22].

We can now combine this intuition with the results of [START_REF] Bena | Giant Tachyons in the Landscape[END_REF] in order to ascertain whether anti-D3 branes at the bottom of the Klebanov-Strassler (KS) solution [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF] 5 can still give rise to metastable vacua when polarized into NS5 branes inside the S 3 at the bottom of the KS solution [START_REF] Kachru | Brane / flux annihilation and the string dual of a nonsupersymmetric field theory[END_REF]. As shown in [START_REF] Bena | Giant Tachyons in the Landscape[END_REF], the "maximum-force" orientation of these NS5 brane shells is inside the S 3 , but the "maximum-tachyon" orientation is in a different direction. Furthermore, the value of the brane-brane-repelling tachyon inside the "maximum-force" plane is exactly zero [START_REF] Bena | Giant Tachyons in the Landscape[END_REF]. It is not hard to see that this NS5 brane will lower its energy by tilting: As one inclines the brane away from the "maximal-force" plane, the coefficient of the cubic term decreases like the square of the tilting angle (given by the phase of z in equation (2.11) in [START_REF] Zamora | Nonsupersymmetric SO(3) invariant deformations of N=1* vacua and their dual string theory description[END_REF] or equation (3.1) in [START_REF] Bena | Giant Tachyons in the Landscape[END_REF]); however, since this brane does not lay inside the maximum-tachyon plane, the coefficient of the quadratic term will decrease linearly with this tilting angle. Since the two terms are of the same magnitude, the overall potential will decrease as the NS5 brane tilts away from the "maximum-force" and settles along another plane that sits between the "maximum-force" and the "maximum-tachyon" planes. Since the tachyon is exactly zero in the "maximum-force" plane [START_REF] Bena | Giant Tachyons in the Landscape[END_REF], and maximal in the "maximum-tachyon" plane, it will have a finite nonzero value in this plane. One can then use the result of our analysis to show that this brane will be unstable. Hence, in the regime of parameters where the antibranes backreact our analysis implies that anti-D3 branes polarized into NS5 branes inside the S 3 at the bottom of the KS solutions are unstable.

Thus, even if our investigation does not address in full all the possible decay channels of all possible polarization channels of tachyonic branes, it shows that the most commonly used onesthe anti-D3 branes polarized into NS5 branes at the bottom of the KS solution -are unstable. It is clearly a very important open question to analyze in detail all polarization channels of these antibranes and to find whether there is any metastable one, or whether they are all unstable. It is also important to understand in general whether brane polarization can cure any branebrane repelling tachyon, or if even the tiniest such tachyon is enough to destabilize all polarized branes. It is also interesting to analyze the polarization of anti-M2 branes in the presence of such tachyons. In [START_REF] Massai | Metastability in Bubbling AdS Space[END_REF] it was found that anti-M2 branes placed in bubbling Lin-Lunin-Maldacena (LLM) geometries [START_REF] Lin | Bubbling AdS space and 1/2 BPS geometries[END_REF] can give rise to metastable vacua in which these anti-M2 branes are polarized into M5 branes. The polarization was induced only by a tachyonic term, and if our analysis extends to that situation it would imply that these metastable anti-M2 LLM vacua are in fact unstable.

Another interesting direction is to explore in general the fate of the P-wave instability of multiple D3 branes polarized into a single D5 brane inside a tachyonic AdS 5 × S 5 solution sourced by these D3 branes. Our calculation gives the polarization potential of a probe in a general tachyonic background and, as explained in [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF], this probe potential can be used to find the potential describing the polarization of all the D3 branes sourcing the solution. When the D3 branes polarize into multiple five-brane shells, the P-wave tachyonic mode we find above corresponds to shifting the centers of these five-brane shells away from each other. However, if one assumes that our calculation of the polarization potential extends to the configuration in which all the D3 branes are polarized into a single five-brane, it is unclear what this P-wave tachyonic mode corresponds to. 6Our analysis explores the effects of the brane-brane-repelling tachyon on the stability of brane polarization shells using the Born-Infeld action of these shells. However, brane polarization can also be described using the degrees of freedom of the branes that polarize [START_REF] Polchinski | The String dual of a confining four-dimensional gauge theory[END_REF][START_REF] Myers | Dielectric branes[END_REF], and this description is valid in the regime of parameters where the backreaction of these branes is not important7 (g s N 1). In the vacua where the branes are polarized, the N × N scalar fields living on the branes become non-commutative. It is not hard to see that upon deforming the Lagrangian of N = 4 SYM with a tachyonic traceless boson bilinear term of the form m 2 Tr(-Φ and these "Higgs" vacua describe the polarization of the D3 branes into D5 branes. Hence, from the point of view of the theory on the branes, a tachyon that causes a runaway behavior on the Coulomb branch (when the scalars commute) can still give rise to Higgs-branch vacua.

The question is then whether these Higgs-branch vacua are metastable, or are tachyonic, and also what is the matrix equivalent of the P-and D-wave instabilities we have found. More generally, to prove that the Higgs-branch vacua are metastable one would have to show that the O(N 2 ) possible perturbations of this vacuum give rise to an O(N 2 × N 2 ) mass matrix that has only positive eigenvalues. At first glance this appears rather unlikely, but if such a metastable vacuum exists it would have quite extraordinary implications for our understanding of vacua in tachyonic non-Abelian gauge theories.

Figure 1 .

 1 Figure 1. The graph shows the a 2 term of the polarization potential v 2 (ω) (with ω ≡ m/C), which is always negative.

  ) one can easily find vacua where these fields become non-commutative[Φ 1 , Φ 2 ] ∼ m Φ 3 + cyclic permutations,(5.1)

	2 1 -Φ 2 2 -Φ 2 3 + Φ 2 4 + Φ 2 5 + Φ 2 6

Note that when a = 0 the function f (ξ) is ξ-independent, as expected for a spherically-symmetric configuration.

Recall that in our conventions m and n are positive.

See for example equation(2.11) in[START_REF] Zamora | Nonsupersymmetric SO(3) invariant deformations of N=1* vacua and their dual string theory description[END_REF] or equation(3.1) in[START_REF] Bena | Giant Tachyons in the Landscape[END_REF].

Which have a brane-brane-repelling tachyon in the regime of parameters where their backreaction is important[START_REF] Bena | Giant Tachyons in the Landscape[END_REF], and the number of anti-D3 branes is larger than the square of the KS three-form flux.

This discussion does not apply to backreacted antibranes, for which we have only shown that a brane-branerepelling tachyon is present in a regime of parameters in which the dipole charge of the polarization shell is greater than one

.[START_REF] Blåbäck | The problematic backreaction of SUSY-breaking branes[END_REF] Although the existence of a brane-brane-repelling tachyon was only established when gsN 1[START_REF] Bena | Giant Tachyons in the Landscape[END_REF], the technology now exists[START_REF] Michel | Remarks on brane and antibrane dynamics[END_REF] for studying whether non-backreacting antibranes also have tachyons; a simple exploration of the SO(6) transformations of the various terms that can appear in the brane-effective-action[START_REF] Michel | Remarks on brane and antibrane dynamics[END_REF] indicates that a brane-brane-repelling tachyon might as well be present in this regime.

Acknowledgments

We would like to thank Johan Blåbäck, Ben Freivogel, Mariana Graña, Praxitelis Ntokos, Silviu Pufu and David Turton for interesting discussions. This work was supported in part by the ERC Starting Grant 240210 String-QCD-BH, by National Science Foundation Grant No. PHYS-1066293 (via the hospitality of the Aspen Center for Physics) by the John Templeton Foundation Grant 48222 and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by Theiss Research).

A Explicit expressions for the coefficients of the tachyonic potentials.

We give below the full expression of the coefficients v 0 (ω), v 2 (ω) and b 1,2,3 (ω) that appear in the P-wave and D-wave brane polarization potentials (3.8) and (4.3):

1053 + 3240ω


Omry Barzilai

Noemi Rebollo

Dani Nadel

Fanny Bocquentin

Reuven Yeshurun

György Lengyel

Gal Bermatov-Paz

Elisabetta Boaretto
email: elisabetta.boaretto@weizmann.ac.il

Radiocarbon dating of human burials from Raqefet Cave and contemporaneous Natufian traditions at Mount Carmel

Keywords: Levant, Israel, Raqefet Cave, el-Wad, Natufian

The Natufian culture (c. 15-11.5

ka cal BP) marks a pivotal step in the transition from hunting and gathering to sedentism and farming in the Near East. Although conventionally divided into Early and Late phases, this internal chronology lacks support from reliable absolute dates. This is now addressed by new AMS dating from two neighbouring Natufian sites at Mount Carmel in Israel:

Raqefet Cave, conventionally assigned to the Late phase of the Natufian; and el-Wad Terrace, spanning the entire Natufian sequence. Results indicate that these two sites were in fact contemporaneous at some point, but with distinct lunate assemblages. Distinguishing between Natufian phases is, therefore, more complex than previously thought; the social implications of diverse but co-existing cultural manifestations must be considered in any future reconstruction of the Natufian.

Introduction

The Natufian was a semi-sedentary hunter-gatherer culture that occupied the Levant during the Terminal Pleistocene (e.g. Belfer-Cohen 1991;Valla 1995). Natufian sites in the Mediterranean woodland area of the southern Levant include curvilinear structures with stone foundations, intensively used cemeteries with diverse burial customs, ground stone tools and bedrock features, decorated art objects and evidence for dog domestication [START_REF] Davis | Evidences for the domestication of the dog in the Natufian of Israel 12,000 years ago[END_REF][START_REF] Belfer-Cohen | The Natufian settlement at Hayonim Cave: a hunter-gatherer band on the threshold of agriculture[END_REF][START_REF] Valla | Les industries de silex de Mallaha (Eynan) et du Natoufien dans le Levant[END_REF][START_REF] Weinstein-Evron | Early Natufian el-Wad revisited[END_REF][START_REF] Bocquentin | Burial practices, biological factors and cultural identities during the Natufian period: a bio-archaeological perspective[END_REF][START_REF] Dubreuil | Long-term trends in Natufian subsistence: a use-wear analysis of ground stone tools[END_REF][START_REF] Rosenberg | The sounds of pounding: boulder mortars and their significance to Natufian burial customs[END_REF]. As such, the Natufian culture was innovative in many ways. Natufian subsistence relied on systematic plant gathering and processing, evidenced by flint sickle blades and ground stone tools, and by the intensified hunting of gazelles and small game [START_REF] Unger-Hamilton | Natufian plant husbandry in the southern Levant and comparison with that of the Neolithic periods: the lithic perspective[END_REF][START_REF] Dubreuil | Long-term trends in Natufian subsistence: a use-wear analysis of ground stone tools[END_REF][START_REF] Munro | Zooarchaeological measures of hunting pressure and occupation intensity in the Natufian[END_REF][START_REF] Edwards | A 14 000 year-old hunter-gatherer's toolkit[END_REF][START_REF] Bar-Oz | Specialized hunting of gazelle in the Natufian: cultural cause or climatic effect?[END_REF][START_REF] Yeshurun | Intensification and sedentism in the Terminal Pleistocene Natufian sequence of el-Wad Terrace (Israel)[END_REF]. These practices are considered to have played a major role in initiating the 'agricultural revolution' in the Near East, thereby lending special importance to the accurate determination of their chronologies (e.g. Valla 1995;[START_REF] Belfer-Cohen | Early sedentism in the Near East-a bumpy ride to village life[END_REF].

The geographic expansion of the Natufian is broadly divided into two provinces: the Mediterranean woodland area and the more arid belt (Belfer-Cohen & Goring-Morris 2013;Goring-Morris & Belfer-Cohen 2013;[START_REF] Richter | Preliminary report on the second season of excavation at Shubayqa 1[END_REF]. The settlement pattern in the Mediterranean area, where Raqefet Cave is situated, includes semi-permanent settlements and other sites that were probably designated for burials (Figure 1). The latter are considered indicators of "boundaries between various regional groups" (Goring-Morris et al. 2009: 205). Some of the Natufian graves display complex funerary practices reflecting an elaborate social system (Garrod & Bate 1937: 14-19;[START_REF] Belfer-Cohen | The Natufian settlement at Hayonim Cave: a hunter-gatherer band on the threshold of agriculture[END_REF][START_REF] Byrd | Death, mortuary ritual, and Natufian social structure[END_REF][START_REF] Bocquentin | Burial practices, biological factors and cultural identities during the Natufian period: a bio-archaeological perspective[END_REF][START_REF] Grosman | A 12,000-year-old shaman burial from the southern Levant (Israel)[END_REF][START_REF] Nadel | Earliest floral grave lining from 13,700-11,700-y-old Natufian burials at Raqefet Cave, Mt Carmel, Israel[END_REF].

Natufian chronology

The rich archaeological record is used for determining the relative Natufian chronology and is conventionally divided into Early and Late phases, although some scholars (e.g. Valla 1995) divide it into three phases: Early, Late and Final. This paper follows the two-phase division that combines the Late and Final into a single phase. The most commonly used criterion for distinguishing between these phases is that of the microlithic lunates, which were originally part of composite hunting tools [START_REF] Bocquentin | Early Natufian remains: evidence for physical conflict from Mt Carmel, Israel[END_REF][START_REF] Yaroshevich | Variability of lunates and changes in projectile weapons technology during the Natufian[END_REF]. Larger lunates shaped by bifacial retouch, known as Helwan lunates, are typically attributed to the Early Natufian, whereas smaller backed lunates characterise the Late Natufian (e.g. [START_REF] Bar-Yosef | L'évolution du Natufien, nouvelles suggestions[END_REF][START_REF] Valla | Les industries de silex de Mallaha (Eynan) et du Natoufien dans le Levant[END_REF][START_REF] Goring-Morris | At the edge: Terminal Pleistocene hunter-gatherers in the Negev and Sinai[END_REF]. This chronological scheme is based upon the stratigraphic sequence of two major sites, Eynan and el-Wad Terrace [START_REF] Garrod | The Stone Age of Mount Carmel. Volume I. Excavations at the Wadi Mughara[END_REF][START_REF] Valla | Les industries de silex de Mallaha (Eynan) et du Natoufien dans le Levant[END_REF][START_REF] Kaufman | The Natufian sequence of el-Wad Terrace: seriating the lunates[END_REF], and is often used for determining the chronology of undated Natufian sites.

Natufian radiocarbon chronology is based on more than 120 radiocarbon dates, which supposedly represent the entire Natufian sequence (Maher et al. 2011: tab. 3;Grosman 2013: appendix). Many dates, however, are on materials from poorly defined contexts or were produced by old dating techniques, such as decay counting. Recent attempts to refine the Natufian chronology based on published data applied several screening criteria to reduce uncertainties and errors. Maher et al. (2011: tab. 3), for example, excluded 34 published dates from their analysis that had large standard deviations or unclear archaeological contexts; single dates and old 14 C determinations were also excluded. Materials problematic for dating (e.g. burnt bones from the old excavations at Raqefet and Hatoula, or charcoal from old excavations at Jericho), however, were not excluded. Moreover, no control was applied regarding charcoal characteristics (i.e. old wood effect), and no details were provided on charcoal preservation.

The chronology proposed by [START_REF] Grosman | The Natufian chronological scheme-new insights and their implications[END_REF] relies on 23 dates attributed to the Early Natufian and 78 dates attributed to the Late Natufian. It is not specified which dates were excluded from the chronological scheme, but the database (Grosman 2013: appendix) is very similar to that of Maher et al. (2011: tab. 3). Dates were excluded from the analysis if they met one of three conditions: a) a large error range (more than 300 years); b) they were prepared before the beginning of the 1980s; or c) the age result is 2000 years older or younger than expected. [START_REF] Blockley | A revised chronology for the adoption of agriculture in the southern Levant and the role of late glacial climatic change[END_REF] rely solely on four Natufian sites for their chronology: Eynan, Hayonim, Nahal Oren (samples from not always clearly defined contexts) and Raqefet Cave (old excavation with unreliable contexts; see [START_REF] Lengyel | Upper Palaeolithic and Epipalaeolithic lithic technologies at Raqefet Cave, Mount Carmel East, Israel[END_REF]. In our view, this scheme, based exclusively on four sites, does not represent the entire range and nuances of the Natufian radiometric chronology.

To avoid such limitations, we chose to focus on recently produced dates from two nearby sites from Mount Carmel as a case study for Natufian chronology, which are from welldefined, high-quality contexts. Accordingly, we here present new dates obtained from the recent excavations at Raqefet Cave and compare them to the published dates from el-Wad Terrace [START_REF] Eckmeier | Radiocarbon dating of the Early Natufian at el-Wad Terrace, Mount Carmel, Israel: methodology and materials characterization[END_REF][START_REF] Weinstein-Evron | New 14 C dates for the early Natufian of el-Wad Terrace, Mount Carmel, Israel[END_REF][START_REF] Caracuta | Charred wood remains in the Natufian sequence of el-Wad Terrace (Israel): new insights into the climatic, environmental and cultural changes at the end of the Pleistocene[END_REF]. The dates from these sites, excavated using similar modern methods, and with samples extracted from secure contexts, enable us to establish a radiometric chronological framework into which the lunate assemblages from these sites can be incorporated. evidence of a lining, composed of a thick layer of plant material, including sage flowers [START_REF] Nadel | Earliest floral grave lining from 13,700-11,700-y-old Natufian burials at Raqefet Cave, Mt Carmel, Israel[END_REF].

A further aspect of note at the site is the wide variety of features hollowed out of the bedrock (Nadel & Lengyel 2009;[START_REF] Nadel | Prehistoric bedrock features: recent advances in 3D characterization and geometrical analyses[END_REF][START_REF] Nadel | An incised pattern inside a Raqefet Cave bedrock mortar: new dimensions to Natufian stone carvings[END_REF]. Stones set on edge were found in a few of the larger, rock-cut mortars (or deep shafts), and phytoliths were recovered from several deep mortars [START_REF] Power | The economic and ritual utilization of plants at the Raqefet Cave Natufian site: the evidence from phytoliths[END_REF].

The flint assemblage contains over 20 000 artefacts, and is currently undergoing detailed analyses [START_REF] Nadel | Raqefet Cave: the 2006 excavation season[END_REF]Lengyel 2009;[START_REF] Lengyel | Natufian foragers in the Levant: Terminal Pleistocene social changes in Western Asia[END_REF]. Two samples from the richest loci (1 and 3) were analysed as part of the current research (Table 1). This corpus of flints was recovered from the immediate surroundings of the burial pits or from within the graves, and should be viewed as representing the cemetery as a whole. In both loci, flakes are the dominant product, comprising just over 50 per cent of the total assemblages. Bladelets are approximately four times more common than blades. Tools encompass 7.9 and 6.4 per cent of the lithic assemblages from loci 1 and 3 respectively. In these loci the lunate assemblage (n = 200) is dominated by the abruptly backed type (89.5 per cent). The abrupt category includes both unipolar and bipolar specimens, and a mixture of both (Figure 6). The Helwan lunates include fully and partially retouched specimens (Figure 7), and eight specimens that have both Helwan and some abrupt retouch. According to the commonly used relative Natufian chronology (e.g. [START_REF] Bar-Yosef | L'évolution du Natufien, nouvelles suggestions[END_REF][START_REF] Valla | Les industries de silex de Mallaha (Eynan) et du Natoufien dans le Levant[END_REF], this proportion of abruptly backed lunates aligns the Raqefet assemblage with the Late Natufian phase.

Comparing complete Helwan (n = 12) and abruptly backed lunate (n = 104) lengths shows that the former lunates are significantly larger than the latter (t (114) = -2.377, p = 0.019) (Table 2).

Flint from the grave fills may not be directly associated with the dated human remains. Such lithics must be either contemporaneous or earlier to have been included in the burials. The relationship of the burials with the materials in the fills was, however, examined taphonomically in the study of the faunal remains from locus 1; these food remains were interpreted as representing funerary feasts, rather than domestic Natufian refuse accumulated prior to the digging of the graves [START_REF] Yeshurun | The social role of food in the Natufian cemetery of Raqefet Cave, Mount Carmel, Israel[END_REF].

The radiocarbon dates

We obtained eight radiocarbon dates (see Table 3 for details) from Raqefet: five from human long bones belonging to five individuals (three adults and two adolescents), and three from charcoal pieces found in association with the burials. The charcoal specimens were taxonomically identified and pre-treated following the standard water-acid-base-acid procedure that was also used for the el-Wad Terrace samples [START_REF] Eckmeier | Radiocarbon dating of the Early Natufian at el-Wad Terrace, Mount Carmel, Israel: methodology and materials characterization[END_REF]. The five bones were pre-screened using Fourier transform infrared spectroscopy, which determined their splitting factor and the preservation of collagen (Rebollo et al. forthcoming). Sediments adhering to the bones were removed before prescreening and their mineralogical composition was analysed. Bone samples were prepared following the ultrafiltration method of Bronk Ramsey et al. (2004). Quality control of the suitability of the dated samples was carried out for each sample. All radiocarbon dates were calibrated using IntCal13 [START_REF] Reimer | IntCal13 and Marine13 radiocarbon age calibration curves 0-50,000 years cal BP[END_REF]OxCal v4.2 (Bronk Ramsey 2009). All samples were prepared and measured at the Dangoor Research Accelerator Mass Spectrometer D-REAMS for Radiocarbon Dating at the Weizmann Institute of Science.

The new set of dates from Raqefet Cave clusters between 14 000 and 13 000 cal BP, except for RTK 6638, which dates to around 12 000 cal BP (Figure 8). The stratigraphy, spatial distribution of the burials and radiocarbon dates indicate that the site was used as a Natufian burial ground over many generations. The earliest burial phase is locus 1 (Figure 3), which saw the burial of at least ten individuals between 14 010 and 13 480 years cal BP. This range is derived from the dating of the lowest burial in the northern cluster (Homo 15, RTK 6481) and the uppermost burial in the southern cluster (Homo 9, RTK 6541). These dates confirm the diachronic progression of the burials in the rock basin, starting from the north, adjacent to the wall of the cave, and continuing towards the south [START_REF] Lengyel | Natufian foragers in the Levant: Terminal Pleistocene social changes in Western Asia[END_REF]. Dating of charcoal from the southern cluster may extend the use of this location to 13 140-12 940 year cal BP. The radiometric results show that this location was used for burial for approximately 500 years, and possibly more. (locus 3, 195-200); 3: bipolar (locus 3, C15d, 190-195); 4: Helwan (locus 3, E15a, 210-214); 5: Helwan and abrupt (locus 3, 193-200).

Other dated graves were double interments in the adjacent locus 3. The double grave containing Homo 18/Homo 19 was dated to 13 640-13 160 cal BP and the double grave of Homo 25/Homo 28 was dated to 12 400-11 600 cal BP.

Discussion

Recent evaluations of available radiocarbon dates have proposed that the Natufian lasted approximately 3500 years, between c. 15 000 and 11 500 cal BP (e.g. Goring-Morris This chronology is supported by comparing the Raqefet Cave results to the neighbouring Natufian site of el-Wad Terrace, located 10km to the west (Figure 1). El-Wad Terrace provides one of the most detailed and best-dated Natufian sequences in the Mediterranean core area [START_REF] Weinstein-Evron | New 14 C dates for the early Natufian of el-Wad Terrace, Mount Carmel, Israel[END_REF][START_REF] Kaufman | The Natufian sequence of el-Wad Terrace: seriating the lunates[END_REF]. The Early Natufian at el-Wad Terrace is represented by a thick layer (>1m) containing structures with living floors, and overlying occupation levels with no architecture. The lithic assemblage from the Early Natufian contexts is clearly dominated by large Helwan lunates [START_REF] Kaufman | The Natufian sequence of el-Wad Terrace: seriating the lunates[END_REF].

Comparison between the lunate assemblages from Raqefet Cave and el-Wad Terrace shows differences between the sites. El-Wad Terrace units 1-2 and W0, which are the two uppermost Natufian levels, are the closest to Raqefet Cave in the Helwan: abrupt ratio. The latter, however, has a higher proportion (almost 25 per cent) of backed lunates (Table 4). Hierarchical cluster analysis of the proportions of the two types of lunates in each assemblage shows two groups (Figure 9). The smaller group includes Raqefet Cave and el-Wad units 1-2 and W0; the rest of the el-Wad assemblages comprise the larger group. The el-Wad phases were recently radiocarbon dated and clearly show a continuous occupation for the Early Natufian between 15 000 and 13 200 cal BP (Figure 7) [START_REF] Eckmeier | Radiocarbon dating of the Early Natufian at el-Wad Terrace, Mount Carmel, Israel: methodology and materials characterization[END_REF][START_REF] Weinstein-Evron | New 14 C dates for the early Natufian of el-Wad Terrace, Mount Carmel, Israel[END_REF]. Evidence for the Late Natufian is sparse and dates to 13 700-11 800 cal BP. The early age of sample RTK-6955 (charcoal) at el-Wad shows overlap there between the Early and the Late Natufian. Being a single date, it was, however, suspected to be an outlier. Now, the new Late Natufian dates from Raqefet Cave suggest the RTK-6955 sample is not an outlier. A comparison between the absolute chronology of Raqefet Cave and the Early Natufian absolute chronology of el-Wad shows an overlap of approximately 1000 years, between around 14 000 and 13 000 cal BP. Thus, the latest Early Natufian of el-Wad Terrace and the earliest Late Natufian of Raqefet Cave are contemporaneous.

The well-established dating results from these two sites raise a question concerning the differences between the contemporaneous lithic assemblages. The distinct lunate compositions (in terms of type frequencies and dimensions) from the two sites are apparently synchronous and thus they cannot be interpreted as reflecting chronological phases. Similarity in ecological setting (e.g. [START_REF] Caracuta | Charred wood remains in the Natufian sequence of el-Wad Terrace (Israel): new insights into the climatic, environmental and cultural changes at the end of the Pleistocene[END_REF] and the limited geographic distance between the sites would seem to exclude the possibility that the two communities did not interact. One plausible explanation for the typological difference is that there was a long intermediate phase between the Early and the Late Natufian during which the two lunate types coexisted [START_REF] Kaufman | The Natufian sequence of el-Wad Terrace: seriating the lunates[END_REF]. Although the presence of the two types together in Natufian sites has usually been interpreted as a mixture of two chronological phases, this a priori assumption has been questioned for several sites in the southern Levant [START_REF] Olszewski | The north Syrian Late Epipalaeolithic and its relationship to the Natufian Complex[END_REF](Olszewski , 1988;;[START_REF] Barzilai | The Natufian site of Nahal Sekher VI: the 2009 excavation season[END_REF]. The use of lunates as a relative chronological marker has been further challenged by the discovery of small lunates modified by Helwan retouch at the open-air sites of Hof Shahaf and Shubayqa 1 [START_REF] Marder | Hof Shahaf: a new Natufian site on the shore of Lake Kinneret[END_REF][START_REF] Richter | Preliminary report on the second season of excavation at Shubayqa 1[END_REF]. Their position within Natufian chronology is currently unknown due to the lack of radiocarbon dating.

Another explanation for the typological differences between Raqefet Cave and el-Wad Terrace may be found in the social sphere. Ethnographic studies show that projectile style can convey social information (e.g. [START_REF] Wiessner | Style and social information in Kalahari San projectile points[END_REF]). Thus, it is possible that the typological differences in lunate types, commonly used as projectiles [START_REF] Yaroshevich | Variability of lunates and changes in projectile weapons technology during the Natufian[END_REF], may attest to social identity. Raqefet Cave functioned primarily as a Natufian burial ground, whereas el-Wad Terrace was a settlement including dwellings, burials and a variety of features [START_REF] Garrod | The Stone Age of Mount Carmel. Volume I. Excavations at the Wadi Mughara[END_REF][START_REF] Weinstein-Evron | Early Natufian el-Wad revisited[END_REF], 2009). That no settlement evidence was found at Raqefet Cave [START_REF] Nadel | Raqefet Cave: the 2006 excavation season[END_REF](Nadel et al. , 2009[START_REF] Nadel | The 2010 and 2011 seasons of excavation at Raqefet Cave[END_REF][START_REF] Nadel | Earliest floral grave lining from 13,700-11,700-y-old Natufian burials at Raqefet Cave, Mt Carmel, Israel[END_REF] suggests that the site must have been used as a burial place for a Natufian settlement nearby. The difference between the lunate assemblages at Raqefet Cave and el-Wad, as well as the presence of burials at el-Wad, suggest that the former was not the burial site of the latter. Thus it is possible that the Raqefet Cave cemetery represents a geographic marker that distinguished between two co-existing Natufian communities, each with its different lithic tradition (i.e. lunate types) within Mount Carmel and maybe also the valleys to the east.

The current case study highlights a common archaeological problem: namely, the integration of relative and absolute chronologies. Natufian sites in the Mediterranean woodland zone are stratigraphically complex and include archaeological contexts that were subjected to a variety of taphonomic processes. It is crucial, therefore, to retrieve materials for absolute and relative dating (e.g. radiocarbon and lithic typologies) from the same secure contexts, such as graves and living floors. Our study shows that the Raqefet Cave cemetery was used for many centuries and that the fossil directeur of the Natufian, the lunate, is not a suitable chronological marker in the Mount Carmel region for the period between 14 000 and 13 000 cal BP.

Our results therefore question the reliability of the commonly used lunate-based relative dating of the Natufian. This applies to the Mediterranean zone, but may have implications for other regions. Hence the use of tool types as a proxy for dating Natufian phases (and probably also phases in other periods and places) should not be considered sufficient to support a high-precision chronology. Furthermore, only specimens retrieved from secure contexts should be incorporated in such relative dating schemes, and wherever possible they should be compared to context-specific radiocarbon dates. It is hoped that additional case studies will enhance the resolution of our dating of the Natufian, and thus further illuminate the complex processes that led to the establishment of sedentary communities and the development of agriculture in the Near East.

Figure 1 .

 1 Figure 1. Map showing major Natufian sites in the Mediterranean core area of the Southern Levant. Raqefet Cave and el-Wad are marked in red.

Figure 2 .

 2 Figure 2. Plan of burials in the first chamber, Raqefet Cave. Bn = dated bone; Ch = dated charcoal.

Figure 3 .

 3 Figure 3. Locus 1 during excavation, looking south-east. Human bones of several individuals are visible. Note the use of stone objects, and the slab with a cupmark on top.

Figure 4 .

 4 Figure 4. The double burial of Homo 25 and Homo 28.

Figure 5 .

 5 Figure 5. Homo 19 during excavation.

Figure 6 .

 6 Figure 6. Lunates from Raqefet Cave. 1: Abrupt (locus 1, C12a, 222-228); 2: abrupt and bipolar(locus 3,(195)(196)(197)(198)(199)(200) 3: bipolar (locus 3, C15d,(190)(191)(192)(193)(194)(195) 4: Helwan (locus 3, E15a,(210)(211)(212)(213)(214) 5: Helwan and abrupt (locus 3,(193)(194)(195)(196)(197)(198)(199)(200).

Figure 8 .

 8 Figure 8. Probability distribution of the calibrated radiocarbon dates from Raqefet Cave and el-Wad Terrace. Colour codes refer to material type: red: human bone; orange: animal bone; black: charcoal.

Figure 9 .

 9 Figure 9. Hierarchical cluster analysis of Raqefet Cave and el-Wad Terrace using the proportions of abruptly backed and Helwan lunates.

Table 1 . A breakdown of flint samples from loci 1 and 3 according to blanks (Raqefet Cave). Note the high similarity between the two loci.

 1

		Blades	Bladelets	Flakes	Cores	Total
	Locus 1
	Debitage	84	439	749	88	1360
	Tools	23	66	28	-	117
	Total	107	505	777	88	1477
	%	7.2%	34.2%	52.6%	6%	100%
	Locus 3
	Debitage	108	400	857	35	1400
	Tools	16	63	17	-	96
	Total	124	463	874	35	1496
	%	8.3%	30.9%	58.4%	2.3%	100%

Table 2 . Dimensions (in mm) of complete lunates from loci 1 and 3, according to type (Raqefet Cave).

 2

	Length	Width	Thickness

Figure 7. SEM image of a Helwan lunate (locus 1, B12d, 223-238).

Table 3 . 14 C dates from Raqefet Cave and el-Wad. The Raqefet Cave dates of Homo 18, 19 and 28 from Nadel et al. (2013), the el-Wad dates from Weinstein-Evron et al. (2012) and Caracuta et al. (2016). Eight samples from el-Wad were excluded from this table as they were either too old (RT 6097-2, RTD 6957 and RTD 6958) or their cultural context was not secure (RTT 6114, 6095-2) (Weinstein-Evron et al. 2012: 820-21).

 3

	14 C age BP	Calibrated	Calibrated

Table 3 .

 3 Continued.

	14 C age BP	Calibrated	Calibrated

Table 4 . Frequencies of Helwan and backed lunates along the el-Wad sequence (Kaufman et al. 2015).

 4

	Lunate types
	Backed	Helwan	Total

© Antiquity Publications Ltd, 2017

Acknowledgements

This work forms part of a post-doctoral research project conducted by O.B., supervised by E.B. (ISF grant 475/10). Fieldwork was carried out under licence numbers G-2004/50, G-34/2006, G-64/2008, G-34/2010 and G-22/2011 of the Israel Antiquities Authority, and permits of the Israel Nature and Parks Authority. The Irene Levi-Sala CARE Archaeological Foundation, the National Geographic Society (grant 8915-11) and the Wenner-Gren Foundation (grant 7481-2008) generously supported the excavation. Teresa Cabellos Panades, Aurore Lambert and Gabrielle Bosset assisted in the excavation of the burials. Michal Birkenfeld prepared Figure 1. Anat Regev-Gisis prepared Figures 2, 6 and7. We thank N. Taha from the Basin Analysis and Petrophysical laboratory (PetroLab) at the University of Haifa, and N. Waldmann and R. Reshef for their help in documenting the lunates. E. Gershtein prepared the photographs of lunates.