H Navelet

R Peschanski

I Saturation From

Qcd Dipoles

Conformal invariant saturation

We show that, in onium-onium scattering at (very) high energy, a transition to saturation happens due to quantum fluctuations of QCD dipoles. This transition starts when the order α 2 correction of the dipole loop is compensated by its faster energy evolution, leading to a negative interference with the tree level amplitude. After a derivation of the the one-loop dipole contribution using conformal invariance of the elastic 4-gluon amplitude in high energy QCD, we obtain an exact expression of the saturation line in the plane (Y,L) where Y is the total rapidity and L, the logarithm of the onium scale ratio. It shows universal features implying the Balitskyi -Fadin -Kuraev -Lipatov (BFKL) evolution kernel and the square of the QCD triple Pomeron vertex. For large L, only the higher BFKL Eigenvalue contributes, leading to a saturation depending on leading log perturbative QCD characteristics. For initial onium scales of same order, however, it involves an unlimited summation over all conformal BFKL Eigenstates. In all cases, conformal invariance is preserved for the saturation mechanism based on dipole loops.

A regime of QCD at small coupling constant where the density of partons begins to saturate has since long been the subject of many studies [1][2][3][4][5][6][7][8][9][10][11][START_REF] Jalilian-Marian | [END_REF][START_REF] Weigert | Unitarity at small Bjorken[END_REF][START_REF] Braun | [END_REF][15][16][17][START_REF] Timneanu | Saturation model for two-photon interactions at high energies[END_REF]. In one class of models, one expects the effect due to the high energy multiplication of partons due to the QCD dynamics. Another way which was proposed to investigate high density partonic effects is the consideration of collisions at high energy on heavy nuclei. Indeed, the number of partons is supposed to be large in the background of the collision providing boundary conditions favorable for the saturation mechanism to happen. In the present paper, we will focus on the first scheme, i.e. the saturation mechanism due to the energy evolution of the parton density in a purely perturbative QCD framework. More precisely, we will consider even "perturbative QCD" boundary conditions, in order to select the features of saturation which could be fully calculated from the QCD lagrangian.

On the experimental ground, saturation has not been clearly identified in the evolution of structure functions in the HERA range, even if some evidence for partial saturation has been found by analyzing vector meson diffractive production in impact parameter space [START_REF] Munier | [END_REF]. The problem of identifying saturation effects in actual or future experiments is very stimulating, since the increase of the energy range is at reach at Tevatron and later at LHC. Even, some lessons will be available from experiments with heavy nuclei at RICH. The goal of our purely perturbative calculations, even if they appear to be far from the experimental realizations, is to give an example of saturation which could be fully computed, without referring to a non-perturbative ansatz. In practice, later on, it will be possible, using factorization properties, to combine our calculations with some reasonable non-perturbative wave functions to get predictions for realistic processes, like hard hadron-hadron interations at Tevatron and LHC.

Our idea is to start from the most elementary hard processes, namely onium-onium scattering at (very) high energies. The reaction can be described [4,5] by the wave-functions of the onia in terms of q q colour singlet dipoles at the time of the (hard) interaction, which then interact through double gluon exchanges. This description, which corresponds to the tree level dipole interactions, is equivalent to the usual Balitsky, Fadin, Kuraev, Lipatov (BFKL) formalism [20] for the same process, as can be proven [21] using the conformal invariant symmetry of the BFKL evolution equation. When energy increases, the wave functions has an ever increasing density of colour dipoles and the tree level dipole interaction is expected to be modified by unitarity corrections which are already present in the perturbative regime. The first order correction is the one-loop dipole contribution which was first considered in Ref. [22] as a formal calculation of next-to-next order contributions in the coupling constant. In the present paper we show that, despite its higher order, this one-loop unitarity correction gives rise to the onset of saturation in oniumonium scattering and thus allows one to give a complete perturbative QCD description of the saturation critical line. This saturation mechanism is to be distinguished from the one which would come from the interaction of a onium (e.g. a collection of independent dipoles) with a large nucleus (see, e.g. [8,[11][START_REF] Jalilian-Marian | [END_REF][START_REF] Weigert | Unitarity at small Bjorken[END_REF]15,17,[START_REF] Timneanu | Saturation model for two-photon interactions at high energies[END_REF]), which, in some cases, can be associated with a non-linear branching process without dipole recombination. In the case we consider, the non-linearities due to the high density of partons correspond to the recombination of QCD dipoles.

We write the onium-onium elastic cross-section

dσ dQ 2 = 1 4π |F Q | 2 , ( 1
)
where Q is the transfer momentum. The amplitude reads

F Q = -i d 2 ρ d 2 ρ ′ 1 0 dz dz ′ Φ(ρ, z)Φ(ρ ′ , z ′ ) f Q (ρ, ρ ′ |Y ) , (2)
where the onium is defined as a heavy qq state with q (resp. q) having the fraction z (resp. 1-z) of the longitudinal momentum in an infinite momentum frame. It is defined by its wave-function Φ(ρ, z) over a basis of dipole states of transverse size ρ ≡ ρ 0ρ 1 , where ρ 0 , ρ 1 are the positions of the dipole's constituants in transverse coordinate space. f Q (ρ, ρ ′ |Y ) is the dipole-dipole elastic amplitude for a full rapidity interval Y and a momentum transfer Q, the calculation of which at one dipole-loop order and saturation properties are the subject of the present paper.

In fact the physical origin of the saturation mechanism that we want to investigate is the following: it is known [4] that the wave function of an Onium in the infinite momentum frame develop a large number of dipole states as a function of the rapidity (which plays the rôle of time in a cascading mechanism). However, the larger is the dipole number, the smaller is their size and thus they do not contribute to saturation. This can be proven rigorously by the fact that the dipole-dipole cross-section add together to give back the BFKL cross-section [21]. It is interesting to note the use of conformal invariance to get this exact bootstrap property between an "s-channel" and "t-channel" description of the same amplitude.

However, now if one computes the dipole-dipole interaction at one-loop level, the system is allowed to take into account quantum fluctuations where virtual dipole pairs may be created during the time of interaction. Moreover, all sizes of dipoles can contribute in the loop, which goes beyond the limitations of the tree-level mechanism. This is expected from perturbative unitarity arguments [4] which expect the damping of the energy dependence of the amplitude by saturation effects due to multi-loop dipole contributions. This is precisely the goal of our work to put this expectation on a quantitative level.

Through unitarity arguments, the QCD saturation phase corresponding to the mechanism we consider requires an estimate of all dipole loop contributions, at least in the high energy approximation. In the present paper we will concentrate on the first step, i.e. the transition to saturation which is expected to be due to the interfernece between the tree and 1-loop level. We expect that the techniques which we were led to consider in this work, relying notably on the conformal invariance properties, will be useful to investigate this intringing phase of matter.

Our main results are the following: i) Up to one-loop QCD dipole level, the dipole-dipole amplitude reads

f Q (ρ, ρ ′ |Y ) ≡ f (0) Q + f (1) Q = α 2 4 |ρρ ′ | dh E h Q (ρ ′ )E h Q (ρ) d(h) e ω(h)Y - 2α π 4 4 π e 2ω( 1 2 )Y ω ′′ ( 1 2 )Y 3 |ρρ ′ | dh (-) n E h Q (ρ ′ )E h Q (ρ) g 3P (h, 1 2 , 1 2 ) χ(h)-2χ( 1 2 ) 2 . (3)
In equation ( 3), the symbolic notation dh ≡ ∞ n=-∞ dν corresponds to the integration over the quantum numbers associated to the continuous unitary irreducible representations of SL(2, C), namely

h = iν + 1 -n 2 ; h = iν + 1 + n 2 ; n ∈ N , ν ∈ R . (4)
The SL(2, C) Eigenvalue of the BFKL kernel is

ω(h) = αN c π χ(h) ≡ αN c π 2Ψ(1) -Ψ (h) -Ψ 1 -h , (5)
where Ψ ≡ (log Γ) ′ , and E h Q (ρ) are the SL(2, C) Eigenfunctions [23]. g 3P (h, h a , h b ) is the known QCD triple Pomeron coupling function [24,25], whose definition is given in Appendix 1, together with E h Q (ρ) and the other ingredients (such as the functions a(h) , d(h)) of the SL(2, C)-invariant formalism.

The symmetry of the tree level amplitude remains valid at one-loop level, leading to a conformal invariant saturation scheme.

ii) The evaluation of formula (3) when the condition L ≡ log ρ ρ ′ ≫ 1 is fulfilled can be done in the forward direction (Q = 0) in terms of the leading Eigenfunction and Eigenvalue of the BFKL kernel, namely with n = 0. By contrast, for similar dipole sizes (L = O(1)) infinitely many Eigenfunctions and Eigenvalues contribute to the loop expression, leading to a quite different structure.

The QCD critical saturation line is derived in the L ≫ 1 regime and is given by:

Ỹ χ(h * ) -χ(h c ) χ( 1 2 ) -2L (h c -h * ) = Ỹc + 5 2 log Ỹ -log L , (6)
where Ỹ ≡ ω(1/2)Y and Ỹc ∼ O log 1/α 2 is a large constant (formula (16) in the text). The transition to saturation occurs when the first member of ( 6) exceeds the second one. Note that using the variable Ỹ means that the rapidity (i.e. energy) dependence of the amplitude is quantified by reference to the BFKL intercept 1 + ω(1/2). Hence the overall Ỹ , L dependence is determined by "universal" numbers, independent of the coupling constant.

In formula ( 6), there appears two critical values of the effective anomalous dimensions h c for the tree level and h * for the one-loop contribution. They are both defined as solution of implicit equations

χ ′ (h c ) χ( 1 2 ) = 2L Ỹ ; χ(h * ) = 2χ(1/2) . ( 7
)
h c is a moving saddle point in the tree-level integrand of (3) while h * ∼ .183 is the fixed location of a double pole singularity in the one-loop integrand of (3). The plan of the paper is the following. The next section II is devoted to the derivation of formula (3) and Section III to that of formula (6). A disussion of the resulting saturation line and an outlook are proposed in section IV. The SL(2, C)-invariant formalism which we use is presented in Appendix 1 and some technical parts of the derivation of the one-loop amplitude are present in Appendix 2 and 3.

II. THE ONE-LOOP DIPOLE AMPLITUDE

At one-loop level, the amplitude f

(1)
Q has a simple and physically appealing formulation [4] in the transverse coordinate space, see Fig. 1. We write

f (1) Q (ρ, ρ ′ |Y ) ≡ 1 2π d 2 r e iQ T (1) (ρ 0 ρ 1 ; ρ ′ 0 ρ ′ 1 |Y ) , (8)
with impact parameter r. The dipole-dipole amplitude in coordinate space reads

T (1) (ρ 0 ρ 1 ; ρ ′ 0 ρ ′ 1 |Y = y +y ′ ) ≡ - y 0 dy y ′ 0 dy ′ d 2 ρ a0 d 2 ρ a1 d 2 ρ b0 d 2 ρ b1 |ρ a ρ b | 2 d 2 ρ a ′ 0 d 2 ρ a ′ 1 d 2 ρ b ′ 0 d 2 ρ b ′ 1 |ρ a ′ ρ b ′ | 2 × n 2 (ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 |y -y, y) n 2 ρ ′ 0 ρ ′ 1 ; ρ ′ a0 ρ ′ a1 , ρ ′ b0 ρ ′ b1 |y ′ -y ′ , y ′ T (ρ a0 ρ a1 , ρ a ′ 0 ρ a ′ 1 ) T (ρ b0 ρ b1 , ρ b ′ 0 ρ b ′ 1 ) (9)
where ρ 0 , ρ 1 are the transverse coordinates of one of the initially colliding dipoles (resp. ρ ′ 0 , ρ ′ 1 for the second one), while ρ a0 ρ a1 and ρ b0 ρ b1 , are the two interacting dipoles emerging from the dipole ρ 0 ρ 1 after evolution in rapidity (resp. ρ {i} → ρ {i ′ } , for the second one).

The overall minus sign comes from the well-known AGK rules [26] relating the amplitudes to the discontinuity here calculated in the c.o.m. frame of the dipole-dipole reaction. The tree level interaction amplitudes T are given in Appendix 1. In formula (9), n 2 (ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 |y -y, y) stands for the probability of producing two dipoles after a mixed rapidity evolution [4,22], namely with a rapidity yy with one-Pomeron type of evolution and a rapidity y with two-Pomeron type of evolution, see Fig. 2. Further on, one has to integrate over y, y ′ . The solution for n 2 is given in Appendix 2. The integration over intermediate variables yields a drastic simplification due to the appearance of quite a few δ-functions. This tedious but straightforward derivation is given in detail in Appendix 3.

Starting with the general formula (A.25) obtained in Appendix 3, we first evaluate the high energy behaviour of the integral over h a,b the dominant contribution of which, as well-known from the ordinary BFKL analysis, is obtained for n a,b = 0. The integrals are dominated by the usual BFKL saddle-points at h a,b = 1 2 ; ω( 1 2 ) = 4 log 2. Taking into account in the saddle-point derivation the logarithmic corrections due to the zeroes in the prefactors at h a,b = 1 2 , one gets:

f (1) Q (ρ, ρ ′ |Y ) ∼ - 2 π 8 α 2π 2 4 π e 2ω( 1 2 )Y ω ′′ ( 1 2 )Y 3 |ρρ ′ | dh (-) n E h Q (ρ ′ )E h Q (ρ) g 3P (h, 1 2 , 1 2 ) χ(h)-2χ( 1 2 ) 2 . (10)
Formula ( 10) is then reported in the full amplitude (3).

A few comments on our result (10) are in order: i) Conformal invariance is explicitly realized, due to the decomposition on the combination of

SL(2, C) Eigenvectors E h Q (ρ ′ )E h Q (ρ)
. This property being shared by the tree-level amplitude, see (A.1), is thus valid for the whole amplitude (3) including one-loop level. Hence, saturation which is expected from the compensation between tree level and one-loop due to the different energy dependence is genuinely a conformal invariant phenomenon. We conjecture that this will be true to all orders, since only tensorial SL(2, C) properties play a rôle in the symmetry properties of the amplitude. Thus the saturation regime based on dipole loop contribution, if mathematically convergent, is expected to respect the conformal symmetry.

ii) The key difference between the tree-level and one-loop amplitude properties lies in the difference in the analytic singularities of the integrands. While the tree level term in ( 3) is dominated by a moving saddle-point at high energy, the one loop amplitude (10) has double poles at χ(h) = 2χ(1/2) = 8 log 2 which depend only on the anomalous dimensions h and thus are independent of the energy. This very peculiar property means that the one-loop contribution may depend (and thus can give information) on BFKL properties which are not reached in the conventional analysis. In particular, all higher conformal spins may contribute, contrary to the tree level contribution. Also, contributions for values of h not limited to the interval [0 + , 1 2 ] can give rise to poles. This may be interpreted as unlimited contributions of higher twist terms of the BFKL solution.

iii) Apart the factor α 4 , which is the natural strength scale of the one-loop amplitude, there is a dynamical factor connected to the square of the function g 3P (h, 1 2 , 1 2 ). This function is the QCD analogue of the triple Pomeron coupling. However, it is not the value g 3P ( 1 2 , 1 2 , 1 2 ) which is here relevant, as was the case in the QCD dipole analysis of the triple pomeron coupling in [25], since h = 1 2 is not solution of the double pole equation. One could also note that the square of g 3P appears, contrary to saturation equations on an extended target, where the dependence is linear [11].

III. SATURATION CRITICAL LINE

As discussed in section I, saturation is expected to take place when the one-loop amplitude, with coupling of order α 4 , becomes comparable in strength with the tree level amplitude of order α 2 . Formula (3) is completely explicit and thus can serve for analyzing saturation depending on transverse momentum or, equivalently using the Fourier transforms (A.4), on impact parameter. For sake of simplicity we will concentrate on the forward contribution at Q = 0, that is (through unitarity) for the total cross-section integrated over impact parameter. In this limit, the SL(2, C) Eigenvectors simplify [23] to give

E h Q (ρ ′ ) E h Q (ρ) → ρ ρ ′ h-1 2 ρ ρ ′ h-1 2 . ( 11
)
Consequently, we may infer that the key "order parameter" for the discussion of saturation will be the scale invariant ratio of incident dipole sizes ρ/ρ ′ , in conformity with the conformal invariance of the amplitude. This will in turn be determined by the ratio of the characteristic scales of the incoming systems which are encoded in their corresponding QCD dipole wave functions, see (2).

The general solution of (3) is easily obtained by a summation over the double pole contributions verifying χ(h i ) = 2χ( 12 ), whose location are caracterized by an infinite number of values h i ) = (n i , γ i ). One finds a formal double series

f (1) Q=0 (ρ, ρ ′ |Y ) ∼ - 2 π 6 α π 2 4 e 2ω( 1 2 )Y ω ′′ ( 1 2 )Y 3 × ni,γi (-1) n g 3P (h, 1 2 , 1 2 ) χ ′ (h i ) 2 ρ ρ ′ hi ρ ρ ′ hi L + ∂ log g 3P (h, 1 2 , 1 2 ) ∂h h=hi , (12)
where, for sake of simplicity, we used the notation L ≡ log |ρ/ρ ′ | .

For incident dipoles of comparable size, L ∼ O(1), we may note that the one loop result then depends on an infinite sum over (n i , γ i ) which is due to the analytic properties of the function χ(h). It even raises the question of the convergence of the double series [START_REF] Jalilian-Marian | [END_REF]. On a physical ground, since the anomalous dimensions h i have no obvious limitation, this situation can be interpreted as due to a dependence of the one loop amplitude on all higher twist contributions, including those with non zero conformal spins n i . This is a quite strange example where higher twist calculations are required to all orders in a purely perturbative framework, in striking analogy with actual non perturbative situations.

Leaving this regime for further work, we will here focus on the case where only the first pole with h * = (γ * i , n * i = 0) contributes, i.e. where the equivalent of a leading twist contribution is sufficient. This approximation is realized in a region where L ≫ 1 (or L ≪ 1 by symmetry).

The dominant double pole in [START_REF] Jalilian-Marian | [END_REF] such that χ(h * ) = 2χ( 1 2 ) = 8 log 2 is h * ≡ (n * = 0, γ * ∼ .183). Consequently, factorizing the tree level coupling, the resulting one loop amplitude can be written

f (1) Q=0 (L, Ỹ ) ∼ α 2 4 |ρρ ′ | exp 2 Ỹ + 2(h * - 1 2 )L -Ỹ1 -3 log Ỹ + log L , (13)
where we define Ỹ ≡ ω( 12 )Y, and

Ỹ1 = log π 12 16α 2 χ ′ (h * ) g 3P (h * , 1 2 , 1 2 ) 2 χ( 1 2 ) χ ′′ ( 1 2 ) 3 , (14)
which is a priori large, being in the perturbative domain α ≪ 1 and constant up to 1/L correction terms. We define the saturation critical line as the line in the ( Ỹ , L) where the tree-level contribution equals the one loop one. It is a simple extension to the conformal invariant configuration of the usual one [1]. Beyond this line, saturation is expected to take place.

We now make use of the well-known saddle-point approximation of the tree level amplitude

f (0) Q=0 (L, Ỹ ) ∼ α 2 4 |ρρ ′ | exp χ(h c ) χ(1/2) Ỹ + 2(h c -1/2)L -log 1 h 2 c (1-h c ) 2 2πχ( 1 2 ) χ ′′ ( 1 2 ) -1/2 log Ỹ , (15)
where the tree level saddle point h c is given by the well known implicit equation given in (7). Inserting the results ( 13) and ( 15) in (3) with a redefinition of the constant

Ỹc = Ỹ1 + log 1 h 2 c (1-h c ) 2 2πχ( 1 2 ) χ ′′ ( 1 2 ) , (16)
we get the equation ( 6) announced in section I for the saturation critical line.

IV. DISCUSSION, OUTLOOK

Let us describe general features of the critical line, when α is very small but otherwise not specified. Following expression (6), it is straightforward to realize that there is an absolute bound when L, Y ≫ 1 given by Ỹ /2L > χ( 12 )/χ ′ (h * ) ∼ .09. Indeed below this bound, the compensation can no more take place between tree level and one loop contributions. It is interesting to note that the bound implies that a renormalization group evolution, with L moving to large values while keeping Y fixed, cannnot lead from a weak density to a saturation region, as expected from the general theoretical picture of the QCD dynamics of partons.

One can delimit three typical regimes in the saturation transition, depending upon the behaviour of the tree level amplitude:

I 0 ≪ L 0 -> L 1 ≪ Ỹ : in this region, saturation is obtained in the region Ỹ > Ỹ0 + 2L 1 2 -h * ∼ Ỹ0 + .63L
, where Ỹ0 is a large effective rapidity scale incorporating the logarithmic prefactors.

II L 0 ≪ L 1 -> L 2 < Ỹ :
It is well-known that the BFKL behaviour discussed in I acquires an extra "diffusion term" (of the gluon transverse momenta along the BFKL ladder contributions) in the intermediate region. Within the saddle-point approximation it leads to the following saturation curve:

Ỹ > Ỹ0 + 2L 1 2 -h * - χ 1 2 χ ′′ 1 2 2L 2 Ỹ , (17)
where the last term is due to diffusion. Quite interestingly, saturation is favoured by diffusion. Indeed, formula (17) approximately interpolates between the behaviour I and II by a smooth decrease of the limiting slope of Ỹ versus L. III L 2 ∼ Ỹ < L : near the bound 2L/ Ỹ < χ( 12 )/χ ′ (h * ) ∼ 10. In this region, one can use the relations

h c ∼ χ( 1 2 ) 2L Ỹ ; χ(h c ) ∼ 1 h c .
Using this and again introducing the effective large rapidity scale Ỹ0 one obtains, within a good approximation at large ( Ỹ , L), the constraint

Ỹ >    8L χ 1 2 + Ỹ0 2    2 . ( 18
)
It is easy to check that ( 18) satisfies the bound Ỹ /2L > χ 1 2 /χ ′ (h * ).

We have not yet discussed the rôle of the logarithmic prefactors. Using our explicit expressions, it will be easy to incorporate them in a more complete study which we leave for the future. They may play a phenomenological rôle. Indeed, it is important to note that the large factor 5 2 log Ỹ of ( 6) is due to the double zeroes appearing in the loop couplings to the intermediate dipoles, see (A.25). This has the important effect of a strong energy dependent damping on the effective triple Pomeron couplings, which could otherwise be large [25]. Another possible important rôle would be, by mere extrapolation to higher dipole loops, to establish a useful hierarchy between these different orders. Indeed the coupling strength expected at order n, namely ∼ (α 2 exp ω( 12 )Y ) n cannot provide a hierarchical structure in the saturation regime. We are thus conjecturing a series of critical saturation lines corresponding to the onset of compensation between a growing number of higher loop order contributions. The study of this regime, necessary to understand the full saturation behaviour in the conformal invariant set-up will be pursued in the near future.

In fact, our work suggests a strategy to attack the full problem of saturation from dipole loops. It seems possible to analyze the n-loop dipole diagrams in the same way as we have shown here for the 2-loop case. It is clear that many diagrams can contribute, depending on the multiple rapidities where new dipoles can branch. However, we may conjecture that the integration over these intermediate rapidity intervals would dominantly lead to the QCD multi-Pomeron exchange and the corresponding 1 → n dipole vertices which were analyzed in Ref. [28]. Then, it will be possible to explore the conformal invariant saturation regime it self and not only the transition to it.

Another point merits some care, when the onium typical sizes are of the same order. A quite interesting regime seems to set in when the sum [START_REF] Jalilian-Marian | [END_REF] over all the double poles in the one-loop integrals becomes relevant. While being still at small coupling constant, this regime shares with the non-perturbative one the features of a summation over higher twist contributions. It would be nice to solve this case in detail.

Finally, the step between our calculations and actual experimental possibilities, which is not done in the present work, could be done either through factorization properties or by the consideration of "hard" subprocesses with very high energies like Mueller-Navelet jets at LHC or γ * -γ * collisions at a future linear accelerator. Relevant studies are certainly deserved in the future.

APPENDIX 1 SL(2, C)-invariant formalism
Using the SL(2, C)-invariant formalism, the solution of the BFKL equation obtained at leading log level of perturbative QCD [20] reads

f (0) Q (ρ, ρ ′ |Y ) = α 2 4 |ρρ ′ | dh E h Q (ρ ′ )E h Q (ρ) d(h) e ω(h)Y , (A.1)
where the SU (3) couplingα is fixed, small but unspecified, in the leading log approximation we consider. In equation (A.1), the symbolic notation dh ≡ ∞ n=-∞ dν corresponds to the integration over the quantum numbers associated to the continuous unitary irreducible representations of SL(2, C), namely

h = iν + 1 -n 2 ; h = iν + 1 + n 2 ; n ∈ N , ν ∈ R . (A.
2)

E h Q (ρ)
and ω(h) are, respectively, the SL(2, C) Eigenfunctions and Eigenvalues of the BFKL kernel [23]. The Eigenvalues read

ω(h) = αN c π χ(h) ≡ αN c π 2Ψ(1) -Ψ (h) -Ψ 1 -h , (A.3)
where Ψ ≡ (log Γ) ′ . The SL(2, C) Eigenvectors are defined1 by

E h Q (ρ) = 2π 2 |ρ| b(h) d 2 r e iQb E h r- ρ 2 , r+ ρ 2 , (A.4)
with

E h r - ρ 2 , r + ρ 2 ≡ |ρ| b(h) 8π 4 d 2 r e -iQr E h Q (ρ) = (-) h-h ρ r 2 -ρ 2 4 h ρ r 2 -ρ 2 4 h , (A.5)
where h = 1h, r is the 2-d impact-parameter, and introducing useful notations

d(h) = 1 ν 2 + (n-1) 2 4 ν 2 + (n+1) 2 4 ; b(h) = π 3 4 h+ h-1 1 2 -h Γ(-iν + 1+|n| 2 ) Γ(iν + 1+|n| 2 ) Γ(iν + |n| 2 ) Γ(-iν + |n| 2 ) ; a(h) = |b(h)| 2 2π 2 = π 4 /2 ν 2 + n 2 4 . (A.6)
We will also make use of the formalism in the full coordinate space, the 4-diple amplitude T (0) being defined at tree level of QCD dipole contributions by

T (0) (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 |Y ) ≡ 1 2π d 2 Q e -irQ f (0) Q (ρ, ρ ′ |Y ) , (A.7)
where the impact parameter is r = 1 2 (ρ ′ 0 +ρ ′ 1 -ρ 0 -ρ 1 ). Inserting (A.1) into (A.7), one finds

T (0) (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 |Y ) = α 2 4 dh G h (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 ) ν 2 + n 2 4 d(h) e ω(h)Y , (A.8)
where , using the SL(2, C) Eigenvectors in coordinate space (A.5), the Green function G h in coordinate space [21,23] can be written as

G h (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 ) ≡ d 2 ρ α E h (ρ 0 ′ α , ρ 1 ′ α ) E h (ρ 0α , ρ 1α ) = |b(h)| 2 π 4 |ρρ ′ | d 2 Qe -ibQ E h Q (ρ ′ )E h Q (ρ) . (A.9)
We will also define for convenience the "instantaneous" (i.e. Y=0) amplitude

T (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 ) ≡ T (0) (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 |Y = 0) . (A.10)
Due to completeness and frame independence of the formalism, many formulae can be simplified using these "instantaneous" amplitudes between dipoles.

The single multiplicity distribution of dipoles n 1 after an evolution characterized by the rapidity Y is simply related [21] to the previous expressions (e.g. (A.9)), namely

n 1 (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 |Y ) = 1 π 2 |ρ ′ | 2 dh G h (ρ 0 , ρ 1 ; ρ ′ 0 , ρ ′ 1 ) ν 2 + n 2 /4 e ω(h)Y . (A.11)
Let us finally note a useful completeness formula [21] valid at tree level:

T (0) (ρ 0 ρ 1 ; ρ ′ 0 ρ ′ 1 |Y = y +y a +y ′ ) ≡ 1 2 2 (2π) 4 d 2 ρ a0 d 2 ρ a1 |ρ a | 2 d 2 ρ a ′ 0 d 2 ρ a ′ 1 |ρ a ′ | 2 × n 1 (ρ 0 ρ 1 ; ρ a0 ρ a1 |y) n 1 ρ ′ 0 ρ ′ 1 ; ρ ′ a0 ρ ′ a1 |y ′ T (0) (ρ a0 ρ a1 , ρ a ′ 0 ρ a ′ 1 |y a ) , (A.12)
where ρ 0 , ρ 1 are the transverse coordinates of one initial dipole (resp. ρ ′ 0 ρ ′ 1 for the second one), ρ a0 , ρ a1 are those of the interacting dipole emerging from the dipole ρ 0 , ρ 1 after evolution in rapidity y (resp. ρ a ′ 0 , ρ a ′ 1 and y ′ for the second one), which interaction is described by the amplitude T (ρ a0 , ρ a1 ; ρ a ′ 0 , ρ a ′ 1 |y a ) as in (A.8). Equation (A.12) has the physical interpretation that, at tree dipole level, the multiplication of dipoles due to rapidity evolution does not lead to saturation, but to the same tree level amplitude in a kind of bootstrap property.

APPENDIX 2

The two-dipole density distribution n2

The two-dipole density distribution n 2 results from the solution of a specific evolution equation which consists of an extension of the mixed evolution of the one formulated in ref. [4,31,24], namely:

n 2 ( ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 | y -y, y) = = αN c 2π 2 dhdh a dh b |ρ a ρ b | 2 dω 1 e ω1(y-y) ω 1 -ω (h) dω e ωy ω (h a ) + ω (h b ) -ω n h,ha,h b 2 (ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 ) , (A.13)
where, by convention, dω i → dω i /2π and

n h,ha,h b 2 (ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 ) = 1 a(h)a(h a )a(h b ) d 2 ρ α d 2 ρ β d 2 ρ γ R h,ha,h b α,β,γ × × E ha (ρ a0α , ρ a1α ) E h b (ρ b0β , ρ b1β ) E h (ρ 0γ , ρ 1γ ) , (A.14) with R h,ha,h b α,β,γ ≡ d 2 r 0 d 2 r 1 d 2 r 2 |r 01 r 02 r 12 | 2 E h (r 0γ , r 1γ )E ha (r 0α , r 2α )E h b (r 1β , r 2β ) (A.15)
where ρ = ρ 0 -ρ 1 , ρ a = ρ a0 -ρ a1 , ρ b = ρ b0 -ρ b1 .

Conformal invariance implies the well-known tensorial reduction [29]

R h,ha,h b α,β,γ ≡ [ρ αβ ] h-ha-h b [ρ βγ ] ha-h b -h [ρ γα ] h b -ha-h ρ αβ h-ha-hb ρ βγ ha-hb -h ρ γα hb -ha- h g 3P (h, h a , h b ) , (A.16)
where g 3P (h, h a , h b ) happens to be [25] the QCD triple Pomeron coupling as obtained in the dipole formulation, namely:

g 3P (h, h a , h b ) = d 2 r 0 d 2 r 1 d 2 r 2 |r 01 r 02 r 12 | 2 [r 01 ] h r 02 r 0 r 2 ha r 12 (1 -r 1 ) (1 -r 2 ) h b [r 01 ] h r 02 r 0 r 2 ha r 12 (1 -r 1 ) (1 -r 2 ) hb . (A.17)
Considering the Fourier transforms (A.4) for the SL(2, C) Eigenvectors E h q in the mixed representation, one writes

n h,ha,h b 2 (ρ 0 ρ 1 ; ρ a0 ρ a1 , ρ b0 ρ b1 ) = g 3P (h, h a , h b ) b(h)b(h a )b(h b ) 2 7 π 10 |ρρ a ρ b | d 2 q a d 2 q b d 2 Q δ (2) (Q+q a +q b ) × E h Q (ρ)E ha qa (ρ a )E h b q b (ρ b ) e -i(qara+q b r b +Qr) d 2 v d 2 w e -i((qa-q b ) v 2 +Q w 2 ) × [v] -1-h-ha-h b [(w-v)/2] -1+h-ha+h b [(w + v)/2] -1+h+ha-h b × {a.h.} , (A.18)
where ρ αβ = v, ρ αγ + ρ βγ = w, 2r a = ρ a0 + ρ a1 , 2r b = ρ b0 + ρ 1 and r is the overall impact parameter. The notation {a.h.} indicates the anti-holomorphic part of the bracketed term in the integrand for which the integration variables are complex conjugates and the exponents h i are replaced by hi . Equivalently, the distribution n 2 for the lower vertex is given by the same equation (A.18) by using prime indices.

APPENDIX 3

The general one-loop amplitude Implementing equations(A.8) and (A.18) in the master equation (3) and first integrating over impact parameters, one gets: (2π) 8 δ(q aq" a )δ(q bq" b )δ(q ′ aq" a )δ(q ′ bq" b ) .

Then, integrating over the intermediate dipole sizes, and using the orthogonality [23] relation

1 4π 2 d 2 ρ |ρ| 2 E h q (ρ)E h ′ q (ρ) = δ (2) (h, h ′ ) + δ (2) (h, 1 -h ′ ) e iφ(h) , (A.19)
where e iφ(h) ≡ E h q (ρ)/E 1-h q (ρ) is just a ρ, q-independent phase factor given in Ref. [23], one finds

(2π) 8 2 4 δ (2) (h a , h" a )δ (2) (h b , h" b )δ (2) (h ′ a , 1-h" a )δ (2) (h ′ b , 1-h" b ), where δ (2) (h, h ′ ) ≡ δ nn ′ δ(ν -ν ′ ).
The integration over q a -q b finally gives (2π)2 δ(v-v ′ ).

Plugging in formulae (8,9) the result obtained in formula (A.18) and integrating over δ-fuctions, one gets:

f (1) Q (ρ, ρ ′ |Y = y +y ′ ) = - α 4 π 2 8 αN c 2π 2 2 dh a dh b d(h a ) |b(h a )| 2 d(h b ) |b(h b )| 2 × dhdh ′ g 3P (h, h a , h b ) b(1-h) g * 3P (h ′ , h a , h b ) b(h ′ ) |ρρ ′ | E h Q (ρ) E h ′ Q (ρ ′ ) H Q (h, h ′ ) × y 0 dy Y -y ′ 0 dy ′ dωdω 1 dω ′ dω ′ 1 e ωy+ω1(y-y) (ω(h a )+ω(h b )-ω)(ω 1 -ω(h)) e ω ′ y ′ +ω ′ 1 (y ′ -y ′ ) (ω(h a )+ω(h b )-ω ′ )(ω ′ 1 -ω(h ′ )) , (A.20)
where, after the change of variables w

= v(1 -2t), w ′ = v(1 -2t ′ ), H Q (h, h ′ ) = d 2 vd 2 td 2 t ′ e iQv(t-t ′ ) v h-h ′ -1 t -1+h+h b -ha (1 -t) -1+h-h b +ha t ′ha-h b -h ′ (1 -t ′ ) -ha+h b -h ′ × a.h. . (A.21)
After integration over v one obtains [30] H

Q (h, h ′ ) = 2 Q h-h ′ 2 Q h-h′ e i π 2 (n-n ′ ) γ(h -h ′ ) d 2 t d 2 t ′ (t -t ′ ) h ′ -h × t -1+h+h b -ha (1 -t) -1+h-h b +ha t ′ha-h b -h ′ (1 -t ′ ) -ha+h b -h ′ × {a.h.} . (A.22)
where, by definition, γ(z) ≡ Γ(z)/Γ(1z) .

The remaining integral can be proven 2 to reduce to

H Q (h, h ′ ) = 2 Q h-h ′ 2 Q h-h′ π 2 (2h-1)(2 h-1) δ(h-h ′ )+(-) n+na+n b γ(h+h a -h b ) γ(1-h ′ -h a +h b ) δ(1-h-h ′ ) .
(A.23)

The integral over h ′ can now easily be performed. The remaining poles at h ′ = h and h ′ = 1h give twice the same contribution due to the over completeness relation of the E h Q generators [23]. One finally obtains: This one-loop amplitude preserves the global conformal invariance of the tree-level BFKL 4-gluon amplitude, since the only scale-dependence on Q is present in the conformal Eigenvectors |ρρ ′ | E h Q (ρ) E h Q (ρ ′ ). It was noted in Ref. [22] that the integration over rapidity variables yields two different contributions depending on the sign of the quantity ω(h a )+ω(h b )-ω(h). Indeed for ω(h a )+ω(h b ) < ω(h), the relevant poles are situated at ω = ω 1 = ω ′ = ω ′ 1 = ω(h), leading to a contribution associated with the single Pomeron dependence e ω(h) Y . In the opposite case, namely ω(h a )+ω(h b ) > ω(h), the relevant poles are situated at ω = ω 1 = ω ′ = ω ′ 1 = ω(h a )+ω(h b ). The resulting amplitude is the one we are interested in for the saturation problem since it possesses the QCD "double Pomeron" energy behaviour e (ω(ha)+ω(h b ))Y > e ω(h) Y . This is precisely the energy behaviour which will compensate for the higher order α 2 in the coupling constant. Notice that the expression depends only on the sum Y = y + y ′ , as it should do from longitudinal boost invariance. One-loop dipole amplitude in rapidity space The initial dipoles ρ 0 ρ 1 , ρ ′ 0 ρ ′ 1 evolve in the rapidity range yy ′ , y ′y ′ with one-Pomeron type of evolution and then with two-Pomeron type of evolution over the range y, y ′ . Note that y + y ′ ≡ Y, is the total rapidity range. After integration over y, y ′ , see Appendix 3, the resulting amplitude depends on the total range Y .

f (1) Q (ρ, ρ ′ |Y ) ∼ - 1 2π 8 α 2π 2 4 |ρρ ′ | dh (-1) n E h Q (ρ ′ )E h Q (ρ) × dh a dh b ( 1 2 -h a )( 1 2 -h b ) h a (1-h a )h b (1-h b )

  f

( 1 )α 4 π 4 8 αN c 2π 2 2 2 dQdy ′ dωdω 1 dω ′ dω ′ 1 e

 1221 Q (ρ, ρ ′ |Y ) =dhdh a dh b g 3P (h, h a , h b ) b(h a )b(h b )b(h) (h a )d(h b ) (-1) n+na+n b | 1 2 -h| 2 |ρρ ′ | E h ωy+ω1(y-y) (ω(h a )+ω(h b )-ω)(ω 1 -ω(h)) e ω ′ y ′ +ω ′ 1 (y ′ -y ′ ) (ω(h a )+ω(h b )-ω ′ )(ω ′ 1 -ω(h)). (A.24)

2 e 2 ,

 22 (ω(ha)+ω(h b ))Y g 3P (h, h a , h b ) χ(h)-χ(h a )-χ(h b )

Figure 1

 1 FIGURES

Note that an analytic expression of the Eigenvectors E h Q (ρ) in the mixed representation has been provided[21] in terms of a combination of products of two Bessel functions. For simplicity, we did not include the impact factors[23,27]. Phenomenologically, the leading contribution to the amplitude (A.1) comes from the n = 0 component which corresponds to the BFKL Pomeron.

One shows using general methods[30] that HQ(h, h ′ ) is a combination of δ functions whose coefficients can then be computed directly on formula (A.22).

Acknowledgements

We acknowledge fruitful discussions with Yuri Kovchegov, Heribert Weigert and Edmond Iancu.


R Peschanski

Unified approach to hard diffraction

Using a combination of S-Matrix and perturbative QCD properties in the small x Bj regime, we propose a formulation of hard diffraction unifying the partonic (Ingelman-Schlein) Pomeron, Soft Colour Interaction and QCD dipole descriptions.

Introduction

In the present paper, we focus on three existing different theoretical approaches of hard diffraction, for which we propose a new, unifying, formulation. The first one we will refer to is the "partonic Pomeron" approach 1 . The hard photon is here supposed to probe the parton distributions of the Pomeron Regge pole considered as a hadronic particle. A second approach 2 is the Soft Colour Interaction one , where hard diffraction is described by a two-step process: during a relatively short "interaction time", the probe initiates a typical hard deepinelastic interaction. Then, at large times/distances, a soft colour interaction is assumed which will decide of the separation between diffractive and nondiffractive events according to a simple probabilistic rule. A third approach is based on the small x Bj regime of perturbative QCD, where the resummation of leading log 1/x Bj leads to the "QCD dipole" approach for hard diffraction 3 .

We show that the three aproaches may find a common formulation. Our main results are the following:

1) The "effective" parameters of the partonic Pomeron are determined from leading log perturbative QCD resummation. They are found to depend not only on Q 2 but also on the ratio η = (Y -y) /y, where Y (resp. y) are the total (resp. gap) rapidity interval.

2) The "effective" parameters of the hard process before modification by the Soft Colour Interaction are determined. Differences with the original model avoid the "Low's theorem" paradox [START_REF] We | Dokshitzer for pointing out to me the problem that a truely soft color interaction should not modify the physical observables[END_REF] .

3) Using S-Matrix properties of triple-Regge contributions, a relation is found between discontinuities of a 3 → 3 amplitude and the three approaches to hard diffraction. In particular, a theoretically founded formulation of a (modified) Soft Colour Interaction approach is proposed as a specific double discontinuity the 3 → 3 forward amplitude.

2 From QCD dipoles to partonic Pomeron Our starting point 5 is the formula for the structure function (inelastic component) for longitudinal and transverse photon:

F Dif f T,L (Q 2 , Y, y) ∼ c+i∞ c-i∞ dγ 1 2iπ dγ 2 2iπ dγ 2iπ δ(1-γ 1 -γ 2 -γ) × Q Q 0 2γ exp {y[∆(γ 1 ) + ∆(γ 2 )] + (Y -y) [∆(γ)]} , (1)
where

∆(γ) ≡ α s N c π {2ψ(1) -2ψ(γ) -2ψ(1 -γ)} ∼ ∆ + ∆ ′′ 2 1 2 -γ 2 (2)
is the BFKL evolution kernel (used in a gaussian approximation), Q 0 a nonperturbative scale associated with the proton. After a triple saddle-point approximation (using the gaussian approximation) justified by the large rapidity gap characteristic of diffraction, one may write

F Dif f T,L ∼ e 2y∆ σ tot γ * -P , with σ tot γ * -P ∼ 2 1+2η exp {(Y -y)ǫ s } Q Q 0 2γs exp   - 2η 1+2η 2 log 2 Q Q0 ∆ ′′ (Y -y)   . (3)
Here, the gaussian saddle-points determine both the new "effective dimension" and "effective intercept":

γ s = η 1 + 2η ; ǫ s = ∆ + ∆ ′′ 8(1 + 2η) (4)
of a "BFKL-like" expression, depending on the variable η = Y-y y . σ tot γ * -P thus defines an effective "partonic Pomeron" derived from the QCD dipole formalism.

From dipoles to Soft Colour Interaction

In Soft Color Interactions models, at least in their formally simpler formulations, one expects the following relation between the total structure function and the overall contribution of hard diffraction at fixed value of x Bj :

F Dif f /tot T,L ≡ xgap xBj dx P F Dif f T,L = 1 N 2 c F hard/tot T,L (5)
where log 1 xgap is the minimal value retained for the rapidity gap. A priori, F hard/tot T,L is supposed not to be modified by the soft color interaction. However a contradiction with "Low's theorem" has been quoted 4 . Indeed, a fully soft color interaction can be proven to give no effect on physical obervables.

In fact, when inserting the QCD dipole expression (1) in relation ( 5), one finds 5

F hard/tot L,T ∼ Q Q 0 2γ hard exp (Y ∆(γ hard )) √ 2π∆ ′′ Y , (6)
which is very similar with the BFKL expression, except that

γ BF KL ∼ 1 2   1 -4 log Q Q0 ∆ ′′ Y   = γ hard = cst ∼ 0.175. (7)
Thus, in some sense, the Soft Color relations are compatible with QCD dipole calculations, but the underlying hard interaction is somewhat different than the assumed mechanism in SCI models. Hence the soft color rearrangement is not simply a "neutralization" of color at long distances which would be forbidden by virtue of "Low's theorem". Apart this modification, formula (5) fixes the relative normalization between diffractive and total structure functions. .

S-Matrix interpretation

Following old results of S-Matrix theory in the Regge domain [START_REF] Mueller | [END_REF] , and as sketched in the figure , one may consider three types of discontinuities of a 3 → 3 amplitude representing hard diffraction. A single discontinuity over the diffractive mass 2 variable, a double discontinuity taking into account also the analytic discontinuity in the subenergy of one of the incident Pomeron exchanges and a triple discontinuity with the discontinuity including the two Pomeron exchanges.

As qualitatively shown in the figure, one can associate the three discontinuity formulae with, respectively, the partonic Pomeron, Soft Color Interaction and QCD dipole approaches. This is schematically depicted in the "timedependent" representation of the figure, where the soft incident Pomerons are represented as living during a "long" time, while the partonic (or QCD dipole) process are initiated by hard interactions. In fact, one can prove 6 the identity of these discontinuities, and thus legitimate the connection realized in the previously mentionned calculations.

Figure 1 :

 1 Figure 1: "Time-dependent" picture of the A(3 → 3) discontinuities. Upper graph: Description of Disc 1 A(3 → 3) (partonic Pomeron approach); Middle graph: Description of Disc 2 A(3 → 3) (candidate for the Soft Colour Interaction approach); Lower graph: Description of Disc 3 A(3 → 3) (QCD dipole approach).

Acknowledgments

This work was done in collaboration with H. Navelet.


Matthieu Viteau

Amodsen Chotia

Daniel Comparat

Duncan A Tate

Thomas F Gallagher

Pierre Pillet

Ionization in a frozen Rydberg gas with attractive or repulsive potentials

Keywords: numbers: 32.80.Ee, 34.20.Cf, 37.10.De

published or not. The documents may come

A cold Rydberg gas is a fascinating system at the boundary of atomic, solid state, and plasma physics. In a MOT, at 100 µK temperature, the atoms move less than 3% of their typical separation, 10 -3 cm, on the 1 µs time scale of the experiments, and such a sample resembles an amorphous solid. Since Rydberg atoms have large dipole moments, scaling as the square of the principal quantum number n, dipole-dipole interactions in the frozen Rydberg gas have a significant effect, even though the atoms are far apart [1,2,3]. For this reason, binary dipoledipole interactions have been proposed as the basis for quantum gates [4,5]. Specifically, the dipole-dipole interaction between a pair of Rydberg atoms can preclude the excitation of the second atom in a sample once the first is excited, a phenomenon termed the dipole blockade. Local, or partial, blockades have been observed in many experiments, such as [6] in Van der Waals case or [7] in dipole one. In addition to binary interactions, there is evidence for many body interactions roughly analogous to the diffusion of spins in a glass [8,9]. This phenomenon is particularly apparent in the dipole-dipole energy transfer tuned into resonance with an electric field, a process often termed the Förster Resonance Energy Transfer (FRET) reaction [10].

In addition a cold Rydberg gas can spontaneously evolve into a plasma. If there is even a very slow ionization process, cold ions are produced, and at some point their macroscopic space charge traps all subsequent electrons produced [11,12,13]. The trapped electrons lead to a collisional avalanche which rapidly redistributes the population initially put into a single Rydberg state. Typically two thirds of the atoms are ionized and one third are driven to lower states to provide the requisite energy [11,14]. The origin of the initial ions is not, however, completely understood [15,16,17]. With a pulsed laser it is possible to excite atoms close enough to each other that they interact strongly, resulting in ionization on a 100 ns time scale, too fast to be a result of motion of the atomic nuclei [18].

With narrow bandwidth, quasi continuous wave (cw) excitation, it is not possible, due to the energy shift pro-duced by the dipole-dipole interaction, to excite atoms which are close together, yet ionization still occurs, although on a time scale of microseconds [19]. One mechanism for this ionization is that pairs of atoms excited to attractive diatomic potential curves collide, resulting in the ionization of one of the atoms. The much more rapid conversion of cold Rb nd atoms to a plasma than ns atoms was attributed by Li et al. to ionizing collisions of pairs of atoms excited to attractive potentials in the nd case but not in the ns case [20]. Using a narrow bandwidth laser Vogt et al. showed that ions are present on attractive Cs npnp potentials [10]. A similar behaviour was noticed by Amthor et al. where they demonstrated that pairs of atoms excited closer together on the Rb 60d60d attractive potential ionized more rapidly than those farther apart. A surprising aspect of their observations is the high ionization rate for the 62s state in spite of the repulsive 62s62s potential [15].

In an effort to isolate the effect of attractive/repulsive potentials from other effects, we have examined the ionization of cold Cs np atoms excited with narrow bandwidth excitation. The interest of the Cs np states is that for n > 42 a pair of Cs np atoms is on an attractive potential, while for n < 42 the potential is repulsive. Thus by varying n from 40 to 45 we switch, at n = 42, from excitation to a repulsive potential to an attractive one, with all other parameters of the system changing only slowly. The results show unambiguously the difference between attractive and repulsive potentials. Perhaps as interesting, in the repulsive case we observe ionization which is nearly linear in the number of excited atoms, but at a rate that is twice that due to blackbody photoionization. Our observation of such a high ionization rate is not unique; others have made similar observations [15]. We suggest here that possible sources of high ionization rates are ionizing collisions, due to motion on attractive iondipole potentials and dipole-dipole potentials activated by state changes stimulated by the blackbody radiation and superradiant decay. In the sections which follow we describe the Cs system we have studied, present our experimental results, and compare them to expectations based on 300K black body population transfer. For convenience we consider the molecular state composed of a pair of Cs Rydberg atoms separated by distance R. A pair of np atoms we term the pp state, and the nearly degenerate pair of ns and (n + 1)s atoms we term the ss ′ state. The energy difference between the molecular pp and ss ′ states at R = ∞ is given by

W pp-ss ′ = 2W np -W ns -W (n+1)s (1)
where W nl is the energy of the Cs nl state. As shown by Fig. 1a, W pp,ss ′ crosses zero at n ∼ 42. For n > 42 the pp state lies below the ss ′ state, while for n < 42 the reverse is true. If we ignore the fine-structure and the angular distribution, the pp and ss ′ states are coupled by the dipole-dipole interaction µµ ′ R 3 , where µ and µ ′ are the dipole matrix elements connecting the np state to the ns and (n + 1)s states, and at finite R the eigenstates are attractive and repulsive linear superpositions of pp and ss ′ . The resulting potential curves are schematically shown in Fig. 1b and Fig. 1c. As R → ∞ the potentials are 1/R 6 van der Waals potentials, but at small R they are 1/R 3 dipole-dipole potentials [START_REF]The ionization time is simply calculated by integrating the dipole-dipole forces between two Rydberg atoms[END_REF]. We excite Cs atoms to the atomic np 3/2 state, or pairs of atoms to the molecular pp state. The repulsive potential is excited for n < 42, the attractive potential for n > 42, while for n = 42 at high atomic density, both potentials are excited.

In the experiment Cs atoms are held in a magneto optical trap (MOT) at a temperature of 100 µK and a number density of up to 5 × 10 10 cm -3 . The atoms are excited to the np 3/2 states via the route

6s 1/2 → 6p 3/2 → 7s 1/2 → np 3/2 (2)
using three cw single frequency lasers. The first laser is the 852nm trap laser, a diode laser which is typically left on continuously. Using acousto optic modulators we form the outputs of the second and third lasers into temporally overlapping pulses at an 80Hz repetition rate. The second laser is a 1470nm diode laser with a typical power of 20mW. It is focused to a beam waist of 100µm. The third laser is a Titanium:Sapphire laser operating near 830nm. It has a beam waist of 70µm, and it crosses the second laser at 67.5 The data are taken by scanning the frequency of the third laser while recording the ion and atom signals, with all other parameters held fixed. In Fig. 2a and Fig. 2b we show typical recordings of the ion and atom signals for the 40p 3/2 and 43p 3/2 states respectively, for two different laser intensities. We note the asymmetrical shoulder on the red side of the line, due to the three step excitation, where direct two and three photon excitations of process (2) can occur. The time delay between the end of the laser pulse and the field ionization pulse is 10 µs. At low power, i.e. at low Rydberg density, we observe in both cases a small number of ions, (7% of Rydberg atoms excited for 40p and 10% for 43p). When we increase the density, for 40p the number of ions is still small (9%) but for 43p we observe, at resonance, complete ionization of the Rydberg sample, the signature of the formation of a plasma. The differences between n = 40 and n = 43 in the frequency and magnitude of the ionic signals are due to the excitation of atoms to the attractive potential in the latter case, as suggested by [20].

The maxima of spectral scans, at a fixed frequency, such as those shown in Fig. 2 for a range of densities, delay times and quantum states give a more comprehensive picture. In Fig. 3 we show the density of ions vs density of Rydberg atoms initially excited, for 0.45, 5, and 10 µs delay times. The number of Rydberg atoms was varied by changing the intensity of the Ti:Sapphire laser. We note that there is a large number of ions present at a delay of 0.45 µs, and we believe they are formed during the 300 ns laser excitation. The central features of Fig. 3 are the following. With a 0.45 µs delay the ionization yields for all n states are essentially the same, but with 5 and 10 µs delays there is a clear difference between the n < 42 and n > 42 states. The only difference between these two cases is that the atoms are excited to a repulsive potential in the former case and an attractive one in the latter case. Two atoms excited to the 43p43p state with a separation of 5 µm collide and ionize in 6 µs [START_REF]The ionization time is simply calculated by integrating the dipole-dipole forces between two Rydberg atoms[END_REF]. For a Rydberg density of 25 × 10 8 cm -3 , corresponding to 5000 Rydberg atoms in the trap, 10% of the atoms are this close together, so the difference between the n < 42 and n > 42 behaviours is evidently due to the dipole-dipole induced collisions. For the 10 µs delay the ion production, for n > 42, starts to become very nonlinear for a ion density of 2×10 8 cm -3 (∼ 400 ions), which we attribute to the trapping of the electrons by the ions and the subsequent ionizing collisions of the electrons with the Rydberg atoms.

For 5 and 10µs delays, the 42p state is between the two sets of curves, corresponding to an excitation to both the repulsive and attractive potentials. While the difference between the attractive and repulsive curves of Fig. 3c is understood, the source of the ionization for the n < 42 states is less evident.

We show two kinds of data for the repulsive case (n < 42). For the 40p and 41p data, 6p cold and hot atoms are present due to the trap lasers. For the data labelled "40p ⋆ " the 6s -6p excitation is turned off at the same time as the second and third lasers are turned off, so there are no 6p atoms with which to interact. In the case of the 40p state with no 6p atoms, we observe an apparently density independent ionization rate of 2400 s -1 . This linear rate also appears to be present in the n > 42 states at low density. The calculated blackbody photoionization rate is 1200 s -1 [20,21]. The observed rate is twice the calculated rate. Our observations are not unique in this regard [15]. The ionization rates for excitation to repulsive curves are generally higher than the calculated blackbody photoionization rates, and we now consider possible causes of this ionization.

One possibility is the black-body transfer of population from the np states to the nearby ns and nd states, producing ns-np and nd-np pairs, with a rate ∼ 10 000s -1 . Roughly half the pairs will be on attractive 1/R 3 potentials and half on the repulsive potentials. Only 5% of those on attractive potentials can collisionally ionize on a 10 µs time scale, corresponding to a rate of ions formation around 250s -1 , which is not fast enough to explain our observations. This rate could be, for short time scales, increased by superadiance [16,22]. In this connection we note that, for some conditions (medium density), the Cs 40p state could exhibits a rapid initial decay, with a ∼ 10 µs decay time, far faster than the 300 K 40p lifetime of 47 µs. But in most of the conditions of this article the presence of the dipolar and ionic dephasing would probably limit this effect.

We suggest here a new contribution to the large ionization rates in the case of the repulsive potentials based on the effect of cold ions, which may attract the nearest Rydberg atom by the ion-dipole interaction. The process could be the following: one ion and one Rydberg could collide to produce a translationally cold Cs + 2 molecule, which in turn attracts another Rydberg atom, a process likely to result in two ions. However, the ions are hot, and the process terminates, so this could increase the ionization rate up to 1000s -1 rate. Taken together the above mechanisms are probably responsible for the observed ionization rates for the n < 42 states. We note, though, that the scenarios outlined above should lead to a nonlinear ionization rate for short times or low densities, which is not apparent in our data, although the fact that the sample of atoms does not have a uniform density may mask such features. The notion of collisions due to ion-Rydberg dipole attractive potentials assumes that the polarizability, of the Rydberg atom, is greater than zero, that is the Stark shift is to lower energy when the electric field is increased. This requirement is met for the Cs 40p and Rb 40s states, but not for the Cs 39d state, and it may be why the ionization rate ∼ 1000 s -1 of the Cs 39d state [23] is close to the calculated blackbody photoionization, but the ionization rates of the Cs 40p and Rb 40s states are not.

In summary, the measurements reported here demonstrate clearly that excitation to an attractive, as opposed to a repulsive potential dramatically increases the initial ionization rate due to dipole-dipole induced collisions. However, atoms excited to repulsive curves also ionize at rates in excess of the expected blackbody ionization rates. We suggest that in this case the attractive atomatom or ion-atom potentials lead to increased ionization. The complete dynamics of the ionization of an ensemble of Rydberg atoms is complex, but the attractive dipoledipole or dipole-ion forces are the main parameter for this problematic.

FIG. 1 :

 1 FIG. 1: (a) The energy difference (W) between the molecular pp and ss' states at R = ∞ for different n. (b) and (c) Schematic potential curves of pair of atoms as function of internuclear distance.

FIG. 2 :

 2 FIG.2: Rydberg and Ion signal with two different laser intensity (42mW and 278mW) for (a) 40p state (repulsive) and (b) 43p (attractive) Data are taken after a time delay of 10 µs, between the end of the laser pulse and the field ionization pulse.

FIG. 3 :

 3 FIG. 3: Ion density as function of initial Rydberg density for different n states and three delay time (a) 0.45 (b) 5 and (c) 10 µs. At 10 µs the "40p ⋆ " is without 6p cold and hot atoms. Each point corresponds to the maxima of spectral scans, such as those shown in Fig.2.

  • , producing a 2 × 10 -3 mm 3 volume of Rydberg atoms. The second and third laser pulses have a 300 ns durations. The inherent limitation of the linewidth of the final transition is imposed by the 54 ns lifetime of the Cs 7s state, the 1 MHz laser linewidth and the Fourier transform of the pulse duration. Subsequent to its production by the laser pulses the cold Rydberg gas is allowed to evolve for times from 450 ns to 50 µs. At this time we analyze the population with a field ionization pulse which rises to 500 V/cm in 50 ns, applied using two grids 1.57 cm apart. Any ions present are ejected from the MOT at the beginning of the field ionization pulse, the Rydberg atoms are ionized, and the resulting ions are ejected later. Both sets of ions strike a microchannel plate detector, producing time resolved ion and Rydberg atom signals for initial states of n < 80, which are registered with two gated integrators and stored in a computer.

The authors acknowledge fruitful discussions with Vladimir Akulin. This work is in the frame of "Institut Francilien de Recherche sur les Atomes Froids" (IFRAF)


'Nothing has changed, it just turned illegal': discourses for the justification of illegal trade and immigration in the Moldovan Republic' Monica Heintz

'NOTHING HAS CHANGED, IT JUST TURNED ILLEGAL': DISCOURSES FOR THE JUSTIFICATION OF ILLEGAL TRADE AND IMMIGRATION IN THE MOLDOVAN REPUBLIC

Monica Heintz University of Paris X-Nanterre

This article envisages the movements of goods and people from the point of view of their real and perceived legal status. It explores the multifaceted reasons and complex arguments used to support legally condemnable activities such as illegal trade and illegal migration. I focus on a Moldovan village situated on the border with Ukraine and present the villagers' discourses for justifying these illegal activities in the context of the severe economic shortage and lack of employment opportunities in the area. The main reasons invoked for justifying continued involvement in economic transactions that became illegal when the USSR collapsed are economic rationality and the tradition of trade between the ex-Soviet republics. As far as labour migration is concerned, discourses for justifying illegal migration are largely absent; the findings of the study indicate that it is not the illegality of labour migration that concerns potential migrants, but only its feasibility and resulting benefits. However, when it comes to the moral dimension of the (illegal) activities in which migrants engage, discourses of justification are produced -not only for officials or the anthropologist, but also for fellow villagers. They need to be persuaded that the standards of morality within the village are distinct from the standards of morality outside the village and that the movement of people and goods should not fall under the jurisdiction of the former.

In the first part of this article I will discuss the economic, political and social circumstances that have led to labour migration and cross-frontier trade in the Republic of Moldova since 1991 and consider the various migration destinations and legal status available to Moldovans. In the second part of the article I will seek to interpret the discourses generated by migrants and their families to justify illegal migration and trade to the community (and the anthropologist).

Why and how do Moldovans migrate?

The National Bureau of Statistics of the Republic of Moldova estimates that around 0.5 million of the republic's 4.35 million citizens are working abroad. They represent as much as 35% of the total active population, which is put at 1.35 million (Anuarul Statistic 2003). Independent mass media make references to one million labour migrants, more than half the active population of the republic and 25% of the total population. It has been estimated that the income sent back to Moldova by migrant labourers solely through Western Union money transfers is already three times the level of the state budget. The difference between official state figures and the figures published by independent media regarding the number of migrants is not only attributable to the different political positions that they seek to advance. Most trade and migration is illegal and thus escapes the control of the state. Although precise statistics on the scale of this phenomenon are lacking, migration is considered to be the most important and visible feature of social life in the country and it is the hottest topic of daily debate.

The causes of migration

The social and economic changes that affected the newly born Republic of Moldova after 1991 were common to most ex-Soviet republics. Union structures were dismantled following the disintegration of the USSR and the newly emerging countries had to build democratic institutions from scratch in an economic and political environment heavily influenced by the Soviet legacy. Certain political and historical circumstances -namely economic dependence on Russia due to Transnistria's breakaway and patterns of employment during the socialist period -aggravated the consequences of the collapse of the USSR in Moldova, resulting in massive unemployment and disproportionate poverty given the county's natural and geographical assets.

The predecessor of the Republic of Moldova, the Moldovan Soviet Socialist Republic (MSSR), came about at the end of the Second World War from the unification of two territories: the Moldovan Autonomous Soviet Socialist Republic (better known by the name of Transnistria, the left bank of the Dniestr river), a republic created in 1924 and functioning within the Ukrainian Soviet Socialist Republic, and a much larger territory, commonly known as Bessarabia (the right bank of the Dniestr river), which was part of Romania before WWII. The Bessarabian region, newly incorporated into the USSR, underwent extensive purges of the cultural and economic elites in the 1940s and remained distrusted throughout the socialist period. During this time political elites were recruited from the Transnistrian part of the republic and all the main industry was built there. The right bank was used largely for agricultural production and light industry (agro-industry and textiles). In general, the Moldovan SSR was tasked with producing cereals and wine for the USSR economy. In June 1990, following a national resurrection movement claiming language rights for the titular nation in the republic, the Moldovans (Romanians), the republic declared its sovereignty within the USSR [START_REF] Fruntasu | O istorie etnopolitica a Basarabiei[END_REF]. In response to the Roumanophone movement, the Russophone left bank of the Dniestr declared its secession from the right bank in September 1990 and formed the Dniestrian Moldovan Republic (Transnistria), a political entity which to this day is still not recognised internationally. Although officially part of the Republic of Moldova (independent since August 1991), Transnistria has its own government, currency, political structures (Troebst 2003;[START_REF] Troebst | The 'Transnistrian Moldovan Republic': From Conflict-Driven State-Building to State-Driven Nation-Building[END_REF]) and, most notably, does not share its economic assets with Moldova. A war between the two banks of the river in 1992 accentuated the disintegration of the economy. The Republic of Moldova has few industrial structures, no sources of energy and gas, and after fifteen years of independence it is still economically dependent on the Russian Federation (which provides these energy resources).

Meanwhile, several USSR enterprises situated on the territory of Moldova were withdrawn at the beginning of the 1990s when the Soviet Union collapsed. The remaining enterprises went through restructuring as in all former socialist states. The result is that today only 12% of the employed population works in industry, with as much as 43% employed in agriculture and 45% in the tertiary sector (Anuarul Statistic 2003). On further analysis of these figures according to the age of the employed population, their ethnic belonging and rural/urban residence, certain patterns of employment emerge that explain the current exodus of the workforce.

In the MSSR collective farms, kolkhozy, were formed in 1949 and the entire adult rural population (except for some specialists) was taken to work in the fields. By the 1970s agriculture was mechanised and there was no need to bring up younger generations on the farms. The young rural population from the right bank of the Dniestr was drawn into light industry. Following the restructuring of industry after 1991, it was the younger generation that was left unemployed. In 1992 decollectivisation began (a process that ended in 1999) and land was redistributed among those who were or had been active in the kolkhoz, i.e. the older generations. Younger generations were left without town jobs and without other means of subsistence. Those with a higher education generally moved to the capital city Chisinau, home to 20% of the republic's total population, where they started small businesses or began to work in the tertiary sector. Many small businesses involve cross-border trade between Moldova and the neighbouring countries, especially within the CSI. It is estimated that 70% of national GDP derives from export and import activities at the borders. Businesses that would generate jobs and income are more difficult to start in the rural milieu, where information, connections and entrepreneurial savoir faire are lacking. Peasants sell their produce in the towns on an individual basis at an insignificant profit and few consider crossing frontiers with their products because of the high cost of bribing the border police. iii New small farms have been set up in villages, pooling the land of several owners. Yet they are few in number and have not been encouraged by the government (led by the Party of Communists since February 2001), which prefers to maintain larger kolkhoz-type farm structures. Not surprisingly, independent farmers and small collective farms are led by men with anti-communist leanings; they make an ideological statement with their involvement in independent farming and their refusal to go to work in Russia.

For the rural population (representing 61% of the total Moldovan population) with no expertise in independent farming and no ideological position to uphold, the only source of work and income appears to be abroad. This would explain why men and women aged 18 to 45-50, with some work experience in factories, can imagine trying labour migration to various destinations that reportedly offer earnings prospects.

Migrant destinations and the question of legality

The destination for migration is primarily linked to a migrant's ability to reach it and only secondarily to the attraction that it exercises on a migrant. Migration towards the West is therefore preferable to migration towards the East, but since the former is more difficult the number of East-ward migrants greatly exceeds the number of West-ward migrants. The length of stay in the host country is also directly proportional to the difficulty of reaching it: migrants do not venture to return to Moldova if they are unsure of their ability to return to the host country afterwards. Consequently, the life of the families left behind takes on a different pace depending on whether or not the members engaged in migration return home on a regular basis.

The first destination for work is Russia (the cities of Moscow and Saint Petersburg), where the monthly wage can be as high as 200-250 dollars.

Since the Republic of Moldova is a member of the Commonwealth of Independent States, the weak successor to the USSR, its citizens had the right to circulate freely in ex-Soviet space in 2003-2004. However, in another legacy of the politics of control of the Soviet Union, foreign citizens are required to register within three days of arrival. Registration costs one hundred dollars for a three-month stay in Russia and does not give the right to work. Given the average monthly wage in Moldova of fifty dollars (a wage that few have the opportunity to earn), not many Moldovan citizens are willing to pay the registration fee. Consequently, after spending three days in Russia most Moldovans stay on illegally and almost all of them work illegally. Work in Russia is facilitated by their knowledge of Russian, a state language during the Soviet era, and by the USSR integration policy that familiarised Moldovans with Russia. iv The feeling of still belonging to the USSR continues to be nurtured in Moldova through material culture, e.g. by keeping the former Soviet passports and bringing various newspapers and magazines back from Moscow. It is also fostered by the Russian minority that denies the changes that took place after 1989 by refusing to learn and speak the state language or to identify with the Moldovan state. Furthermore, the nostalgia for bolshevita v led many citizens to vote for the return of the Communist Party with its pro-USSR programme in the 2001 elections. vi However, few have experience of actually travelling in Russia before they need to seek work abroad. Moldovan men are easily spotted as foreigners in large Russian metropoles, suggesting that their former Soviet identity does not help to disguise their contemporary appearance as the 'foreigner'. Often the police catch them and fine them, i.e. they extract money from them in exchange for not denouncing their illegal residence status in Russia. vii Staying in Russia is risky -their illegal status confines workers to their place of employment (where they also live) and prevents them from walking on the street during the day. viii Travelling freely back and forth between Russia and Moldova is possible due to the ID card (no record is kept of border crossings) and the low (CIS-protected ix ) travel prices. This enables rural inhabitants to leave quite spontaneously for Russia when there is less work to do in their households and they lack cash. Relatively few labour migrants go to Russia for periods longer than a few months in a row because immigrants aged 18 to 50 have children at home and households to keep. Work opportunities in Russia may also not be continuous.

Work in Russia is not entirely disruptive for family life since migrating members are able to return on a regular basis. It is also less disruptive for community life because the connection between villagers is kept up when several villagers work in the same Russian town and communicate news from home to each other or their families. However, no reconstitution of the community abroad is possible due to the illegal status that restricts Moldovans to their workplaces (those working in construction tend to live on the building site; those who take care of older people or children live with the host families) and because the size of the cities in which they work obliges them to live miles apart.

The second and most sought-after destination is of course Western Europe. This is because the earnings are higher, ranging from 800 to 1000 euros per month. Western Europe is the destination for longer-term migration because of the difficulty of access and hence it requires careful planning. Schengen visas are almost inaccessible to potential labour migrants, who are easily spotted by the European embassies in Chisinau and refused visas. The solution often lies in paying for extremely expensive visas, costing around two thousand euros, which some travel agencies organise illegally. x Italy is the Western European country with the largest Moldovan population, with the most popular employment there being caring for the elderly (especially among female migrants). Such jobs finally seem to have been accepted by Italian officials, because after 2-3 years some Moldovans are able to obtain papers enabling them to travel back and forth to Moldova. In addition to the appealing similarity of the Italian and Romanian languages, Italy is a common destination because of the channels opened by the first immigrants and certainly also because of a more indulgent attitude on the part of the carabinieri. For those who do not succeed in legalising their status, temporary returns are not foreseen and a definitive return is dictated simply by eventual deportation from the Western state. In the case of Western European destinations family networks function best, while for Russian destinations neighbourhood ties seem to suffice for spreading information about job opportunities and ways of coping with the border guards and local police.

Labour migration to Western Europe is facilitated in the western part of the Republic of Moldova by the fact that many Moldovans can travel with Romanian passports. xi Romanian legislation allows former citizens of Romania and their descendents to regain Romanian citizenship. Until 2002 a residence requirement on the territory of Romania made it difficult for Moldovans living further away from the Moldovan/Romanian border to regain Romanian citizenship. Obtaining Romanian passports has become more difficult over the years as Romania has had to come into line with European Union regulations in preparation for its EU integration, yet this has also made their acquisition more tempting (Marin 2006). Economic disparities have appeared between regions: villages close to the Romanian border are richer than those close to the Ukrainian frontier due to the availability of work in the West as opposed to opportunities in the East (tall new houses are material proof of this). Romanian passports permit travel twice a year for three months at a time within countries that have signed the Schengen Agreement, albeit without the right to work. The constraint of having to return every three months for a three-month period is easily circumvented by bribing the border police to change the return date -a bribe my Romanian informants put at one hundred euros, a reasonable amount to spend when working within the European Union. For Romanian passport holders, travelling to the West becomes as spontaneous and need-oriented as travelling to Russia for Moldovan passport holders. xii Money brought back to the Republic of Moldova is mainly used for building houses. When new needs such as for decorating a room or bathroom, one member of the family leaves again to work until the necessary amount of money has been earned. As one middle-aged woman complained, this process never stops because new needs or new ambitions always arise. Yet as a younger male migrant explained to me, while he was still young enough to be active he could not retire and live on what he had earned because he was certain that sometime in the future he would again experience a lack, possibly at a stage in his life when he would not be able to get work so easily. What pushes villagers to migrate, therefore, are not just needs and ambitions, but also certain ideas about life and work.

Despite the advantages of working in the West, the long-term absence is more disruptive for family life and in many cases several years of absence cause the destruction of the family. Often, those working in the West do not go back to their home village, but upon return settle in Chisinau. Many migrate permanently, establishing new families abroad or migrating with their entire nuclear families. Children of permanently migrating families are often taken care of by grandparents until they reach school age, before following their parents abroad.

The migration destination has the capacity to fashion a certain identity and civic attitude, despite the weak link existing between the host country and the migrant due to her/his illegal status. Consequently, newspapers often present migrant labour in Western Europe as able to generate a European identity for Moldovans, notwithstanding whether the migrants travel on Moldovan or Romanian passports (i.e. more or less legally). Such a European identity manifests itself when they are back in Moldova through the desire for integration into the European Union, voting for pro-European parties and against the Communists and increased participation in public life. This sympathy for the West is borne of practical considerations: Western European countries offer better living and working conditions than Moldova or Russia. Lack of access to information about living standards in the European Union prompts many to regard conditions in Russia as the best there could be (inasmuch as they are the best that they have witnessed or heard about) and hence cooperation with Russia is viewed as the most desirable political position. Travelling and working in Western Europe opens up fresh perspectives and renders other identifications attractive. Migrants to the West return to Moldova with different notions of public life (for instance, with new expectations regarding the protective role of the state, the freedom of mass media) and with visions of desirable futures that differ from those of migrants to the East [START_REF] Heintz | Introducere: Statul si cetatenia moldoveneasca[END_REF].

Discourses of justification for illegal and/or immoral actions

In 2003/2004 I conducted eight months of fieldwork in Satu Vechi, a rural community of some 3,000 inhabitants situated on the Moldovan/Ukrainian border. In August 2004 I tried to measure the scale of the migration phenomenon based on a sample of 75 households drawn from the same neighbourhood. The survey revealed that 22 households (30%) had at least one migrant labourer away working at the time of my interview, a total of 30 workers out of almost 300 inhabitants (10%). Of these, 26 migrant workers were working in Russia, one in Turkey, one in the Czech Republic, two in Italy (one was there temporarily on a Romanian passport, the other had not returned for the past three years because his status there was illegal). If the population included in the sample between the ages of 18 and 45 is regarded as capable of working (many have medical problems for which they receive pensions), almost one third had gone to work abroad. Relatively speaking, this is a village considered to have been impacted less by migration than others, and it would be impossible to extrapolate or estimate the extent of the national migration phenomenon from this sample. At the local level, though, the number of emigrants significantly influences village life in that it imposes unusual family arrangements: children have to be brought up by aunts and uncles or great-aunts and -uncles, siblings are divided between several households, grandparents are separated while one grandparent takes residence in the children's house to look after the grandchildren while the other continues taking care of the farm etc.

It is hardly any wonder, then, that labour migration is a topic of lively interest and discussion in the village. Most conversations turn around those who work abroad, when they will return to pay a visit, whether the (nuclear) family can still hold together when the husband/father or wife/mother is away, what jobs could be found and how much they bring in, including many anecdotes linked to the illegal nature of their employment. While politics is almost taboo in the village, most people still being afraid to openly criticise the government, nobody hesitates to talk about the tricks employed to get around the border police or the local Russian or faraway Italian police. However, migrants avoid being precise about the type of jobs they do abroad and the conditions in which they live, unless specifically asked to do so. In general, they present themselves as hard workers, living in better conditions and with greater dignity than their family left behind at homeexcept for occasional accounts of being on the run. xiii I will discuss below, firstly, the (lack of) discourses used to justify the illegal aspects of migration and, secondly, the strategies for justifying immoral activities.

The lack of justification provided for illegal activities

Direct discourses for justifying illegal migration are lacking; people do not believe that there is anything to justify in front of a fellow villager or anthropologist because they do not consider that there is anything wrong with what they are doing. What are the reasons for this absence?

The first reason for this lack of a need for justification is simply the scale of immigration, which reassures both migrants and their families that what they are doing is 'normal'. xiv We have seen the large (estimated) figures for the scale of migration. It should be borne in mind that during the Soviet period just what was politically dangerous was never known exactly, but by not standing out from the crowd and by doing the same things as everyone else one could stay on the safe side. xv In the village labour migration is 'normal': it is travelling far away to find a job if none is available closer to home.

The second reason is a lack of responsibility in the face of abstract laws governing migration and labour. As most interviewees put it, why was a Moldovan allowed to work in Russia until 1991 but not now? In their eyes, this is just a matter of politics -of which they are the victims. Faced with this perceived injustice, the potential migrant chooses not to shoulder any responsibility. It is easy to make comparisons with the Soviet period and conclude that it cannot be fundamentally wrong to work in Russia.

The same reasoning applies to work in Western

Europe if a Moldovan compares himself with a Latvian or an Estonian, similarly former Soviet citizens. As people who consider themselves hardworking, Moldovans do not feel guilty about the illegality of their work and their sole concern is with not being caught, since this costs them money. This lack of concern was obvious to me when people I scarcely knew but who had learnt that I was a European Union citizen did not hesitate to tell me about their illegal work in the EU and their ways of tricking the police there, without fearing that I would have a negative opinion about this. They would also openly declare their desire to obtain a Romanian passport so as to travel more easily to their destination. In Romania the intelligentsia used to condemn Romanian illegal workers for endangering their possibility to circulate freely in Europe through their disrespect of travel conditions. Consequently, labour migrants from Romania show a greater awareness of the illegality of their work and are more careful in their discourse, at least in front of strangers. In my previous work in Romania in 1999/2000 and again in 2003/2004, I heard many discourses of justification for illegal trade and immigration from Romanians (Heintz 2006: 156). They were a response to the accusations of the independent press and the government, which continuously criticised these illegal actions for compromising Romania's prospects of European integration. xvi The independent press in Moldova, on the other hand, displays sympathy towards labour migrants and uses their departure to reproach the government's lack of efficiency in implementing economic change. The 'blame the victim' position is not popular.

The third reason for the absence of justifications has to do with the fact that migration is seen as the last solution, one which is embraced out of despair rather than rational reasoning. A young divorced woman, who works in Saint Petersburg in order to save money for building a house, told me that she would like to be able to go to Italy, despite the danger of being caught and deported and not allowed to travel abroad for a number of years. She explained that if she worked there for two years and earned as much as she could in Russia in ten years, she would not need to travel abroad any more and could do without a passport. The important thing for her is to get enough money to build a house. In 2003 newspapers were full of scandalised reports concerning young Moldovan women who, after they were caught and imprisoned while working illegally in Italy, declared to Moldovan journalists that they preferred their life in detention to their life in Moldova. (This could again be taken as a strong criticism of the policies of the government in power.) Single or divorced women as well as those who have experienced domestic violence are especially willing to take the risk of migrating to Italy or Turkey because the life they leave behind and their status in the rural community are not attractive. Rural communities remain traditional in their way of judging broken families and condemn single women, despite the increased number of divorces and second marriages among the younger generation (many due precisely to labour migration). Returning home, these women do not resettle in the village but generally seek refuge in towns.

All these reasons 'naturalise' the illegality of migration, which becomes acceptable and fails to trigger the shame associated with the fact of being caught by the police or sent to prison that is usually felt in traditional Moldovan society. Justifications become unnecessary as the Moldovan unemployed are presented as victims of historical circumstances, not particularly bound to the legal frame of any state since no state provides them with the expected social or employment security -an idea of entitlement that is a typical socialist legacy (Heintz 2006: 93).

Justifications of "immoral" activities

While the illegality of work abroad is not judged by locals, there are ample discussions and comments within the community about what types of jobs young women hold in Southern European countries (a question less often formulated for those who work in Russia). Publicly, these concerns are expressed in collective paternalistic terms (along the lines of 'why are our beautiful girls obliged to go to Italy?') and they are often rhetorically linked to demographic rather than moral anxieties. There is plenty of first-hand evidence from the village that many female villagers have engaged in prostitution abroad, a condemnable activity by local standards. If no need is felt to justify illegal trade and immigration (at least the lack of justifications would suggest this is the case), there is secrecy and a need for justification in private circles with respect to what would be regarded in the village as 'immoral' activities. Are migrants involved in activities disapproved of as 'immoral' rejected by the community, or are there justifications for their actions powerful enough to enable them to gain acceptance from the community?

Having becoming involved in such activities abroad, many young women -fearing rejection from the community -do not dare to return to the village but settle further away in towns. Their families, however, remain in the village and have to justify their daughters' behaviour and also explain why they maintain relationships with them. One of my closest informants recounted how her daughter (who has now gone to Russia) had been trapped and forced into prostitution in Turkey a few years before and explained how she had invented (or the daughter had invented for her) a story in which the daughter had escaped with the aid of a benevolent Turk without prostituting herself. This story is quite telling of the shame that such an activity might entail. Similar stories of forcibly trafficked young women were used to explain how and why the daughters of the village ended up in prostitution. Most 'lost daughters' involved in prostitution were forgiven by their parents who accepted them back into their home, but their stories are not forgotten and I collected a few of them myself unsolicited from their parents. It might be that these parents (mothers) thought that I already knew their daughters' stories from somewhere else and they tried to avoid its negative reflections on their families by dissociating themselves from their daughters. They both condemned the daughters' actions and explained their reasons for forgiving them. Keeping one's reputation in the village is important, especially as one grows older.

The fact that all these women were presented as innocent victims may be challenged as being nothing more than a justification strategy. However, the village of Satu Vechi was special in that one of its residents was an efficient female trafficker who had helped women from several neighbouring villages to get work as prostitutes in Turkey for almost ten years. The female trafficker (in her late twenties at the time of the interview) prided herself on being the first woman to have left for Turkey ten years earlier and the one who had been able to facilitate the departure of other girls afterwards. She told me that her business was to arrange visas for young girls who wanted to work abroad, and hence she felt that many women owed her a great deal. Thus she had her own justification strategy, but could the villagers even accept the justification of a trafficker?

In the village the female trafficker was ironically said to have returned from Turkey because she was too old (she was 28) and nobody wanted her sexual offers any longer. Yet despite this irony and comments such as 'many mothers hate her' (as a close female informant expressed it), most families kept good relations with the trafficker and her family. The reason for this is that relations within and outside the village are considered distinct, as is the type of logic governing them. On the other hand, adultery that is uncovered in the village could lead to a break in relations. For instance, a case of adultery that occurred three or four years before my fieldtrip was recounted to me by five different individuals from different households despite my lack of close involvement with the families of the adulterous couple, thus showing that collective memory acted as a gatekeeper for immoral behaviour in the village. 'Forgetting' to pay back borrowed money could also lead to a break-up in relations between householdsbut not the fact of being trafficked or the act of trafficking human beings. Discourses of justification tend to separate the realm of local moral laws from what happens abroad, suggesting that actions taking place abroad -which is defined as being not within the borders of the Republic of Moldova -are somehow exceptional and should not be judged according to local standards.

The case of the woman trafficker plainly shows that the strategy used for accepting migrants who had been involved in what is perceived as an immoral activity abroad (notably prostitution) is one of dissociation of the two worlds of home and abroad. This is done through discourses of justification that emphasise the exceptional character of life abroad and the lack of agency with which emigrants are endowed. The exercise of will is somehow seen as being impossible in circumstances that are unknown and unintelligible to villagers.

Conclusion

In this article I reviewed some of the reasons for and conditions of labour migration from the Republic of Moldova in the light of laws regulating migration. It was shown that the issue of illegality is discussed not in moral terms but uniquely in terms of the constraints and risks it entails. While there is no need to justify to the community or the anthropologist why one migrates illegally, it remains important to justify how and why one engages abroad in 'immoral' activities such as prostitution. The strategy used in these discourses of justification is to present the conditions of migrants abroad as being so incommensurably different (and difficult) that local standards of morality could hardly apply. The morality of the village should be put to one side and local judgments should not be applied to villagers' actions abroad.

Notes

iii In the frontier village where I have done fieldwork there is a tradition of growing vegetables to be sold in Ukraine, where prices are higher; people mainly complain about the new border regulations that prevent this trade from being as profitable as it used to be. Although they continue crossing to Ukraine for commercial sale of the produce in the town over the Dniestr, this tradition is gradually fading away. iv Russia is not considered to be 'abroad' (zagranitsa in Russian) because Russia and Moldova used to belong to the same state in the past and also because there are material objects that keep Russia in this category today: the 'passport for going abroad' is not needed for trips to Russia (an ID card suffices), and 0.5 million Moldovan inhabitants had old Soviet passports until the end of 2003, when the passport regime finally changed and Moldovan passports became compulsory for all Moldovan citizens. v A play on words equating the Bolshevik life with a good life, dolce vita. vi One middle-aged educated woman in the Moldovan village, who had never worked in Russia, once revised her USSR discourse of belonging by saying 'Here, in our country, in the USSR, sorry, Russia'. vii One migrant woman and her family back home expressed their pride that she was never stopped by the police because she was elegant and 'looked like a Russian'. viii The degrees of risk are discussed in the village before people make their choices. Moscow is said to be more dangerous than St. Petersburg, where the police check IDs more often. ix An airfare of 60 euros was offered for Chisinau-Moscow-Chisinau in 2003, contrasting with 300 euros for Chisinau-Vienna-Chisinau (a shorter journey).

x Many travel agencies only obtain visas for Slovenia; the labour migrants are left at the frontier with Italy and have to cross the border on foot at night. xi 300,000-500,000 Moldovan citizens are said to possess Romanian passports. The figures are not accurate because many of those who have regained Romanian passports are no longer Moldovan residents. xii This is also similar to the behaviour of the rural population from Romania working in the West.

xiii If asked, one will recount how he spent a winter's night in a tree out of fear that a Russian policeman might take his money; others might tell how they have been tricked over payments; another woman will explain how she hides the money she has earned in the bus seat because she is thoroughly searched at the border, only to recover it days later from the driver. xiv The word 'normal' is frequently used to mean 'the usual state of things' and often has a pejorative connotation. xv In Soviet times, dissidents, those standing out of the crowd, were considered somehow abnormal [START_REF] Yurchak | The Cynical Reason of Late Capitalism: Power, Pretence and the Anekdot[END_REF]).

We would note that this concern with illegality is linked to practical rather than moral judgements.