Holographic Description of Noncommutative Schwinger Effect

We consider the phenomenon of spontaneous pair production in presence of an external electric field for noncommutative Yang Mills theories. Using Maldacena's holographic conjecture the threshold electric field for pair production is computed from the quark-antiquark potential for noncommutative theories. As an effect of noncommutativity, the threshold electric field is seen to be smaller than its commutative counterpart. We also estimate the correction to the pair production rate of quark-antiquark pairs to the first order of the noncommutative deformation parameter. Our result bears resemblance with an earlier work (based on field theoretic methods).


Introduction
Quantum Field Theory is primarily studied in its perturbative regime. However there exists quite some novel non-perturbative features of quantum field theories amongst which the Schwinger Effect [1] stands its ground 1 . The vacaum of Quantum Electrodynamics is a bath of e + e − virtual pairs which gets created and annihilated instantaneously. However in presence of an external electric field, the e + e − pairs spontaneously become real and their production rate in the weak-coupling weak field approximation is given by [1].
Intuitively the result can be understood in the following way. For the virtual electronpositron pairs to be real, they should gain atleast an enegry equal to sum of their rest mass (2m) plus their coulombic interaction. The virtual particles become real at seperation distances where the external electrostatic potential equals the self energy and coulombic attraction, EL * = 2m + V coulomb (L * ) i.e. at the distance when effective potential barrier below which the virtual particles become real via tunelling proces, vanishes. This is exactly the content of the Schwinger formula (1.1). For electric field when the effective potential becomes monotonically decreasing (exhibits replusive behaviour) for all seperation distance the tunelling becomes insignificant altogether. This value of electric field is called the threshold value at which the production rate is not expotentially supressed anymore.
The idea of noncommutative quantum field theories [3] [4], where space-time itself ceases to commute was originally proposed to curb the UV/shortdistance divergences appearing in interacting Quantum Field Theories. The idea received revival when noncommutative field theories were found to be low energy limit of open string theories on a Dp Brane with a contant NS-NS two-form B µν , the noncommutative feature being a dynamical consequence of quantization [5] [6]. There exsists noncommutative generalisations of Riemannian geometry [7] on which the standard model can be defined, wherein the parameters of the theory are interpreted as geometric invariants. It is also known that in presence of a background magnetic field the effective theory is best described by a noncommutative physics [4], which are generically Lorentz violating though their effects are quite small to be detected in practical experiments [8]. The transition from commutative theories to its noncommutative counterpart is acheived via the moyal/star product( ).
The above equation implies [x µ , x ν ] = iθ µν , signifying nonvanishing commutation relations between spacetime cordinates itself. The paramenter θ depends on the strength of the background magnetic field. The Schwinger Effect in noncommutative QED has been calculated in [9] where a correction to the pair production rate has been found leading to a decrement in the threshold electric field as a consequence of noncommutativity. However to carry on the same kind of analysis for strong coupling in general becomes an uphill task and the presence of noncommutativity makes matters worse.
The Gauge/Gravity (holographic) correspodence [10] which links a strongly coupled gauge theory to classical gravity is an important tool in these kind of scenario. As presently understood the correspondence has firm ground for a strongly coupled large N gauge theory. There exists holographic models of large N QCD [11] [12][13] [14] which has been able to calculate viscosity coefficients, the η s ratio of the deconfined quark gloun plasma to an appreciable degree of accuracy. As such confinement in quantum field theories have been a long standing issue of interest and it is generally acknowledged to be a non-perturbative phenomenon. It is quite probable during heavy ion collisions, strong electromagnetic fields are produced due to scattering of charged particles in presence of which the qq pairs are produced via Schwinger Mechanism leading to a new explanation of deconfinement. It is also known that holographic configning theories admit a new kind of critical electric field below which Schwinger pair production does not occur. In this prespective it is interesting to see how noncommutativity would effect the threshold electric field of strongly coupled gauge theories. The Schwinger effect as such is not an obeserved phenomenon as the value of the threshold field is much higher than accessible regimes. This paper is organised as follows. In section 2 we review the derivation of Schwinger effect in a super-conformal SU(N) gauge theory by relating the same to the expectation value of circular wilson loop. Also in the same section the basics of noncommutativity in string theory is reviewed for the sake of clarity. Section 3 is devoted to the analysis of effective potential for virtual particles in NCYM plasma, from where the explicit form of the thershold electric field is found out by analytical means. In section 4 we compute the expectation value of circular wilson loop by perturbation over the commutative (AdS) result and hence find the first order correction to the pair production rate of Schwinger particles. We close this paper by conclusions in section 5.

Pair Production in SYM and the Wilson Loop
In its most prominent avatar [10] [15], the holographic duality relates N = 4 Super Yang Mills theory to type IIB String Theory in AdS 5 × S 5 . To study Schwinger effect one has to account for "massive" matter (corresponding to a probe brane) in fundamental representation and an U(1) gauge field. The way to do so is to break the symmetry group of the problem from SU(N+1) to SU(N) × U(1) with the higgs mechanism. Such methods were first introduced in [16], the following closely follows [17] [18]. For more sophisticated treatment refer to [19] [20]. The bosonic part of N = 4 SYM for the SU(N+1) theory in eucledian signature reads as.
F µν is the field strength of the SU(N+1) gauge fieldÂ µ .Φ i (i = 1, ..., 6) collectively denotes six scalars in the adjoint representation of SU(N+1). The gauge group is broken as, The non-diagonal parts, ω µ and ω i transform in the fundamental representation of SU(N) and form the so called W-boson multiplet. The VEV of the SU(N+1) scalar fields is supposed to be of the form, As a result of the decomposition (2.2) the SU(N+1) action (2.1) breaks up into three parts of the following form [16]Ŝ is basically the free QED action constructed out of the gauge field a µ . S W governs the dynamics of the W bosons and its coupling to the gauge fields. Disregarding the ω µ 's and higher order terms the W boson action reads, D µ is equipped both with the SU(N) gauge field A µ and also with the U(1) gauge field a µ . By expanding the action S W and choosing φ i = (0, 0, 0, 0, 0, 1), the mass term for ω 6 vanishes while those for ω i , i = 1, ..., 5 remain. The pair production rate is given by the imaginary part of vacuum energy density [21]. For the present scenario the SU(N) gauge field A µ is a dynamical field and the U(1) gauge field a µ is a "fixed external" field of the form a By using Schwinger's parametrization and worldline techniques [22] , one can express the pair production rate (2.6) as a path integral for a particle subject to an appropriate Hamiltonian under boundary conditions, Using saddle point approximations as in [17] [18] and assuming the mass to be heavy i.e. m 2 >> E , (2.7) becomes proportional to the path integral of the particle subjected to the "external" gauge field a (E) µ times a phase factor, namely the SU(N) Wilson loop.
Evaluating (2.8) by the method of steepest descent the "classical" trajectory becomes a circle . So the production rate is proportional to the expectation value of the non-abelian circular Wilson Loop and can be computed via holographic conjecture in the large N limit.

Noncommutativity from String Theory
The effective worldsheet action in presence of B µν field is given by, The equations of motions along with the boundary conditions when dB = 0 are The boundary conditions (2.12) are neither Neumann nor Dirichlet, one can indeed try to diagonalise (2.12) to be Neumann like by redefining the fundamental variables leading to the so called "open string metric" [6] [23] . However a more direct attack along the lines of [24] is to solve the equation of motion (2.11) first and constrain the solution by (2.12). For B = B 23 dX 2 ∧ dX 3 , the solution of (2.11) compatible with (2.12) is A similar solution accompanies X 3 (t, s), the forms of which encode the nontrivial boundary conditions. The solutions for the other co-ordinates are the usual ones [25]. The canonical momentum of the action (2.11) is by the virtue of mode expansion (2.13), 14) The fact that the current scenario leads to noncommutativity was first recognised in [26] [27] as the canonical momentum at the ends of the string become fuctions of the spatial derivatives of the string cordinates as per (2.12) and (2.14). From the symplectic 2-form the canonical commutation relation of the modes are, From the mode expansion (2.13) and the relations (2.15)-(2.17) one has The second term in (2.18) sums up to zero when s + s = 0, 2π. Therefore the end points of the string become noncommutative. The nontrivial part of the normal ordered virasoro constraints and the total momentum which accompanies (2.10) are given by It is clear from the above that the mass of the particle becomes dependent on the value of the B µν field. However the noncommutaive field theories contructed out of Moyal product leave the mass of the particle (quadratic part of the Lagrangian) unchanged. Since the string equations of motoion and boundary conditions are linear equations, one can redefine the operators to be in terms of which the mass of the theory remains unaltered [28].
It terms of the which the only nontrivial commutation relation become It has been checked that perturbative string theory in the present backdrop corresponds to Noncommutative Yang Mills at one loop. For further details see [28] and references therewithin.

Potential Analysis of Noncommutative Schwinger Effect
We start with a brief description of the holographic dual of Noncommutative Yang Mills (NCYM) [29] [30]. In the spirit of AdS/CFT correspondence one looks for supergravity solutions with a non zero aymptotic value of the B field. Such a solution is the D1-D3 solution which in the string frame looks like, The solution (3.1) is aymptotically flat and represents N (1) D1 branes dissolved per unit covolume of N D3 branes. The information of the D1 branes is stored in the relation tan ψ = N . It can also be seen that the asymptotic value of the B field is B ∞ 23 = tan ψ while R is related to the other parameters via R 4 = 4πg s N cos ψ.
The proper decoupling limit of the above stated solution resembles the field theoretic limit of the noncommutative open string [6] [28], for which the asymptotic value of B 23 has to be scaled to infinity in a certain way.
With the above scaling and keeping x (2,3) , u,ĝ, θ fixed the resulting metric and field configurations are given by The above is the holographic dual to NCYM with gauge group SU(N) and Yang Mills coupling constant g N CY M = √ 4πĝ which captures the dynamics of NCYM . Due to the noncommutativity the symmetry group of the theory becomes SO(1, 1) ⊗ SO (2). The gravity dual to NCYM at finite temperature T is found from the near horizon limit of the black D1-D3 solution and reads The most rigourous approach to study Schwinger effect is to find the expectation value of circular Wilson loop and relate it to pair production rate. However one may think of the vacaum to be made of qq pairs bound under an attractive potential and study how an external electric field modifies this potential. This is the essence of potential analysis which was first put forward in [31]. To compute the interquark potential, one needs to look at expectation value of rectangular Wilson Loop when the loop contour is regarded as trajectory of particles under consideration in the x 0 −x 3 plane where x 3 is the direction of qq orientation. As pointed out in [32][33] one places a probe D brane at a finite position instead of the boundary to get a W boson of finite mass. 2 As per the holographic procedure the vev of the Wilson Loop of a gauge theory is given by the partition function of a fundamental string in the background of the holographic dual with the ends of the string anchored on the probe D brane along the contour of the Wilson Loop C (2.9) i.e.
Where S[X, h] indicates the action of the fundamental string 3 . In the classical limit which is realised when the string length α is small (or the 't Hooft coupling is big ) the above expression is domitated by the on shell value of the Polyakov/Nambu Goto action. Thus the prescrtiption of computing Wilson loops is reduced to computing the area of the world sheet of the fundamental string which end on the specified profile at the probe D Brane [16] [34] [35], situated at a finite radial position in the dual geometry for the present case.
2 A string with Drichitlet conditions at both ends has the following canonical Hamiltonian [25], where first term indicates the potential energy of the stretched string and is the analogue of mass created due to symmetry breaking.

Potentatial Analysis at Zero Temperature
In this section we will study the properties of the modified potential of NCYM in presence on a constant external electric field for quark-antiquark pairs along one of the noncommutative directions (x 3 ) . The appropriate holographic dual as pointed above is given by (3.3). To calculate the area we take the string world sheet to be parametrised by s a ≡ (s, t). The Nambu Goto action reads In the above G ab is the induced metric on the worldsheet and G µν is the metric of the target spacetime/holographic dual(3.3)(3.4). The above action exhibits two diffeomorphism symmetries whith the help of which one can set two of the embedding functions to arbitary values provided that the resulting profile matches with the contour of our choosing on the probe brane. One usually chooses the so called static gauge for which the profile reads 4 In the above the extremisation of the Nambu Goto action is given by the functional form of u(s) .The Ω i are the co-rordinates of S 5 . For present purposes x 3 ≡ s is assumed to to range between [−L, L], when 2L indicates the interquark seperation on the probe brane with the boundary condition u(±L) = u B where u B indicates the position of the probe brane along the holographic direction. Again the temporal direction is assumed to range from [−T , T ] with the further assumption that T L. This is because the rectangular Wilson Loop gives sensible results when one assumes the interaction between dipole is adiabatically switched on and off as illutrated in Figure 1.
Before proceeding further let us address the issue of the B field. For noncommutative gauge theories the B field is excited (3.3)(3.4) and is present in the string action via the Wess-Zumino term dtdsB µν ∂ t x µ ∂ s x ν along with the usual Polyakov/Nambu Goto part. However the gauge choice given above (3.7) cancells the contribution of the Wess-Zumino part of the action. It is possible to consider the qq pairs at a velocity in the x 2 direction and take into account the contribution of the B µν term as in [36]. However in the present case where the virtual particles in vacaum are modelled as qq dipoles, such a configuration seems hardly sensible.
As per the above gauge choice the induced metric/line element on the world sheet reads Using the above form of the induced metric in the action (3.6), we get Extremisation of the above Lagrangian is equivalent to solving the Euler Lagrange equation with the effective Lagrangian L N G = du ds 2 + u 4 1+λθ 2 u 4 when u = u(s) along with the the boundary condition u(s ≡ x 3 = ±L) = u B as the countour profile is already taken into account by the gauge choice. One can indeed solve the relevant problem and find the explicit form of u(s ≡ x 3 ). However since the Lagrangian in (3.9) does not explicitly depend on the parameter s 5 , by Noether's theorem there exists a conserved quatitiy (Q) for the solution.
As indicated in [35] the fundamental string is assumed to be carring charges at its two ends and are otherwise symmetric about its origin. Thus one can choose u(s) to be an even function of s ≡ x 3 . In the present case this means the x 3 direction of NCYM is symmetric around its origin which holds true inspite of its noncommutative nature. From the form of (3.10) the solution of du ds involves both positive and negative signs. Thus there exist a value of the parameter s = s 0 for which du ds (s 0 ) = 0. This is the turning point of the string profile as indicated in Figure 1. Simplifying (3.10) and introducing a rescaled holographic co-ordinate y = u u 0 one obtains the following differential equation In the above u 0 indicates the value of u(s) at s = s 0 and the gauge choice x 3 (s, t) = s has been used. The above equation is obtained from evaluating the l.h.s of (3.10) at the turning point. From the equation obtained one can estimate the separation length (2L) of the qq dipole by integration both sides It is worthwhile to point out that if one tries to take u B → ∞ in (3.12) the dipole length diverges due to the second integral (which is absent in the commutative counterpart where θ = 0). However unlike the generic quark-antiquark potential calculation where a divergence is attributed to the self energy of infintely massive quarks, the present situation cannot be remedied by such arguments. The fact that the holographic dual of a NCYM does not live at radial infinity has been reported in [29] where it has been shown a slight perturbation on the string profile at infinity destabalises it completly. An alternative has been advocated in [36] where the string profile is allowed to have a velocity in the x 2 and the B µν term in the string action contributes to the interquark length unlike the present case. It can be shown that for a certain velocity of the qq pair in the tranverse direction the dipole can be consistently taken to radial infinity. As indicated before we avoid such a configuration for the present case.
The mass of the fundamental matter (qq pairs) is given by the self energy of a stretched string from the probe to the interior [37]. For detemining the same the relevant gauge is x 0 = t, u = s, x 3 =constant. Thus the mass is given by The dipole separation length of the test particles (3.12) can be analytically integrated and in terms of the parameter a = u 0 u B one has 14) The sum of the potential and the static energy of the qq pairs is given by the (time averaged) onshell value of the Nambu Goto action which by the virture of (3.9),(3.12) is From the above one can look at the limit when L → ∞ which is the same as taking the limit a → 0. This fact is evident from the above expression. This is the situation when the string steches to the interior i.e. u 0 → 0 . Using the relation 2 F 1 (− 1 4 , 1 2 , 3 4 , 0) = 1 the leading dependence of the interquark length (3.14) is given by L = 1 2m λ π Γ(3/4) a Γ(1/4) , and the interquark potential (3.15) becomes The the usual Coulomb law is recovered at large distances . However for arbitary speration there is a rapid modification from the Coulomb law. This can be attributed due to two reasons. Firstly as is evident from (3.14) and (3.15) for arbitary values of the parameter a, the noncommutative effects creep in which breaks the conformal symmetry and hence coulombic dependence. Secondly as found in [31], even for a commutative theory the potential profile is altered from and is finite even at short distances. This is becuase in presence of a mass term the theory is not conformal anymore as can be understood from the presence of (2.5) which is coupled the the usual SU(N) action. To get a better view of the same we look at a →1 limit of (3.15) and (3.14). It is evident from the integrals that both of them vanishes in the above said limit. Looking at the limiting values one has, In the above the O(1 − a) 2 terms have been neglected. Similarly for the interquark distance one has, , it can be easily seen that for short interquark separation .
Thus increment in the value of noncommutaive parameter θ results into decrement of the the slope of the potential curve at small separation. This is because of repulsive forces due to noncommutativity, signalling the force needed to detach the noncommutative qq pair should be smaller than its commutative counterpart.
Till now we have calculated the interquark potential. However in presence of and external electric field the charged qq pairs develop an electrostatic potential of their own. The total potential is given by the sum of the two. Defining the effective potential to be , It can be guessed from (3.19) that in the presence of an external electric field of strength, the qq pair overcomes linear barrier of the potential profile in Figure 2. However it is still to be seen whether the above mentioned electric field is enough to get out of the tunneling phase for all values of interquark separation L. One has from (3.12) and (3.15), In the above we have reinstated the turning point u 0 and have introduced the ratio r = E E T for simplicity where E T , the threshold electric field is given by (3.21). It is apparent from (3.22) , at the value r = 1 (applied field being of threshold value) the first term vanishes. Thus the potential profile is governed fully by the second term G (L(u 0 )) , and will cease to put up a tunneling barrier if the function is monotonically decreasing and is vanishing at the origin. At L = 0 which is realised if u 0 = u B it is evident from (3.22) that G (L = 0) = 0. Indeed this is the case for U ef f ective (L = 0) too.
Moreover one can show from (3.22) that, The first term comes when the differential operator acts on the lower limit of integration in (3.22). A regulator ε whose physical meaning is rather vague has been put in the expression as the first term is actually divergent. By a similar procedure one has .
Though the above two terms are actually divergent one can see from (3.24) and (3.25) that their ratio isn't.
It is easily seen that G(L(u 0 )) is a monotonically decreasing function w.r.t L(u 0 ) i.e for u B ≥ u 0 . From the above it is clear that the net potential/force due to the applied field has two components , (a) (1 − r)E T L(u 0 ) : This is the part which creates the potential barrier for r < 1 i.e the attractive force between the qq pairs in an external electric field. At r = 1 this part ceases to contribute and for r > 1 the force corresponding to this part becomes repulsive.
(b) G (L(u 0 )) : This part contributes to bringing down the potential barrier. As can be seen from (3.26), the associated force due to this is repulsive for all values of L(u 0 ) except at the origin where it vanishes. Thus at threshold point (r = 1) where the first component (part a) becomes irrelevant the slope of the net potential for all nonzero values L is negative as can be seen in Figure 4 confirming the prediction of (3.21).
It is also interesting to see whether the effective potential admits a confining phase where the tunelling behaviour is totally absent. This amounts to showing the existence/nonexistence of a intermediate value of u 0 for r < 1 where the total potential as in (3.20), (3.22) vanishes. Alternatively one can check the values of the slope of the potential at extreme points and look for a sadle point of the same. It is easy to see that at L = 0/u 0 = u B the slope (3.23) is given by (1 − r)E T which is positive for r < 1. However at u 0 = 0/L → ∞ the slope of the potential is given by, Thus we see that the slope of the potential curve is negative (force between qq pairs is repulsive) at large distances for all values of the applied electric field unlike the situations in [38] [39], indicating the present case of not being configning. It is also clear from (3.23) and (3.26) that the maximal potential barrier is encountered when, This is the point when the effective qq potential admits a phase transition i.e. the potential becomes repulsive rather than being attractive. It can be seen from the blue line in Figure 4 that the maximal height of the effective potential barrier decreases at electric fields close to its threshold value, a fact which can be attributed to the replusive forces due to noncommutativity. However lengthscale (interquark separation) associated with the same is increased since L(a) is a decreasing function of the parameter a (atleast for low values of noncommutative parameter) as in Figure 3.

Potential Analysis at Finite Temperature
The finite temperature case closely resembles the above calculation. For the present situation the gravity dual is given by (3.4). The thermal mass of fundamental qq pairs is given by Very much like the above analysis the Nambu Goto action in static gauge reduces to Quite similar to the previous case the conserved quatity arising from the Lagrangian (3.30) is Demanding the profile admits a zero slope at the turning point u 0 (the turning point should be greater than the horizon radius i.e. u 0 ≥ πT ), we have From the above equation the separation length between test particles can be integrated out to be The above equation is written in terms of redefined variables : y = u u 0 ; a = u 0 u B , the parameter m is the mass at zero temperature. The intequark potential at finite temperature for noncommutative theories is obtained from (3.30) and (3.33) to be, It is not possible to integrate the above two equations analytically. However due to the presence of a finite temperature the interquark potential ceases to be coulombic even at large intequark separation which can be explicitly checked by computing (3.34) for small temperature using binomial approximation. This phenomenon can be attributed to the breakdown of conformal symmetry (for the commutative case) in finite temperature.
In presence of an external electric field E, the effective potential experienced by the qq pairs gets modified to In the first step we have added and subtracted the term "E th · L T " where E th , the threshold electric field at finite temperature is to be found out. The slope of the potential profile (3.35) for fixed values of the physical parameters is given by, At the thershold point where R = 1, the first term vanishes and one is left with the second term alone which itself consists of the thershold value (3.35). However at the threshold point the slope the potential should be negative for all allowed values of the parameter a (and henceforth the separation L T ). An explicit calculation leads to Similarly (after quite some algebraic manupulations) one finds , It is clear from (3.33) that as the parameter a → 1 the intequark separation L T → 0 . Moreover at a = 1 the effective potential (3.35) vanishes too. At L T = 0 (a = 1) the effective force (3.39) on the qq pairs should be zero at the threshold condition. Thus, Thus we see that effect of finite temperature is to decrease the threshold electric field. However noncommutative effects donot mix up with influence of finite temperature. Similar inference can be drawn from studying the quasinormal modes of scalar perturbations in presence of noncommutativity as in [40] 6 which shows enhancement of the dissipation rate in accordance to decreament of threshold field. From (3.40) and (3.39) we have, The monotonicity of the second term w.r.t. is quite clear in the allowed range of a . Thus the value of E th so found suffices to cause vacaum decay at the threshold point (R=1).

Pair Production Rate of Noncommutative Schwinger Effect
In this section we would like to estimate the rate of production of qq pairs interacting with NCYM in presence of an external electric field along a noncommutative direction. As indicated in (2.8) the production rate is proportional to the Wilson loop of the classical eucledian trajectory of particles under the presence of the electric field i.e. a cirlce in the x 0 − x 3 plane 7 . An explicit solution of the circular string profile for the gravity dual of N = 4 SYM is given in [41] and in [32]. Here we state the same for later purposes.
The solution holds true in the conformal gauge of the Polyakov action which is equivalent to the Nambu Goto action at the classical level. In the above R indicates the radius of the Wilson loop on the probe brane. The parameter s in one of the co-ordinate of the 2 dimensional string worldsheet and its value on the probe brane is given by s 0 , t parametrises the circular contour on the probe brane and thus has range [0, 2π]. Moreover one can obtain the relation, sinh s 0 = 1 Ru B which connects the allowed range of the worldsheet parameter to the physical quantities like mass and external electric field. For the present purpose the relevant gravity dual is given by (3.3). The Polakov action in conformal gauge looks Both the worldsheet and target spacetime have been continued to eucledian signature. We have redefined α ≡ λθ 2 and have negelected the terms involving X 1,2 . The equations of motion correspoding to (4.2) are given by These equations are to be supplemented with the condition X 2 0 (t, s 0 ) + X 2 3 (t, s 0 ) = R 2 . The set of equations in (4.3)-(4.5) form a system of coupled second order non-linear differential equations and in general is impossible to solve. In the context of Gauge/String duality, solution of Wilson loops in general background has been an perplexing issue . A certain way has been suggested in [42] based on employing a "circular ansatz", but it can be checked that such methods are valid only if the background has SO(2) isometries in the plane of the Wilson loop. However if relevant the background is a continuous parametric deformation of AdS one can describe the string profile as X µ (t, s; σ i ) where σ i collectively indicates the 7 Under eucledian continuation the distinction between electric and magnetic field vanishes. The classical eucledian trajectory is the well known cyclotron trajectory in contant magnetic fields. The fact that pair production rate is given by "circular" wilson loops holds true only in eucledian signature. deformation parameters. Expanding in power series, X µ (t, s; σ i ) = X µ (t, s; σ i = 0) − σ i · ∂ σ i X µ (t, s; σ i = 0) + O(σ 2 i ) and noting that X µ (t, s; σ i = 0) is the known AdS solution the nonlinear equations become simplified. In the present context the deformation parameter is σ ≡ α = λθ 2 . 8 Using the expansion X 0 (t, s) = K cos t sech s − αχ 0 (t, s) One obtains the following equations at order O(α) − 2 coth 4 s sech s (sin t ∂ t χ 0 + cos t tanh s ∂ s χ 0 + sin t tanh s ∂ s χ 3 − cos t ∂ t χ 3 ) (4.9) In deriving the above, (4.3)-(4.5) has been linearised using 9 1 1+αU 4 ≈ 1 − αU 4 and then (4.6) has been used keeping in mind that terms of O(α 0 ) are AdS equations which are automatically zero. Moreover we have assumed αU 4 (t, s) ≈ α K 4 coth 4 s upto first order in α. Equations (4.7)-(4.9) though being simplied than before are still daunting. Using the ansatz ξ(t, s) = ξ(s), χ 0 (t, s) = χ 0 (s) cos t, χ 3 (t, s) = χ 3 (s) sin t in (4.7)and (4.8) one has ∂ s χ 0 = 1 2 sinh 2 s coth s (∂ 2 s χ 0 − χ 0 ) − ξ sinh 2 s sech 3 s − sech s tanh s sinh 2 s ∂ s ξ (4.10) Since the set (4.7)-(4.9) are coupled differential equations the solution of the first two has to satisfy the other one. Substituing (4.10), (4.11) in (4.9) and noting that the reuslting equation has to be satisfied for all values of parameter t one obtains the following three equations tanh s sinh s ∂ 2 s χ 0 − 2 sech s χ 3 − tanh s sinh s χ 0 + 3 K 4 coth 2 s csch 2 s = 0 (4.12) coth s ∂ 2 s ξ − 2ξ coth s csch 2 s (1 + 3 tanh 2 s) + 2 (csch 2 s − 1) ∂ s ξ = 0 (4.14) It can be checked that (4.14) has no real solution, further more (4.14) being a linear equation permits a solution of the form ξ = 0. Thus we are left with the first two equations (4.12),(4.13) which are coupled differential equations themselves. To simplify those we define the following variables whose significance is rather obscure.
In terms of the above one has Digressing a bit from the main discussion let us see the first order correction to the onshell action in light of the perturbation theory set up. From the decomposition (4.6) one has upto O(α) (∂ s X 0 ) 2 = K 2 cos 2 t (sech 2 s tanh 2 s + 2α sech s tanh s ∂ s χ 0 ) (4.20) (∂ s X 3 ) 2 = K 2 sin 2 t (sech 2 s tanh 2 s + 2α sech s tanh s ∂ s χ 3 ) (4.21) Using the above in the Polyakov action in presence of the NC dual (4.2) and approximating 1 1+ α K 4 coth 4 s ≈ 1 − α K 4 coth 4 s one has in the first order of the effective noncommutative parameter.
Using the equations (4.12),(4.13) and the fact that ξ(s) = 0 one can reduce the first order correction to above expression to δ α L onshell = −α cosh s sin 2 t 2 csch 2 s χ 0 + 3 K 4 coth 2 s sech s csch 2 s − ∂ 2 s χ 3 + χ 3 + cos 2 t 2 csch 2 s χ 3 − 1 K 4 coth 4 s sech s csch 2 s − ∂ 2 s χ 0 + χ 0 = − 2α K 4 coth 2 csch 2 s 1 + csch 2 s cos 2 t (4.23) In the last line (4.12),(4.13) have been put to use. Thus we see that the equations of motions alone determine the first order correction of the onshell action from the commutative counterpart. In the context of holographic entaglement entropy similar methods have been presented in [42]. For finding out the limit of the integration and its connection to physical variables one has to solve the equations of motion. Returning to our main discussion, the real part of the solution of (4.16) is The equation (4.17) which dictates the deviation of the circular symmetry cannot be solved by analytical means. It is worthwhile to note that (4.17) is not a linear equation and does not admit a solution χ − (s) = 0. However since it is a second order differential equation it certainly admits a solution with χ − (s = s 0 ) = 0 for a specific value s 0 . 10 Thus at the point given by s = s 0 the profile is circular and the variable χ + is twice the radius of the loop(R). Putting the above in mathematical language From the above one gets, Ru B = csch s 0 1 + α 12 u 4 B . This relation serves to define the integration limits and also connects the onshell value of the action to physical parameters. From the above one has in first order of the noncommutative parameter, coth s 0 = In presence of an external electric field in the x 3 direction the on shell value of the action gets modified to 11 In the above we replaced the contants by hyperbolic functions (4.25) and have defined (1 + 5 2 η) 2 E. Note that the dependence on the radius (R) is now encoded in the hyperbolic functions themselves. Quite similar to arguments in [32], [44] at large value of R ( large E) the production rate (2.8),(3.5),(4.26) is dominated by Γ ∼ exp √ λR 2 E similar to the phase (in potential analysis) when V sch does not permit a tunneling barrier. However for small R (πR 2 E not dominating the other terms) the approximate production rate is 10 There exist no known methods to solve a generic second order partial differential equation. We donot claim that the first derivative of χ− is zero at s0 11 U(1) gauge fields contribute to the string Lagrangian via a boundary term. For constant electromagnetic field the string equations of motion are unchanged but the boundary conditions are altered (Robin). For inhomogenous fields this is not the case. Schwinger Effect for inhomogenous fields have been explored via holographic methods in [43] It is evident that as R varies one moves from a tunneling or damped production phase (when the first term in (4.27) dominates) to a spontaneous production phase (when the second term dominates). This is quite synonymous to the potential analysis with the identification Γ ∼ exp(−V ef f ective ). The potential barrier vanishes when both terms in (4.27) cancells each other which happens at π 4 m 4 θ 2 λ (4.28) To get the production rate one needs to extremise the onshell action with respect to R for reasons mentioned before. Instead of extremising w.r.t. R one can extremise w.r.t. csch s 0 , however doing so one is left to solve a sextic equation. To simplify the situation note that the value of coth s 0 is proportional to the mass of the quark (W boson) via the relation derived earlier . Thus for heavy mass, (actually λθ 2 m 4 m 6 1) the contribution of the second and third term of (4.26) is negligibe. Under those circustances one has Extremising the above w.r.t. csch s 0 (i.e. R) one is lead to coth s 0 = 1−3η 2E which is a condition connecting R and E. From the relation thus obtained the on shell value of the action is Thus the production rate which is proportional to the (negative) exponential of the onshell action (2.8),(3.5),(4.30) is given by Where we have restored the physical parameters via the relation η = 8π 4 m 4 θ 2 15λ . In the above the threshold electric field is given by π 4 m 4 θ 2 λ (4.32) At low electric field , E E T , the second term in (4.31) ceases to contribute and one is left with One can compare (4.33) to the result of [9], and both of them show the same pattern with the identification of − → B ∼ θ where − → B indicates an external magnetic field in [9]. As indicated in the introduction in presence of strong magnetic fields commutativity is lost and the theory is described by noncommutaive physics. A reason for concern may be the extra ( θ dependent) term in (4.31) and (4.33) which is independent of the electric field. We belive this is an effect of our simplification of solving a quadratic equation instead of a sextic one (see above).

Conclusions
In this paper we have performed an interquark potential analysis to find the effective potential barrier in presence of an external electric field in noncommutative gauge theory. From the same we have shown that the threshold electric field is decreased from it's commutative counterpart. In presence of noncommutativity there exist strong repulsive forces between the particles at short distances i.e. the coulombic interaction developes a short distance repulsive correction. This implies the electrostatic potentatial energy needed to tear out the virtual particles is less than usual explaning the result found. We have also argued that noncommutative does not lead to confinement as at large distances the behaviour of the potential is essentially coulombic as demonstrated. We also have found out the thermal corrections to the above and have seen that finite temperature effects don't entangle with (space-space) noncommutative ones as expected. We have also perturbatively computed the corrections to circular wilson loop over the known commutative result in the first order of the noncommutative deformation parameter, and hence the decay rate has been found out from which the decrement of the threshold value is also clear. At low electric field our result shares the same pattern with that of Chair and Sheikh-Jabbari for noncommutative U(1) gauge theory.