We consider the phenomenon of spontaneous pair production in the 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 production rate of quark/antiquark pairs to the first order of the noncommutative deformation parameter. Our result bears resemblance with an earlier related work (based on field-theoretic methods).

1. Introduction

Quantum field theory is primarily studied in its perturbative regime. However, there exist quite some novel nonperturbative features of quantum field theories amongst which the Schwinger effect [1] stands its ground (for a recent review, see [2]). The vacuum of quantum electrodynamics is a bath of virtual pairs which gets created and annihilated instantaneously. However, in the presence of an external electric field, the pairs spontaneously become real and their production rate in the weak-coupling weak field approximation is given by [1].

This expression holds for weak coupling and weak electric fields only. The exponential suppression hints that the pair production process can be modeled as quantum mechanical tunneling in a certain potential barrier. For the electron-positron pairs to become real, they should gain at least an energy equal to sum of their rest masses (). However, in the presence of an external electric field, the virtual particles gather an energy of via electromagnetic interactions, being their separation distance. To understand the situation better, let us assume that the positron is located at distance and the electron at distance from the origin along the direction parallel to the electric field. To become physical particles, the electron has to climb through a potential barrier of height (same for the positron) by gaining energy from the external electric field. If the virtual pairs are separated by a distance such that then the electron (and the positron) becomes a real particle as it now has the required rest mass energy. This value is the width of the potential barrier. Thus, the transmission coefficient is approximately . This is exactly the content of the Schwinger formula (1).

Motivated by this analogy, let us look from the perspective of a “virtual” dipole. When the “virtual” dipole has a separation , the total effective potential barrier they encounter can be estimated to be of the form

In this picture, the virtual particles become real by tunneling through the above said potential barrier. The first two terms indicate the self-energy including the Coulombic interaction between pairs. For small separation, the Coulomb term dominates the expression and the potential is negative. At large values of , the effect of electric field takes dominance making the potential negative too. For small electric field, i.e., , there exist two zero points of the potential profile and the potential is positive in intermediate regimes of separation . In this case, the particles become real by tunneling through this barrier and the production rate is exponentially suppressed as described by the Schwinger formula. However, for electric fields , the potential becomes negative all along and ceases to put up a barrier, indicating a catastrophic instability of vacuum where the are spontaneously produced. The value of electric field for which the potential changes its character is called the “threshold electric field” .

The idea of noncommutative quantum field theories [3, 4] where spacetime position viewed as operators itself ceases to commute was originally proposed to curb the UV divergences appearing in interacting quantum field theories. The idea received revival when some noncommutative field theories were found to be effective low energy limit of open string theories on a Dp brane with a constant NS-NS two-form , the noncommutative feature being a dynamical consequence of quantization [5, 6]. However, the low energy limit of these string theories turns out to be quantum field theories defined on a “noncommutative” spacetime. There exist noncommutative generalizations of Riemannian geometry [7] on which the standard model can be defined, wherein the parameters of the theory are interpreted as geometric invariants. Mathematically, this amounts to abandoning the smooth structure of spacetime in favor of a space equipped with a noncommutative algebra of real-valued functions much like the transition from classical to quantum physics via phase-space methods. The transition from commutative theories to its noncommutative counterpart along the lines stated above is achieved via replacing the ordinary product between functions by the Moyal/star product .

The above equation implies , signifying nonvanishing commutation relations between spacetime coordinates (viewed as operators) itself. Noncommutative quantum field theories are generically Lorentz violating (due to the presence of a noncommutative parameter ), the effects of which are small to be detected in practical experiments with current experimental bound on the noncommutative parameter to be around by conservative estimates [8].

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) correspondence [10] which links a strongly coupled gauge theory to classical gravity is an important tool in these kinds of scenarios. The Schwinger mechanism has been argued in the holographic setting in [11]. It has also been shown via holographic methods [12] that the Schwinger effect in a large N confining gauge theory admits a “new kind” of critical electric field (apart from the usual threshold value). If the external electric field is less than the confining string tension , the pair production is prohibited as the effective potential barrier remains positive instead of dumping out at large separation. However, when the electric field is larger than , pair production is allowed as a tunneling process. Thus, at this value () of electric field, a confinement/deconfinement transition happens.

In this paper, we like to study the holographic Schwinger effect for “quarks” coupled to a large noncommutative gauge theory in the presence of external gauge field. The large noncommutative gauge theory (NCYM) is realized by its relevant holographic dual geometry (to be mentioned later) while the quark is modeled as massive strings extending from the interior to a large but finite position in the holographic direction so that the mass of the quarks is not infinite. We evaluate the interquark potential (both numerically and analytically when possible) for the large NCYM from the rectangular Wilson loop by calculating the extremal area of a string worldsheet ending along a rectangular contour at the boundary. The effective potential for describing the Schwinger effect is then calculated by introducing an external electric field. We analytically find out the value of the external electric field for which the effective potential puts up a tunneling behavior, i.e., the threshold electric field. We repeat the same calculation for finite temperature NCYM and observe that the thermal contribution to the threshold electric field does not mix up the noncommutative ones. We proceed to find out the decay rate by finding out the on-shell value of the Polyakov action coupled to an external electric field for a string with a circular contour at the boundary. We use perturbations over the known result of circular holographic Wilson loop of ordinary YM and hence find out the correction of the Schwinger decay rate [11] up to the first order of the noncommutative deformation parameter.

This paper is organized as follows. In Section 2, we review the derivation of the Schwinger effect in a superconformal gauge theory by relating the same to the expectation value of the circular Wilson loop. Also, in the same section, the basics of noncommutativity in string theory are 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 threshold electric field is found out by analytical means. In Section 4, we compute the first order noncommutative correction to the pair production rate of Schwinger particles. We close this paper by conclusions in Section 5.

2. Pair Production and Noncommutativity

2.1. Pair Production in SYM and the Wilson Loop

In its most prominent avatar [10, 13], the holographic duality relates super Yang-Mills theory to type IIB string theory in . To study the Schwinger effect, one has to account for “massive” matter (corresponding to a probe brane) in fundamental representation and a gauge field. The way to do so is to break the symmetry group of the problem from to with the Higgs mechanism. Such methods were first introduced in [14]; the following closely follows [15, 16]. For more sophisticated treatment, refer to [17, 18]. The bosonic part of SYM for the theory in Euclidean signature reads as where is the field strength of the gauge field . collectively denotes six scalars in the adjoint representation of . The gauge group is broken as

The nondiagonal parts and transform in the fundamental representation of and form the so-called -boson multiplet. The VEV of the scalar fields is supposed to be of the form

As a result of the decomposition (5), the action (4) breaks up into three parts of the following form [14]: where is basically the free QED action constructed out of the gauge field . governs the dynamics of the bosons and its coupling to the gauge fields. Disregarding the ’s (The s start coupling to the via . The effect of these couplings to the vacuum energy density of the is damped by at least. Thus, in the large limit, the are neglected for the present study.) and higher-order terms the boson action reads where is equipped both with the gauge field and also with the gauge field , i.e., . By expanding the action and choosing , the mass term for vanishes while those for , , remain. For the present scenario, the gauge field is a dynamical field and the gauge field is a “fixed external” field of the form . The external gauge field contributes to the vacuum energy density via the covariant derivative in as mentioned above. The pair production rate is given by the imaginary part of vacuum energy density [19].

The factor of comes from the number of ’s. By using Schwinger’s parametrization and worldline techniques [20], one can express the pair production rate (9) as a path integral for a particle subject to an appropriate Hamiltonian under boundary conditions,

Using saddle point approximations as in [15, 16] and assuming the mass to be heavy, i.e., , (10) becomes proportional to the path integral of the particle subjected to the “external” gauge field times a phase factor, namely, the Wilson loop.

In the above, indicates the path ordering of the time parameter and the nonabelian index is handled with matrix trace, . Evaluating (11) by the method of steepest descent, the “classical” trajectory becomes a circle. So the production rate is proportional to the expectation value of the circular Wilson loop and can be computed via holographic conjecture in the large limit.

From expression (11), it naively seems that the pair production rate is nonzero in general implying the vacuum energy (of an gauge theory) density to have an imaginary part even in the absence of an external electric field. However, this is not the case. The expression in front of the Wilson loop in (11) can be evaluated by steepest descent methods and leads to a multiplicative factor of . Thus, the pair production rate (and the imaginary part of vacuum energy density) indeed goes to zero when the external electric field vanishes. A detailed calculation of the same is presented in [15, 17].

2.2. Noncommutativity from String Theory

The effective worldsheet action in the presence of field is given by

The equations of motions along with the boundary conditions when are

The boundary conditions (15) are neither Neumann nor Dirichlet; one can indeed try to diagonalize (15) to be Neumann-like by redefining the fundamental variables leading to the so-called “open string metric” [6]. However, a more direct attack along the lines of [21] is to solve the equation of motion (14) first and constrain the solution by (15). For , the solution of (15) compatible with (14) is

A similar solution accompanies , the forms of which encode the nontrivial boundary conditions. The solutions for the other coordinates are the usual ones [22]. The canonical momentum of the action (14) is by the virtue of mode expansion (16)

The fact that the current scenario leads to noncommutativity was first recognized in [23, 24] as the canonical momentum at the ends of the string becomes functions of the spatial derivatives of the string coordinates as per (15) and (17). From the symplectic -form, the canonical commutation relation of the modes is

From the mode expansion (16) and the relations (18)–(20), one has

The second term in (21) sums up to zero when . Therefore, the end points of the string become noncommutative. The nontrivial part of the normal ordered Virasoro constraints and the total momentum which accompanies (13) are given by

It is clear from the above that the mass of the particle becomes dependent on the value of the field. However, the noncommutative field theories constructed out of Moyal product leave the mass of the particle (quadratic part of the Lagrangian) unchanged. Since the string equations of motion and boundary conditions are linear equations, one can redefine the operators to be in terms of which the mass of the theory remains unaltered [25].

It terms of which the only nontrivial commutation relation becomes

It has been checked that perturbative string theory in the present backdrop corresponds to noncommutative Yang-Mills at one loop. For further details, see [25] and references there-within.

3. Potential Analysis of the Noncommutative Schwinger Effect

We start with a brief description of the holographic dual of noncommutative Yang-Mills (NCYM) [26, 27]. In the spirit of AdS/CFT correspondence, one looks for supergravity solutions with a nonzero asymptotic value of the field. Such a solution is the D1-D3 solution which in the string frame looks like

The solution (25) is asymptotically flat and represents D1 branes dissolved per unit covolume of D3 branes. The information of the D1 branes is stored in the relation . It can also be seen that the asymptotic value of the field is while is related to the other parameters via .

The proper decoupling limit of the above-stated solution resembles the field-theoretic limit of the noncommutative open string [6, 25], for which the asymptotic value of has to be scaled to infinity in a certain way.

With the above scaling and keeping , , , fixed the resulting metric and field configurations are given by

The above is the holographic dual to NCYM with gauge group and Yang-Mills coupling constant which captures the dynamics of NCYM. As the holographic direction tends to infinity, the first two directions scale as and the directions develop a dependence in the metric. Due to the noncommutativity, the symmetry group of the theory becomes as can be clearly seen from the isometry of (27) [28]. The gravity dual to NCYM at finite temperature is found from the near horizon limit of the black D1-D3 solution and reads

The most rigorous approach to study the Schwinger effect in the holographic setting is to find the expectation value of the circular Wilson loop and relate it to the pair production rate as in [11]. However, one may think of the vacuum to be made of 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 [29]. To compute the interquark potential, one needs to look at the expectation value of the rectangular Wilson loop when the loop contour is regarded as the trajectory of particles under consideration in the plane where is the direction of orientation. As pointed out in [11, 30], one places a probe D brane at a finite position instead of the boundary to get a boson of finite mass. A string with Dirichlet conditions at both ends has the following canonical Hamiltonian [22], where the first term indicates the potential energy of the stretched string and is the analogue of mass created due to symmetry breaking.

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 (12), i.e., where indicates the action of the fundamental string (The boundary conditions are given by the trajectory of the string at the probe brane, . Stated more explicitly, where . The functions vanishes at the boundary which is given by a specific value , being the parametric representation of the contour of the Wilson loop.) which has been Wick rotated to Euclidean signature, as is usually done in string theory [22]. In the classical limit which is realized when the string length is small (or the ‘t Hooft coupling is big), the above expression is dominated by the on shell value of the Polyakov/Nambu-Goto action. Thus, the prescription of computing Wilson loops is reduced to computing the area of the world sheet of the fundamental string which ends on the specified profile at the probe D brane [14, 31, 32], situated at a finite radial position in the dual geometry for the present case.

3.1. Potential Analysis at Zero Temperature

In this section, we will study the properties of the modified potential of NCYM in the presence on a constant external electric field for quark/antiquark pairs along one of the noncommutative directions (). The appropriate holographic dual as pointed above is given by (27). To calculate the extremal area of the string configuration ending of the probe brane, we take the string world sheet to be parametrized by . The Nambu-Goto action reads

In the above, is the induced metric on the worldsheet and is the metric of the target spacetime/holographic dual (27) and (28). The above action exhibits two diffeomorphism symmetries with the help of which one can set two of the embedding functions to arbitrary 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 (Strictly speaking, the embedding function which extremizes the Nambu-Goto function may not respect such a gauge choice globally and may lead us to a local minimum of the Nambu-Goto action. One can hope the results found will converge to the true value if perturbations are added.)

In the above, the extremization of the Nambu-Goto action is given by the functional form of . The are the coordinates of . For present purposes, is assumed to range between , when indicates the interquark separation on the probe brane with the boundary condition where indicates the position of the probe brane along the holographic direction. Again, the temporal direction is assumed to range from with the further assumption that , meaning time scale of the problem (within which the quarks/antiquark pairs remain separated) to be much larger than the length scale. This is because the rectangular Wilson loop gives the interquark potential when one assumes the interaction between dipole is adiabatically switched on and off as illustrated in Figure 1 (Intuitively, one thinks, two quarks to be separated at distant past and then reunite at distant future. Thus, the worldline becomes a rectangular closed contour whose area can be identified with the interquark potential .).

Before proceeding further, let us address the issue of the field. For noncommutative gauge theories, the field is excited (27) and (28) and is present in the string action via the Wess-Zumino term along with the usual Polyakov/Nambu-Goto part. However, the gauge choice given above (32) cancels the contribution of the Wess-Zumino part of the action. It is possible to consider the pairs at a velocity in the direction and take into account the contribution of the term as in [33]. However, in the present case where the virtual particles in vacuum are modeled as 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 (31), we get

Extremization of the above Lagrangian is equivalent to solving the Euler-Lagrange equation with the effective Lagrangian when along with the boundary condition as the contour profile is already taken into account by the gauge choice. One can indeed solve the relevant problem and find the explicit form of . However, since the Lagrangian in (34) does not explicitly depend on the parameter (In principle, one considers a profile where as the two diffeomorphism symmetries are exhausted by (32). However, such a choice would reflect nonadiabatic interactions, i.e., the interpretation of Wilson loops of measuring the interaction potential of pairs at rest as in Figure 1 would be invalid.), by Noether’s theorem, there exists a conserved quantity () for the solution.

As indicated in [32], the fundamental string is assumed to be carrying charges at its two ends and is otherwise symmetric about its origin. Thus, one can choose to be an even function of . In the present case, this means the direction of NCYM is symmetric around its origin which holds true in spite of its noncommutative nature. From the form of (35), the solution of involves both positive and negative signs. Thus, there exist a value of the parameter for which . This is the turning point of the string profile as indicated in Figure 1. Simplifying (35) and introducing a rescaled holographic coordinate , one obtains the following differential equation:

In the above, indicates the value of at and the gauge choice has been used. The above equation is obtained from evaluating the l.h.s of (35) at the turning point. From the equation obtained, one can estimate the separation length () of the dipole by integration both sides

It is worthwhile to point out that if one tries to take in (37), the dipole length diverges due to the second integral (which is absent in the commutative counterpart where ). However, unlike the generic quark/antiquark potential calculation where a divergence is attributed to the self-energy of infinitely 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 [26] where it has been shown a slight perturbation on the string profile at infinity destabilizes it completely. An alternative has been advocated in [33] where the string profile is allowed to have a velocity in the and the 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 pair in the transverse 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 ( pairs) is given by the self-energy of a stretched string from the probe to the interior [34]. For determining the same, the relevant gauge is , , . Thus, the mass is given by

The dipole separation length of the test particles (37) can be analytically integrated, and in terms of the parameter , one has

The sum of the potential and the static energy of the pairs is given by the (time averaged) on-shell value of the Nambu-Goto action which by the virtue of (34) and (37) is

From the above, one can look at the limit when which is the same as taking the limit . This fact is evident from the above expression, see Figure 2. This is the situation when the string stretches to the interior, i.e., . Using the relation , the leading dependence of the interquark length (39) is given by , and the interquark potential (40) becomes

Thus, the usual Coulomb law is recovered at large distances. However, for arbitrary separation, there is a rapid modification from the Coulomb law. This can be attributed due to two reasons. Firstly, as is evident from (39) and (40) for arbitrary values of the parameter , the noncommutative effects creep in which breaks the conformal symmetry and hence Coulombic dependence. Secondly, as found in [29], even for a commutative theory, the potential profile is altered from and is finite even at short distances. This is because in the presence of a mass term, the theory is not conformal anymore as can be understood from the presence of (8) which is coupled to the usual action. To get a better view of the same, we look at a limit of (40) and (39). 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 terms have been neglected. Similarly for the interquark distance, one has

Comparing (42) and (43) and using the identity , it can be easily seen that for short interquark separation,

Thus, increment in the value of noncommutative parameter results into decrement of the slope of the potential curve at small separation. This is because of repulsive forces due to noncommutativity, signaling the force needed to detach the noncommutative pair should be smaller than its commutative counterpart.

Till now, we have calculated the interquark potential. However, in the presence of an external electric field, the charged 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 (44) that in the presence of an external electric field of strength, the pair overcomes the linear barrier of the potential profile in Figure 3. 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 . One has from (37) and (40),

In the above, we have reinstated the turning point and have introduced the ratio for simplicity where the threshold electric field is given by (46). It is apparent from (47), at the value (applied field being of threshold value), the first term vanishes. Thus, the potential profile is governed fully by the second term and will cease to put up a tunneling barrier if the function is monotonically decreasing and is vanishing at the origin. At which is realized if , it is evident from (47) that . Indeed, this is the case for too.

Moreover, one can show from (47) that

The first term comes when the differential operator acts on the lower limit of integration in (47). A regulator (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 (49) and (50) that their ratio is not.

It is easily seen that is a monotonically decreasing function w.r.t , i.e., for . From the above, it is clear that the net potential/force due to the applied field has two components as follows: (a): this is the part which creates the potential barrier for , i.e., the attractive force between the pairs in an external electric field. At , this part ceases to contribute, and for , the force corresponding to this part becomes repulsive(b): this part contributes to bringing down the potential barrier. As can be seen from (51), the associated force due to this is repulsive for all values of except at the origin where it vanishes. Thus, at threshold point () where the first component (part a) becomes irrelevant, the slope of the net potential for all nonzero values is negative as can be seen in Figure 4 confirming the prediction of (46)

It is also interesting to see whether the effective potential admits a confining phase where the tunneling behavior is totally absent. This amounts to showing the existence of an intermediate value of for where the total potential as in (45) and (47) vanishes. Alternatively, one can check the values of the slope of the potential at extreme points and look for a saddle point of the same. It is easy to see that at , the slope (48) is given by which is positive for . However, at , the slope of the potential is given by

Thus, we see that the slope of the potential curve is negative (force between pairs is repulsive) at large distances for all values of the applied electric field unlike the situations in [12, 35], indicating the present case of not being confining. It is also clear from (48) and (51) that the maximal potential barrier is encountered when

This is the point when the effective potential becomes repulsive rather than being attractive.

3.2. Potential Analysis at Finite Temperature

The finite temperature case closely resembles the above calculation. For the present situation, the gravity dual is given by (28). The thermal mass of fundamental 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 quality arising from the Lagrangian (55) is

Demanding the profile admits a zero slope at the turning point (the turning point should be greater than the horizon radius, i.e., ), 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: ; the parameter is the mass at zero temperature. The interquark potential at finite temperature for noncommutative theories is obtained from (55) and (58) to be

It is not possible to integrate the above two equations analytically. The separation however due to the presence of a finite temperature, the interquark potential ceases to be Coulombic even at large interquark separation which can be explicitly checked by computing (59) 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 the presence of an external electric field , the effective potential experienced by the pairs gets modified to

In the first step, we have added and subtracted the term “” where the threshold electric field at finite temperature is to be found out. The slope of the potential profile (60) for fixed values of the physical parameters is given by

At the threshold point where , the first term vanishes and one is left with the second term alone which itself consists of the threshold value (60). However, at the threshold point, the slope the potential should be negative for all allowed values of the parameter (and henceforth the separation ). An explicit calculation leads to

Similarly (after quite some algebraic manipulations), one finds

It is clear from (58) that as the parameter , the interquark separation . Moreover, at , the effective potential (60) vanishes too. At (), the effective force (64) on the pairs should be zero at the threshold condition. Thus,

Thus, we see that the effect of finite temperature is to decrease the threshold electric field. However, noncommutative effects do not mix up with the influence of finite temperature (in the sense that there are no “” mixed terms in the expression of the threshold electric field). Similar inference can be drawn from studying the quasinormal modes of scalar perturbations in the presence of noncommutativity as in [36] (We thank Juan F. Pedraza for bringing this to our notice.) which shows enhancement of the dissipation rate in accordance to decrement of threshold field. Also, as shown in [37, 38], the NCYM is less viscous than its commutative counterpart. It is of interest to wonder whether viscosity and threshold electric field has anything to do with each other. From (65) and (64), we have

The monotonic nature of the second term w.r.t. the parameter is quite clear in the allowed range of . Thus, the value of so found suffices to cause vacuum decay at the threshold point ().

4. Pair Production Rate of the Noncommutative Schwinger Effect

In this section, we would like to estimate the rate of production of pairs interacting with NCYM in the presence of an external electric field along a noncommutative direction. As indicated in (11), the production rate is proportional to the Wilson loop of the classical Euclidean trajectory of particles under the presence of the electric field, i.e., a circle in the plane. An explicit solution of the circular string profile for the gravity dual of SYM is given in [39] and in [11]. 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, indicates the radius of the Wilson loop on the probe brane. The parameter in one of the coordinates of the 2-dimensional string worldsheet and its value on the probe brane is given by ; parametrizes the circular contour on the probe brane and thus has range . Moreover, one can obtain the relation, , 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 (27). The Polyakov action in conformal gauge looks

Both the worldsheet and target spacetime have been continued to Euclidean signature. We have redefined and have neglected the terms involving . The equations of motion corresponding to (68) are given by

These equations are to be supplemented with the condition . The set of equations in (69)–(71) form a system of coupled second-order nonlinear differential equations and in general is impossible to solve. In the context of gauge/string duality, the solution of Wilson loops in general background has been a perplexing issue. A certain way has been suggested in [40] based on employing a “circular ansatz,” but it can be checked that such methods are valid only if the background has isometries in the plane of the Wilson loop (which in our case, i.e., plane is absent). However, if relevant the background is a continuous parametric deformation of AdS, one can describe the string profile as where collectively indicates the deformation parameters.

Expanding in power series, , and noting that is the known AdS solution, the nonlinear equations become simplified. In the present context, the deformation parameter is . Crudely speaking, this amounts to treating NCYM as a perturbation over YM. Using the expansion,

One obtains the following equations at order :

In deriving the above, (69)–(71) has been linearized using (This is possible because in the present case, we have an upper bound of ; in using the binomial expansion, we have assumed to be small.) , and then, (72) has been used keeping in mind that terms of are AdS equations which are automatically zero. Moreover, we have assumed up to first order in . Equations (73)–(75) though being simplified than before are still daunting. Using the ansatz in (74) and (75), one has