Holographic Pomeron and the Schwinger Mechanism

Holographic Pomeron and the Schwinger Mechanism

Gökçe Başar basar@tonic.physics.sunysb.edu Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA    Dmitri E. Kharzeev Dmitri.Kharzeev@stonybrook.edu Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA    Ho-Ung Yee hyee@tonic.physics.sunysb.edu Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA    Ismail Zahed zahed@tonic.physics.sunysb.edu Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA
July 13, 2019
Abstract

We revisit the problem of dipole-dipole scattering via exchanges of soft Pomerons in the context of holographic QCD. We show that a single closed string exchange contribution to the eikonalized dipole-dipole scattering amplitude yields a Regge behavior of the elastic amplitude; the corresponding slope and intercept are different from previous results obtained by a variational analysis of semi-classical surfaces. We provide a physical interpretation of the semi-classical worldsheets driving the Regge behavior for in terms of worldsheet instantons. The latter describe the Schwinger mechanism for string pair creation by an electric field, where the longitudinal electric field at the origin of this non-perturbative mechanism is induced by the relative rapidity of the scattering dipoles. Our analysis naturally explains the diffusion in the impact parameter space encoded in the Pomeron exchange; in our picture, it is due to the Unruh temperature of accelerated strings under the electric field. We also argue for the existence of a ”micro-fireball” in the middle of the transverse space due to the soft Pomeron exchange, which may be at the origin of the thermal character of multiparticle production in ep/pp collisions. After summing over uncorrelated multi-Pomeron exchanges, we find that the total dipole-dipole cross section obeys the Froissart unitarity bound.

pacs:
12.39.Fe,12.38.Qk,13.40.Em

I Introduction

Near-forward parton-parton and dipole-dipole scattering at high energies is sensitive to the infared aspects of QCD. General QCD arguments show that the resummation of a class of t-channel exchange gluons may account for the reggeized form of the scattering amplitude BFKL (), qualitatively consistent with the observed growth of the scattering amplitude EXPERIMENT (). Nevertheless some empirical features of the hadron-hadron scattering (e.g. the Pomeron slope) point to the importance of non-perturbative effects.

A nonperturbative formulation of high-energy scattering in QCD was originally suggested by Nachtmann NACHTMANN () and others VERLINDE (); KORCHEMSKY () using arguments in Minkowski space. At high energy, the near-forward scattering amplitude can be reduced to a correlation function of two Wilson lines (parton-parton) or Wilson loops (dipole-dipole) in the QCD vacuum. The pertinent correlation was assessed in leading order using a two-dimensional sigma model with conformal symmetry VERLINDE (), and also the anomalous dimension of the cross-singularity between the two Wilson lines KORCHEMSKY (). Both analyses were carried out in Minkowski geometry, with a close relation to QCD perturbation theory.

An Euclidean formulation was used within the stochastic vacuum model through a cumulant expansion in DOSCH () to assess the Wilson loop correlators in Euclidean space. Their phenomenological relevance to proton-proton scatering was pursued in PIRNER (). The instanton vacuum approach to the parton-parton and dipole-dipole scattering amplitudes was used in SHURYAK () to estimate the role of instanton-antinstanton configurations in both the elastic and inelastic amplitudes. In particular, a class of singular gauge configurations reminiscent of QCD sphalerons were shown to be at the origin of the inelasticities. The smallness of the pomeron intercept was shown to follow from the smallness of the instanton packing fraction in the QCD vacuum. The “instanton ladder” has been argued to generate the soft Pomeron both at weak  Kharzeev:1999vh (); KHARZEEV () and strong coupling, through D-instantons Kharzeev:2009pa (). First principle considerations of the Wilson-line correlators in Euclidean lattice gauge theory have now appeared in MEGGIOLARO () which may support the arguments for non-perturbative physics in diffractive processes.

Elastic and inelastic scattering in holography have been addressed initially in the context of the conformally symmetric AdS setting using Minkowskian string surface exchanges between the Wilson-line/loops in the eikonal approximation Rho:1999jm (). This approach was further exploited in Janik:2000aj (); Janik:2000pp (); Janik:2001sc (); Giordano:2011sn () to address the same problem in holographic QCD with confinement WITTEN () for quark-antiquark scattering. In the confined Euclidean background geometry, it was assumed that the most part of string worldsheet stays at the infrared (IR) end point in the holographic direction, so that the problem effectively reduces to the flat space one with an effective string tension at the IR end point. The helicoidal surface was argued as the minimal string surface between two Wilson lines for large impact parameter. The inelasticities (a deviation of the amplitude from being a pure phase) were identified through a multibranch structure in analytic continuation from Euclidean to Minkowski space. However, the physics picture behind this multibranch structure has been somewhat mysterious. One-loop string fluctuations around the helicoidal surface have shown to be important for addressing key aspects of the Pomeron and Reggeon physics such as intercepts.

A more thorough study of the Pomeron problem in the context of holography has been performed in TAN (). Specifically, the Pomeron was argued to follow from a full string amplitude in a curved geometry of holographic QCD, including fluctuations in the holographic radial direction. One of our motivations for the present work is to clarify the relation between the approaches in Rho:1999jm (); Janik:2000aj (); Janik:2000pp (); Janik:2001sc (); Giordano:2011sn () and the one in TAN (), identifying the valid regime of approximations in the analysis of the former.

A compelling picture of the role of the holographic radial direction as one varies was presented in TAN (). As , where denotes a mass scale of confinement, the string worldsheet was shown to be pushed to the UV regime along the holographic direction where the behavior of Pomeron kernel becomes similar to the BFKL Lipatov:1976zz (); Kuraev:1977fs (); Balitsky:1978ic (). The regime is however more model-dependent, and the string worldsheet can in principle stay close to the IR end point. It is in this regime (soft Pomeron regime) that the flat-space approximation in Janik:2000aj (); Janik:2000pp (); Janik:2001sc (); Giordano:2011sn () can be justified.

Based on the same flat-space approximation for soft Pomerons, we will attempt to compute a full closed string exchange amplitude between two Wilson loops in dipole-dipole scattering. The two Wilson loops with large relative rapidity set the relevant asymptotic states in the high energy eikonal formulation, and provide an effective boundary condition for the exchanged closed strings. For a small dipole size , this boundary condition will be argued to be similar to the one in the D0 brane scattering problem, which allows us to compute, modulo a few subtle differences, all the essential features of the expected Reggeized amplitude in soft Pomeron regime.

The Regge behavior of closed string exchange in flat space has been known for a while from the simplest Virasoro-Shapiro amplitude of scattering. Irrespective of the details of the external states, the closed string exchange gives rise to a universal Pomeron kernel

(1)

where the in the intercept should be replaced by for purely bosonic string, where is the number of massless bosonic worldsheet fluctuations minus two from ghosts. The universality of the above kernel can be understood from the fact that it arises through semi-classical worldsheets connecting the two high energy string states, whose lengths in the impact parameter space are of order in units of  Gross:1987kza (). As is large, the details of the external states are not relevant in (1). A similar conclusion will be reached in our case of dipole Wilson loops specifying the external states, as it should.

In this work, we intend to clarify a number of physical issues of relevance to dipole-dipole scattering in the diffractive regime based on stringy holography:

  1. We would like to clarify the results of Janik:2000aj (); Janik:2000pp (); Janik:2001sc (), especially the ambiguity of the multibranch structure in the variational approach. The universal result in (1) indicates that only the minimal branch cut is physical, while higher winding contributions are artifacts of the variational method. Also, the slope and the intercept in their results seem to differ from (1).

  2. We would like to understand the regime of validity of the approximations used in those works, such as using the flat-space approximation and neglecting the massive worldsheet fermions. We also would like to clarify the relation to the analysis in TAN (). We will elaborate on this in section VI.

  3. We would like to understand the physical origin of the universal semi-classical worldsheets that are responsible for the Pomeron kernel (1). The effective D0 brane scattering analogue we have for a small dipole size turns out to be useful for this. Indeed, via worldsheet T-duality, we show that these semi-classical worldsheets map to the stringy analogue of worldline instantons (anti-instantons) in the Schwinger mechanism of pair creation under an external electric field, where the effective electric field in our T-dual picture is induced by the relative rapidity of the original Wilson loops. This gives us more insight onto the nature of the semi-classical worldsheets. Moreover, we will argue that the Schwinger mechanism description indicates the existence of a “micro-fireball” in the middle of the created string due to the Unruh temperature of an accelerated string worldsheet, which may explain the observed thermal multiplicity in pp collisions as well as the diffusion behavior implied by the soft Pomeron.

  4. Beyond the universal kernel (1), we also include the dependence of the full amplitude on the dipole size , in a reasonable approximation. This is a question of prefactor multiplying (1) that depends on the size (virtuality) of the external dipole states. Our result is reminiscent of the phenomenological dipole parameterization of the cross section of deep-inelastic scattering in terms of a dipole size and the saturation momentum GBW ().

  5. We also consider the case of dipole Wilson loops of higher representations. We show that this case allows some of the multi-winding contributions with winding number , where depends on the representation. When becomes comparable to , the N-ality becomes important and the correct objects exchanged should be -strings described by D-branes, instead of simple overlapping number of strings, so that the results for large should be modified.

In view of computing the connected expectation value of two largely separated Wilson loops, we comment on one aspect of our result. Indeed, we note that the semi-classical worldsheets responsible for (1) exist for large impact parameters. They are sustained by the rapidity of the two Wilson loops. In the case of zero rapidity, that is, for a pair of static Wilson loops, it has been known that there is a phase transition at large distance where semi-classical worldsheets connecting the two Wilson loops cease to exist Gross:1998gk (), and one necessarily goes to the perturbative supergravity mode exchanges. Interestingly, this Gross-Ooguri transition is removed in our case by a finite rapidity difference between the two Wilson loops: there always exist semi-classical worldsheets between the two Wilson loops with rapidity angle. In the T-dual picture, this is due to the fact that a finite electric field always admits stringy worldsheet instantons for any separation of two end points of the string. One can also check, for example in (35), that these contributions disappear in a static limit , conforming to the perturbative supergravity exchange regime, thanks to the occurence of an essential singularity.

Although our results are based on general features of holographic models with confinement, it is nonetheless useful to have a reference model, expecially when we discuss the regime of validity of our approximations. We will consider the double Wick-rotated non-extremal D4-brane geometry by Witten WITTEN (). This holographic QCD with D4 branes offers a nonperturbative framework for discussing Wilson loops in the double limit of large number of colors and t’ Hooft coupling . The effective string tension at the IR end point is given by (or ), although the expression is model-dependent and not essential for our purposes.

The outline of the paper is as follows: in section II we set the definitions for the eikonalized dipole-dipole scattering amplitudes in the impact parameter space representation, and review their analysis in Euclidean perturbation theory. In section III we compute the string amplitude of t-channel closed string exchange between the two dipole Wilson loops in holographic QCD, based on a few reasonable assumptions. The Wilson-loop correlation function is shown to pick up a real part (corresponding to inelasticity) from the pole contributions generated by the rapidity twisting of the bosonic zero modes. We then identify these contributions with the semi-classical worldsheet instantons in the Schwinger mechanism in the T-dual picture, where the electric field is induced by the relative rapidity. We also argue that not all contributions from multiple windings are physical due to a difference between real D0 brane and our Wilson loops, and one necessarily needs to truncate the sum up to depending on the representation of the Wilson loops: for the fundamental representation, . We discuss a related interesting issue of -ality and -strings in our picture. In section IV, we obtain our elastic dipole-dipole scattering amplitude from soft Pomeron exchange in the momentum space, and discuss the phenomenology of our results. The parallel between our Pomeron and the empirical soft Pomeron advocated by Donnachie ans Landshoff are detailed in section V. In section VI, we examine the validity regime of our assumptions taken in the computation. We then show in section VII that the total cross section from the eikonal exponentiation of our results obeys the Froissart unitarity bound. Our conclusions are in section VIII.

Ii Perturbation Theory

We consider an elastic dipole-dipole scattering

(2)

with a dipole size , and , , . The color and spin of the incoming/outgoing quarks inside the dipoles are traced over.

In Euclidean signature, the kinematics is fixed by noting that the Lorentz contraction factor translates to

(3)

where is the Euclidean angle between the two high energy trajectories in the longitudinal space and in the center of mass frame. Scattering at high-energy in Minkowski geometry follows from analytically continuing in the regime  THETAMEGGIOLARO (). It is convenient to consider the trajectories in the impact space representation as , , and , where is the t-channel momentum () and is the impact parameter. (The first two coordinates are longitudinal space and the collectively means the transverse two dimensional impact parameter space).

Using the eikonal approximation, LSZ reduction and the analytic continuation discussed above, the dipole-dipole scattering amplitude in Euclidean space takes the following form NACHTMANN ()

where

(5)

is the normalized Wilson loop for a dipole, . In Euclidean geometry is a closed rectangular loop of width that is slopped at an angle with respect to the vertical imaginary time direction (see FIG. 1). The two dimensional integral in (LABEL:4X) is over the impact parameter with , and the averaging is over the gauge configurations using the QCD action.

In (LABEL:4X-5), the dipole sizes are fixed; as such is their scattering amplitude. In NACHTMANN (), this amplitude is folded with the target/projectile dipole distributions to generate the pertinent hadron-hadron scattering amplitude. We note their size is generic for either longitudinal () or transverse () dipole size. In general, the dipole-dipole scattering amplitude depends on the orientation of the dipoles. We expect the amplitude to be of the form:

(6)

After analytic continuation to Minkowski space, the longitudinal orientation is suppressed by a power of which is just the Lorentz contraction factor. Throughout, will refer to as the longitudinal dipole orientation is suppressed at large .

Figure 1: Dipole-dipole scattering configuration in Euclidean space. The dipoles have size and are apart. The dipoles are tilted by (Euclidean rapidity) in the longitudinal plane.

We will assume that the impact parameter is large in comparison to the typical time characteristic of the Coulomb interaction inside the dipole, i.e. . As a result the dipoles are color neutral, and the amplitude in perturbation theory is dominated by 2 gluon exchange. Thus SHURYAK ()

(7)

for two identical dipoles of size with polarizations along the impact parameter . The analytic continuation shows that , leading to a finite total cross section. We note that , and thus subleading at large .

Iii Holographic Computation and the Schwinger Mechanism

In this section, diffractive dipole-dipole scattering in holographic QCD will be pursued through closed string exchanges between the two dipole Wilson loops. Instead of working in the semi-classical approximation as originally proposed in Rho:1999jm (); Janik:2000pp (); Janik:2000aj (); Janik:2001sc () and dictated by the tenets of holography, in the present approach we will attempt to compute a full string partition function with reasonable approximations. As a consequence some of our results include subleading -corrections such as the intercept, although the main focus of our discussion is on the leading large contributions dominated by semi-classical worldsheets. Our motivation is to identify these contributions via a more rigorous computation compared to the variational approaches taken in Janik:2000pp (); Janik:2000aj (); Janik:2001sc (), resolving some of the issues related to the multibranch structures in them. Also, our computation will give us more physicsal insight on the nature of these semi-classical worldsheets in terms of a stringy version of the Schwinger mechanism with an electric field induced by the probes relative rapidity.

For small dipoles and large impact parameter , we assume that most of the string worldsheet stays at the IR end point, so that we have effectively a flat-space with an effective string tension neglecting fluctuations along the holographic direction. This approximation is based on the generic form of the confining metric

(8)

where is the 4 dimensional flat metric and stands for an extra compact space depending on a particular string theory compactification which is not important for our argument. For confinement, the function has a zero at some finite in the holographic direction. In order to minimize its area, the string worldsheet connecting the dipoles that are placed on the boundary and separated by a large impact parameter b, rapidly falls down to the IR end-point . At the horizon where the string lives, the string area is measured in units set by the effective string tension . For example, for Witten’s WITTEN () confining metric we have . In fact, this flat-space approximation is valid only in the regime of the soft Pomeron where  TAN (), and this will be assumed throughout our paper.

Also, we will neglect the fermionic degrees of freedom on the string worldsheet, which is a deviating point from the analysis inTAN (). This is a question of worldsheet one-loop determinant corrections to the leading semi-classical string partition function. It is motivated by the results in KINAR () for the standard Wilson loop, where it was shown that for the static Wilson loop (), the worldsheet one-loop contribution to the quark-antiquark Wilson loop is dominated by massless bosonic degrees of freedom giving a Lüscher-type contribution, whereby the bosonic mode along holographic direction and all worldsheet the fermionic modes become massive and give only subdominant contributions. In section VI, a more precise condition for this to be valid in our case will be presented, especially in comparison to the “locality” assumption in TAN () which breaks down for sufficiently large . Based on these approximations, our problem effectively reduces to the one in the flat space bosonic string theory. However, when we discuss dipoles of higher representations at the end of the section, the nature of gauge/gravity correspondence of holographic QCD will be important.

Figure 2: (a) Closed string exchange as a funnel contribution (b) Approximation similar to D0 brane scattering with subtle differences explained in the text. The coordinates are the same as in FIG. 1.

The Euclidean connected dipole Wilson loops correlator

(9)

appearing in (LABEL:4X) gets the leading large contribution from the exchange of one closed string as in FIG. 2(a): the closed string makes a funnel connecting the two dipole Wilson loops. Note that the funnel has been proposed long time ago as the geometry underlying the Pomeron exchange within the framework of the “topological expansion” Veneziano:1976wm (). We would like to compute the string partition function summing over all possible fluctuations within the same topology. This problem is different from the closed string exchange between D-branes in a number of ways:

  1. In our case of funnels, the area inside the funnel is empty so that the string action is reduced by that amount, whereas for the D-brane case, there is no such effect.

  2. In the D-brane case, multi-winding of the cylinder topology is allowed without further large suppression, while it is not in the case of emission from a string worldsheet. To have multi-winding, the string genus has to increase leading to further suppression. This point will be relevant later when we discuss the truncation of the multi-winding contributions and the dipoles of higher representations.

As this is a difficult problem in string theory due to a finite dipole size , we necessarily have to make reasonable approximations that would allow us to proceed while still giving us all essential features of the expected result. For a small dipole size , the two boundaries of the funnel will be highly pinched along the dipole direction, so that they effectively lie on two straight lines aligned along the direction of the Wilson loop trajectories as depicted in FIG. 2(b). This leads to a reasonable approximation of treating these boundaries strictly sitting on two straight lines inside the two dipole Wilson loops, and the string partition function over these restricted configurations can be computed in a similar manner as in the case of D0 brane scattering. After that, the locations of the two boundary lines inside each dipole will be integrated over with a measure naturally obtained from the Polyakov string action, which gives us the final amplitude with dependency on the dipole size . As our final result contains all the expected behaviors of Regge trajectory and the intercept, the subset of full configurations that we have chosen seems to be large enough to contain all essential configurations relevant in the Regge regime.

As discussed before, there are differences between the real D0 brane scattering amplitude and the amplitude we would like to compute. The first point in regard to the area reduction inside the funnels is fine in our approximation because the D0 brane cannot have a non-zero area anyway. However, the second point is relevant and we have to discard all higher winding contributions in our final result as they are artifacts of D0 brane and are not the allowed configurations in our original problem. They will be relevant for scattering of dipoles in higher color representations.

With these in mind, the Euclidean correlator as a string partition function of one closed string exchange is given by

(10)

where

(11)

is the string partition function on the cylinder topology with modulus ( is the circumference of cylinder when its length is normalized to 1) with suitable boundary conditions that we just discussed above, and the Polyakov string action is

(12)

in a conformal gauge for the worldsheet metric. The ghosts contributions follow from the diagonal gauge-fixing of the metric, and for the bosonic string it amounts to two longitudinal ghosts. The dot refers to and the prime refers to . The measure is the well-known measure of conformal classes of worldsheet metrics on the cylinder, and the factor is due to the relative genus in comparison to the unconnected Wilson loops. For Witten’s geometry, at the IR end point is

(13)

whose precise form is model-dependent, but the suppression is universal.

The integration in (11) is over periodic configurations

that stretch between the twisted dipole surfaces in Euclidean space as shown in FIG. 2(b), with

(14)

Here, we take the origin in the logitudinal space as the intersection point of the two trajectories at projected to the longitudinal space. As we shift the locations of these two trajectories along the dipole separation width , the intersection point will move, but the relative geometry of the two trajectories is the same, and it is easily seen that the amplitude does not depend on these shifts. Therefore, integrating over the dipole size will simply give us from the two dipoles, up to some unknown constant measure factor that we will discuss later. Note that this dependence is a consequence of our approximation of the pinched boundaries for the closed string funnels, but it will be shown to be more general by the fact that the relevant worldsheet instantons have a small width of order so that it is justified as long as for a large . Our result is reminiscent of the dipole parametrization GBW () of the deep-inelastic cross section.

The twisted boundary conditions (14) are readily implemented through

(15)

with , and the ordinary Neumann (Dirichlet) boundary condition for (). The longitudinal coordinates follow from the coordinates by a local rotation on the world-sheet, which implements successive boost transformations on the world-sheet. (15) is a ruled transformation at the origin of the helicoidal geometry. The above transformation is useful because the string fluctuation modes become purely quadratic in terms of the coordinates. The Jacobian of this transformation is 1.

iii.1 Mode Decomposition

The untwisted coordinates satisfy both the periodic and usual Neumann/Dirichlet boundary conditions on the dipole surfaces. The quadratic action in (11) is easily diagonalized using

(16)

which shows how two parallel dipoles with (potential problem) get to the twisted dipoles with (scattering problem). A similar mode decomposition for the potential problem using the Nambu-Goto string was originally discussed in ARVIS (). The transverse coordinates are untwisted with . They obey both the periodic and Dirichlet boundary conditions. Their mode decomposition is

(17)

with the impact parameter being two dimensional. Note that the total is eight dimensional.

We note that includes zero modes which are constant over the coordinate,

(18)

They induce the zero modes in the original coordinates as

(19)

which is a periodic form of the helicoidal surface. We now use the mode decompositions (16) and (17) to compute the string propagator (11).

iii.2

Since the Polyakov action is quadratic with the above mode expansions on the world-sheet, it can be factored out into its basic contributors,

(20)

where , represent the longitudinal zero and non-zero mode contributions respectively. is the transverse mode contribution, and is the extra ghost contribution required by the covariant gauge fixing. Below we will provide a detailed description of the calculation of using standard zeta function regularization. The other contributions follow similarly, and will only be quoted as final results.

The transverse mode decomposition (17) once inserted in (12) yields products of Gaussian integrals for the transverse modes

(21)

The infinite products in (21) can be evaluated by using zeta function regularization technique. Indeed, the infinite product of a constant can be written as

(22)

where is the Riemann zeta function. In particular this leads to

(23)

since the zero mode and non-zero mode contributions cancel. Similarly from analytic continuations and we get

(24)

Finally by using the product formula for

(25)

the transverse contribution can be put into the form,

(26)

and the second identity in (24) can be used to express in the standard form

(27)

where is the Dedekind eta function,

(28)

The longitudinal mode contribution to the string propagator follows similarly by inserting (16) in the Polyakov action (12) and carrying out the Gaussian integration. This contribution can be separated into the zero mode contribution as given by (18) and the non-zero mode longitudinal contribution. Specifically, the longitudinal zero mode part contributes

(29)

while the longitudinal non-zero mode part contributes

(30)

Notice that for the longitudinal modes, plays the role of a Bohm-Aharonov phase that modifies the azimuthal quantum number . This observation shows that the sloping or twisting of the Wilson lines in Euclidean space is dual to an ”electric/magnetic” field in the longitudinal directions (electric and magnetic fields are indistinguishable in Euclidean space). It is this longitudinal electric field that is at the origin of the Schwinger mechanism. A more direct way of seeing the Schwinger mechanism via worldsheet T-duality and its worldsheet instantons will be explained shortly.

The ghost contribution tags to the two longitudinal non-zero mode contributions and is unaffected by the twist. Its contribution to (20) is

(31)

Combining all the terms (27-31) in (20) leads to the full periodic propagator

(32)

where we include the factor from integrating over the dipole width . The first contribution in (32) stems from the longitudinal zero modes, the second contribution is from the longitudinal non-zero modes including the ghost fields, and the final contribution arises from the transverse modes.

By dimensional reasoning, there must be a multiplying which should come from the integration measure. The overall unknown numeric constant of this measure can be reabsorbed into our definition of the dipole parameter .

iii.3

The contribution of (32) to the elastic dipole-dipole amplitude at fixed impact parameter follows from inserting it into (9). For small dipoles for which our approximations are justified, takes the form

(33)

which shows that the elastic amplitude vanishes as the dipole size . The phenomenological relevance of (33) to deep-inelastic scattering including the possible connection to the saturation phenomena will be discussed elsewhere.

Using (32) after the analytic continuation to Minkowski space gives

(34)

The zero mode contribution in (34) developes poles along the real T-axis for or . Feynman prescriptions for the elastic scattering amplitude requires deforming the contour above the negative poles and below the positive poles. For , the contribution at the poles is purely real

(35)

which gives the inelasticity that we are interested in. We note that (35) displays an essential singularity as , which is a hallmark of tunneling. This is related to the fact that they are generated through worldsheet instantons via the Schwinger mechanism as we detail below.

Since we are interested in the limit, the above expression, which is written in the open string viewpoint, is not suitable to correctly identify the limit, and one needs to transform it to a closed string viewpoint by using the modular relation of the Dedekind eta function, APOSTOL (). We have

(36)

Also

(37)

exhibits a harmonic spectrum. It is the generating function of the bosonic string level density. Asymptotically FUBINI ()

(38)

The exponentially rising density (38) is a hallmark of string excitations. We note that .

iii.4 Schwinger mechanism

In this section, we will provide a physical understanding of the non-perturbative contributions in the exponent of (35), that is the terms which drive the Regge behavior in momentum space for as we will see in section IV. Recall that the ’th contribution comes from the pole at , which will be important later. The nature of these contributions indicates that they should arise from semi-classical worldsheet instantons. These world-sheet instantons bear some similarities to the instantons/sphalerons advocated in SHURYAK (); KHARZEEV () to play an important role in diffractive scattering.

We will shed more light on this by showing that upon worldsheet T-duality these worldsheet instantons map to a stringy version of well-known instantons in the Schwinger mechanism of pair creation under external electric field, where in our case the electric fields acting on the end points of the open string is triggered by the relative rapidity of the probes via T-duality. We will find the analytic solutions for these worldsheet instantons in the stringy Schwinger mechanism, and show that the ’th contribution arises from a -wrapping worldsheet instanton solution, much like point particles. Moreover, we will show that these tunneling configurations last a time and carry an on-shell action .

We start from our assumption that the two boundaries of the cylinder worldsheet sit on the straight lines with rapidity angles and for respectively. These are effectively the same boundary conditions as in the D0 brane scattering set up. At (the analysis for will be similar and we will simply present the final result later) the boundary condition can be written explicitly as

(39)

We then invoke a worldsheet T-duality along the direction ,

(40)

to have a dual description in terms of . Note that the worldsheet instantons we will present shortly are in the zero winding/momentum sector, so that the compactification of the direction and its radius transformation in T-duality is not relevant for our purposes. This is a technical tool to find the worldsheet instantons in the original problem. The boundary condition (39) then becomes

(41)

which is easily shown to be equivalent to putting a boundary term to the Polyakov action,

(42)

with

(43)

being an electric field along the direction, . This aspect is a well-known feature of T-duality in D-brane physics. Note that the electric field acts on the two end points of the open strings stretching between the two dipoles. The signs of the electric fields on both ends are opposite due to the opposite direction of motions of the two dipoles, but the two end points of a string carry opposite charges, so that there is a net acceleration. This explains the existence of a Schwinger mechanism of pair creation of strings in high energy collisions.

To find the worldsheet instantons of this stringy version of the Schwinger mechanism, we proceed to the Euclidean description with an action

(44)

where we have changed the variable , and are defined by

We have to find saddle the points of the action (44) with respect to both the -integral and the worldsheet fields , based on the largeness of .

A similar problem was solved in SCHUBERT (), and we follow the same steps to find the explicit solutions. The -dependence is algebraic, and it is easy to find its saddle point as

(46)

Inserting this back into (44) gives

whose equations of motion are

(48)

with the boundary conditions

(49)

The Dirichlet boundary condition for fixes its solution as , and for we write a -wrapping Ansatz as

(50)

with a -dependent radius function to be determined. With (50), we have

and the equation of motion for is

(52)

with the boundary condition

(53)

The unique consistent solution of (LABEL:Iu2), (52), and (53) is possible only for , and is given by

(54)

with

(55)

where we have used (43), . From (46) we see that this corresponds to confirming our expectation. The value of the on-shell action in (44) is also easily computed as

(56)

which precisely agrees with the negative of the exponent in . These are convincing evidences that the Regge behavior of soft Pomeron exchange is indeed driven by a Schwinger mechanism of pair creating strings, where the effective electric fields are induced by the rapidity of the projectiles.

The circular motion of the string instanton on the Euclidean longitudinal plane with the -dependent radius becomes an accelerating hyperbolic motion in Minkowski spacetime. The resulting -dependent acceleration is

(57)

which has a maximum at the center of the string, . Due to this acceleration, the string feels a -dependent Unruh temperature,

(58)

with a maximum at the center. The temperature quickly drops to a small value around the two end points. The existence of this finite temperature may naturally explain the diffusion-like phenomena noted in Pomeron physics, the details of which will be discussed elsewhere.

It is very interesting to compare the Unruh temperature with the Hagedorn temperature and/or deconfinement transition temperature which characterize the transition temperature to a plasma phase. The effective Hagedorn temperature is given by

(59)

where the last expression is for the Witten’s geometry. The deconfinement temperature of the same model is  Aharony:2006da (), thus at strong coupling. In section IV where we go to the momentum space Regge behavior, we will see that the dominant contribution in the -integral for a fixed comes from a region where

(60)

so that we have two different cases:

  1. When , we have , so , i.e. the middle region of the string feels a temperature greater than the deconfinement temperature when

    (61)
  2. In the other case of , we have , and when

    (62)

Note that our soft Pomeron picture is valid when , so that can easily satisfy both (61) and (62).

Figure 3: In the middle of the transverse string worldsheet, there exists a micro-fireball with an effective Unruh temperature that exceeds the deconfinement temperature.

To summarize, in the middle of the created string there is a small region where the temperature is higher than the deconfinement temperature and the string description should be replaced by a plasma phase. As the temperature quickly drops away from the center, the plasma size is small: we call it “micro-fireball”. See FIG. 3. A simple computation gives its transverse size as

(63)

for the case 1, and

(64)

for the case 2. For large , both become small.

This is an important observation. The existence of a micro-fireball from a single soft Pomeron exchange can naturally explain the observed apparent thermal nature of multiparticle production in high energy collisions. The Unruh radiation in QCD was previously argued to be responsible for the apparent thermalization in Refs. Kharzeev:2005iz (); Kharzeev:2006zm (); Castorina:2007eb (); Zhitnitsky:2012im (). This phenomenon may also give a new insight on the origin of the diffusion-like behavior in the impact parameter space (“Gribov diffusion”) and in the transverse momentum space. The micro-fireball is a consequence of the non-perturbative aspects of QCD with soft Pomerons, which is not clearly seen in the regime of perturbative QCD.

This point can be made more transparent after inserting (36-38) into (35), leading

(65)

and noting that

(66)

is the normalized diffusion propagator in dimensions,

(67)

The diffusion constant in rapidity space is . For long strings, the diffusion propagator (66) emerges as the natural version of the periodic string propagator in (10-11) in the diffusive regime .

iii.5 Truncation of the -sum and dipoles of higher representations

Since (65) is ultimatly tied with the total cross section in impact parameter space as in (93) below, it behooves us to interpret the appearance of the in the k-ality sum. Schematically, the sum can be written as

(68)

So the odd k-ality sum yields instantons, while the even k-ality sum yields antinstantons. Indeed, the instantons produce pair of close strings by tunneling forward while the antinstantons annihilate pair of close strings by tunneling backward. This back-and-forth process is allowed because there is no constaint on the bosonic pair creation process in the Schwinger mechanism. This is not true for the fermionic pair creation process. Incidentally, this back-and-forth process reminiscent of instanton-anti-instanton dynamics may be the world-sheet analogue of the sphaleron mechanism suggested in SHURYAK (). Indeed, standard instantons contribute to the tunneling amplitude and to the probability, prompting us to rewrite the exponents in (68) as which is a sphaleron probability.

It is clear that the ’th contribution comes from the semi-classical worldsheet which wraps the configuration times. Although these multi-winding contributions are perfectly fine in the case of a real D0 brane scattering, they are in fact not allowed topologically in our case of closed string emission/absorption from the string worldsheets of two dipole Wilson loops. To understand this we note that we are originally summing over funnels, and having multiple funnels on top of each other changes the genus of the total string worldsheet, which entails further suppressions. This means that only the contribution in (65) is physical while higher winding contributions are artifacts of our D0 brane analogue. It is interesting that a similar kind of ambiguity appeared in the variational approach in Janik:2000aj (); Janik:2000pp (); Janik:2001sc (), where one gets similar contributions from the multibranch structure of the minimized Nambu-Goto action. Our discussion indicates that these multibranch contributions arise from worldsheet configurations that are prohibited by topology without further suppression, and hence should be discarded at large .

Figure 4: If the dipoles belong to a representation with central charge , then they can exchange strings with k-ality . Charge conservation dictates that the part of the string worldsheet inside the dipole has k-ality .

Although the above conclusion is true for the Wilson loops in the fundamental representation, the situation can change if one considers dipole Wilson loops of higher representations. Intuitively it is clear that the worldsheet that a Wilson loop of higher representation bounds should be a composite object made of multiple overlapping fundamental string worldsheets. When the representation is constructed from a product of fundamental representations, the corresponding worldsheet that bounds the Wilson loop should be a composite object made of fundamental strings. When , the distinction between this object and the simple non-interacting fundamental strings is small, whereas for the composite object is quite different from the simple sum of fundamental strings, and it is typically described by D-branes wrapping appropriate cycles. For example, in Witten’s geometry, the -antisymmetrized representation, corresponding to -string, is described by D4 brane wrapping the internal cycle, whose string tension features Casimir scaling Callan:1999zf ()

(69)

although the precise form of the string tension is model-dependent SINELAW ().

On these composite worldsheets made of fundamental strings, it is indeed possible to attach multi-winding worldsheets of fundamental strings up to . It is easy to understand this as in FIG. 4. For example, if dipole the Wilson loops in the -antisymmetrized representation emit/absorb multi-wound strings, the interior of the funnel should be a -string worldsheet by string charge conservation. This gives an inequality . Therefore, in the sum (65) one might keep the terms up to