A The Wilson loop calculations

The abelian confinement mechanism revisited: new aspects of the Georgi-Glashow model

Abstract:

The confinement problem remains one of the most difficult problems in theoretical physics. An important step toward the solution of this problem is the Polyakov’s work on abelian confinement. The Georgi-Glashow model is a natural testing ground for this mechanism which has been surprising us by its richness and wide applicability. In this work, we shed light on two new aspects of this model in D. First, we develop a many-body description of the effective degrees of freedom. Namely, we consider a non-relativistic gas of W-bosons in the background of monopole-instanton plasma. Many-body treatment is a standard toolkit in condensed matter physics. However, we add a new twist by supplying the monopole-instantons as external background field. Using this construction, we calculate the exact form of the potential between two electric probes as a function of their separation. This potential is expressed in terms of the Meijer-G function which interpolates between logarithmic and linear behavior at small and large distances, respectively. Second, we develop a systematic approach to integrate out the effect of the W-bosons at finite temperature in the range , where is the W-boson mass, starting from the full relativistic partition function of the Georgi-Glashow model. Using a heat kernel expansion that takes into account the non-trivial thermal holonomy, we show that the partition function describes a three-dimensional two-component Coulomb gas. We repeat our analysis using the many-body description which yields the same result and provides a check on our formalism. At temperatures close to the deconfinement temperature, the gas becomes essentially two-dimensional recovering the partition function of the dual sine-Gordon model that was considered in a previous work.

1 Introduction

After almost 60 years since Yang and Mills formulated their theory [1], color confinement remains one of the greatest puzzles in theoretical physics. Eventhough lattice gauge theories have been successful in demonstrating quark confinement in computer simulations, it is safe to say that up to date there is no analytical understanding of the confinement mechanism in dimensions. 2 A breakthrough idea toward the solution of the confinement problem was introduced in the pioneering work of Polyakov [4], who showed that the proliferation of monopole-instantons in the vacuum of compact QED in dimensions leads to the confinement of electric charges. These monopoles are solution to the Euclidean classical equations of motion, and result due to the compact nature of the gauge group. Immediately after this work, Polyakov showed that the same confinement mechanism is at work in the Georgi-Glashow model 3 in D [5]. This model consists of an Yang-Mills theory coupled to a triplet of scalar fields. 4 Previously, it was shown in [8, 9] that this model admits finite non-singular and nonperturbative solutions to the classical equations of motion. These are the Polyakov ’t-Hooft magnetic monopoles in D, and monopole-instantons in the Euclidean setup in D. The monopole-instantons are charged under abelian group. This is the unbroken compact subgroup of the original , which breaks spontaneously upon giving a vacuum expectation value to the scalars. The monopole-instantons interact via Coulombic forces and form a gas of monopole plasma. In order to test the effect of the monopole proliferation on two external electric test charges, one computes the behavior of the Wilson loop in the background of the monopole plasma. The Wilson loop is a gauge invariant order parameter for confinement. Usually, we take our loops to be rectangles which represent the creation, propagation and annihilation of two charged probes separated a distance for time . In the confinement phase, the expectation value of the Wilson loop experiences an area-law behavior

(1)

where is the area of the loop enclosed by the curve , and is a proportionality constant interpreted as the string tension. In order to understand the area-law behavior, we can think of the Wilson loop as a current loop which itself generates a magnetic field. The monopole and anti-monopoles will line up along the area of the rectangular loop to screen out the generated magnetic field. Therefore, the area-law behavior is associated with the screening sheet of monopoles along the area of the loop.

A complementary way of thinking about the confinement mechanism is to consider the dual superconductivity picture advocated by ’t Hooft and Mandelstam [10, 11]. In superconductors, the condensation of electric charges, known as Cooper pairs, breaks the of electromagnetism spontaneously which in turn gives a mass to the photon. This is the Meissner effect which is responsible for screening the magnetic flux lines in superconductors. However, in type II superconductors magnetic field lines are allowed to exist in the form of flux tubes known as Abrikosov vortices [12]. If we place two magnetic monopoles in the bulk of a type II superconductor, the magnetic flux lines can not spread everywhere in the bulk. Instead, they collimate into a thin flux tube which can be though of a string connecting the two monopoles. Hence, at distance much larger than the screening length of the superconductor the potential between the probes will behave as , where is the string tension. According to ’t Hooft and Mandelstam, the vacuum of Yang-Mills behaves as type II superconductor except with a reversal of the rules of electric and magnetic charges. In fact, the Polyakov works [4, 5] are the first demonstration of the dual superconductivity picture. Later on, a beautiful implementation of this picture was worked out by Seiberg and Witten in D in a supersymmetric context [13, 14].

Since the pioneering work of Polyakov, the Georgi-Glashow model has been a testing ground not only for the confinement phenomena, but also for the deconfinement transition. Pure Yang-Mills theory in D experiences a deconfinement transition at strong coupling which hinders a full understanding of the transition [15]. Hence, one needs a simpler theory that resembles the original one, yet under analytic control. The finite temperature effects in the D Georgi-Glashow model were first considered in [16]. There, it was shown that a confinement-deconfinement transition happens at temperature , where is the Yang-Mills coupling constant. However, the authors in [16] ignored the effect of the W-bosons which plays an important role near the transition region, as was shown later on in [17]. Taking the W-bosons into consideration, the authors in [17] argued that the partition function near the transition region is that of a two-dimensional double Coulomb gas of monopole-instantons and W-bosons. This gas can be mapped into a dual sine-Gordon model which is further studied using bosonization/fermionization techniques. With such technology, it was shown that the inclusion of the W-bosons modifies the transition temperature to and that the transition is second order and belongs to the 2D Ising universality class. 5

In this paper, we shed light on some issues that have not been considered before in the Georgi-Glashow model. In the first part of this work, we answer an important question which concerns the behavior of the potential between two external electric probes at intermediate distances. At distances much shorter then the screening length of the monopole-instanton plasma , the potential between the probes is logarithmic. On the other hand, at distances much larger than , the potential is linear. However, an analytic expression for the behavior of the potential in the intermediate region between the logarithmic and linear potential is still lacking. For this purpose, we develop a Euclidean many-body description of the partition function of the system. In this formalism, we consider the external electric probes as well as W-bosons in the background of the field generated by an arbitrary number of monopole-instantons. Since we are interested only in temperatures much lower than the mass of the W-bosons, we can limit ourselves to a non-relativistic description. Many-body treatment is a powerful tool in condensed matter systems. In the present work, we adapt this method to take into account the effect of monopole-instantons which, to the best of our knowledge, has not been considered elsewhere. At zero temperature, the W-bosons do not play any role thanks to the Boltzmann suppression factor . In this case, we can work only with a partition function that describes the external probes in the background of monopole-instanton gas. Further, we assume that the density of the instantons obey a Gaussian distribution. Although this introduces an error in the string tension, such an assumption considerably simplifies our computations. We find that the potential between two probes of charges separated a distance is expressed in terms of the Meijer-G function

(2)

which smoothly transits from logarithmic behavior at distances to a linear potential at distances much larger than the screening length. This behavior is illustrated in Figure (1) where we plot the potential and electric field as a function of the separation distance .

Figure 1: The potential and electric field profiles using the expression (2). The potential is logarithmic for distances , and linear for distances . The transition from logarithmic to linear behavior happens at a distance where the electric field profile levels off.

In the second part of our work, we perform a systematic study of the finite-temperature partition function of the Georgi-Glashow model in the temperature range , where is the W-boson mass. In this regard, we take two different approaches. In the first approach, we start our treatment from the full relativistic partition function which treats the monopole-instantons as external background field, and then apply a heat kernel expansion technique which takes into account the non-trivial thermal holonomy. This results in an effective action that contains both relevant and irrelevant operators. Ignoring the irrelevant ones, we show that the partition function of the system takes the form of the grand canonical distribution of a non-relativistic three-dimensional double Coulomb gas:

The gas consists of W-bosons and monopole-instantons with fugacities and , respectively. Two W-bosons carrying charges and located at interact logarithmically (which is the two-dimensional potential) at all temperatures in the range . While two monopole-instantons with charges and three-dimensional positions interact via , where is the Green’s function of the Laplacian operator on , and is the thermal circle. In addition, W-bosons interact with monopole-instantons via the Aharonov-Bohm phase . At temperatures much larger than the inverse distance between two-monopole instantons, yet much lower than the deconfinement temperature, the Green’s function reduces to a logarithmic function and the gas becomes essentially two-dimensional recovering the same partition function considered before in [17]. In the second approach, we start from the non-relativistic many-body description and then integrate out the W-bosons to recover (LABEL:final_expression_for_Z_in_the_introduction). This also works as an independent check which ensures the validity of our many-body description.

This paper is organized as follows. In Section 2, after a quick review of the Georgi-Glashow model, we write down the relativistic partition function of the system taking into account the monopole-instantons as background field. Then, motivated by a physical picture, we give a non-relativistic many-body description of the system. In Section 3, we use the many-body partition function, aided by a mean-field approach, to derive the potential between two static electric probes at zero temperature. In Section 4, we use both the relativistic and non-relativistic partition functions to show that the finite-temperature Georgi-Glashow model can be thought of as a three-dimensional double Coulomb gas as explained above. Finally, we conclude in Section 4 and provide directions for future research. The paper contains three appendices displaying miscellaneous calculations.

2 Theory and formulation

2.1 The Georgi-Glashow model: perturbative treatment

We consider the Lagrangian of Georgi-Glashow model in

(4)

where is the three-dimensional coupling constant, and are matter fields in the adjoint representation of the group. The Greek indices run from to and the color indices run from to . The field strength tensor and the covariant derivative are given by

(5)

The field acquires a vacuum expectation value (say in the direction) which breaks the down to . Then, we write the third component of the Higgs field as , where is the physical excitation of the Higgs field. The third color component of the gauge field remains massless, and the other two components form massive vector bosons:

(6)

The massive vector bosons, or W-bosons for short, carry electric charge , where . We also define the charged Goldstone Bosons as

(7)

Substituting (6) and (7) into (4), we find that the Lagrangian (4) can be rewritten as the sum of quadratic and interaction pieces :

(8)

and

(9)

where , and . At this stage, let us emphasize that the field does not only describe the photon fluctuations, but it can also include any external background fields, like the monopole-instanton field, as we will explain shortly.

The last line in the quadratic Lagrangian (8) has three terms that may add difficulties to our analysis. We can use integration by parts in the first term to get which is not in the form of a Klein-Gordon operator, i.e. . The other two terms couple the W-bosons to the Goldston bosons. Fortunately, one can get rid of these three terms by entertaining the fact that our Lagrangian has a gauge freedom. Therefore, by choosing an appropriate gauge we can eliminate the unwanted terms. To this end, we add the gauge fixing Lagrangian , where

(10)

Thus

(11)

It is clear that this gauge fixing Lagrangian eliminates the last three terms in (8). After gauge fixing, we also need to include the ghost contribution , where are the ghost fields. To calculate the quantity we proceed as follows. First, we note that the fields and transform under the infinitesimal gauge transformation as and . Then, we decompose into , the gauge parameter of the unbroken group, and along the and color directions. Hence, we obtain

(12)

Then, we find .

In the following, it is more appropriate to work in Euclidean space. A Euclidean version of the Lagrangian can be obtained by performing a Wick rotation. Adding the contribution from the gauge fixing and ghost terms, the total Lagrangian reads (from here on, we do not distinguish between upper and lower indices)

(13)

where , and we have used where is the W-boson mass. Notice that the Lagrangian is invariant under the electromagnetic gauge group since the gauge fixing we have used leaves the subgroup of the intact.

This ends our treatment of the perturbative part. However, the Georgi-Glashow model contains also nonperturbative solutions. These are monopole-instantos that were first discovered by Polyakov [9] and ’t Hooft [8] as solitons in the Georgi-Glashow model in D. According to the path integral formulation of field theory, the grand partition function of the system is obtained by summing over all trajectories, that take us from one point in the field space to the other, weighted by their action:

(14)

This sum must include contributions from both perturbative and nonperturbative sectors. Before writing down the partition function of the Georgi-Glashow model, in the following section we review the nonperturbative solutions of the theory at hand.

2.2 Nonperturbative effects: adding monopole-instantons

In addition to the perturbative excitations described above, the three-dimensional Georgi-Glashow model admits nonperturbative objects. These are monopole-instantons allowed by the non-trivial homotopy , and are obtained as classical solutions to the Euclidean non-abelian equations of motion. These solutions have to be included in the path integral formulation of the field theory, which can have dramatic effects on the physics. Monopole-instantons are particle-like objects localized in space and time, have internal structure and mediate long range force, thanks to the unbroken . Although a single instanton solution satisfies the equations of motion, two or more instantons do not. However, if these objects are well separated, then a solution that is a superposition of many instantons can still be a good approximate solution to the equations of motion. In a reliable semi-classical treatment, one includes an arbitrary number of these objects in the path integral provided that they are well separated, or in other words, their density is low. This is known as the dilute gas approximation. In such approximation, the internal structure of the instantons does not play any role, and for all purposes we can replace the non-abelian field solution with an abelian one.

The abelian field of a single monopole-instanton localized at the origin is given by

(15)

where and are respectively the spatial and Euclidean time coordinates, and is the spherical-polar radius. The above solution is singular at . This is the Dirac string that stems from the location of the monopole-instanton at the origin and extends all the way along the negative -axis. This string is not physical; it is just a gauge artifact as can be shown directly by calculating the monopole field . The magnetic charge carried by a single monopole-instanton is defined as the surface integral of the monopole magnetic field over a 2-sphere, divided by :

(16)

In the dilute gas approximation, we add the contribution from an arbitrary number of these monopole-instantons that are randomly distributed all over the spacetime. This results in the total field

(17)

where is the position of the monopole-instanton and is its charge. Since monopole-instantons carry charges, they will interact via Coulombic forces. The form of interaction can be obtained from the action

(18)

where , and we have introduced the magnetic coupling

(19)

The electric and magnetic couplings are related by the Dirac quantization condition . This is twice the minimal value allowed for since the -bosons have twice the minimal charge.

Now, we are ready to include both perturbative and nonperturbative sectors in the path integral sum (14).

2.3 The grand partition function

The grand partition function of the system is obtained as a path integral over the fields , , , , and . Then, one includes the contribution from the nonperturbative sector as a sum over an arbitrary number of positive and negative monopole-instantons. Hence, the grand partition function reads

(20)

where the monopole fugacity is given by 6

(21)

and is a function of the ratio between the W-boson mass and the Higgs mass. This function tends to unity in the Bogomolny-Prasad-Sommerfield (BPS) limit [21, 22], , and tends to in the opposite limit [23]. In the following, we assume that the Higgs mass is heavy and hence the Higgs field is short ranged and we can neglect its effects in our analysis. As we mentioned above, in order for the partition function to make sense, the monopole-instanton gas has to be dilute, or in other words . This in turn requires that we work in the weak coupling limit . Given the monopole fugacity , the average distance between two monopole-instantons is , apart from a dimensionfull pre-exponential factor.

It is very important to stress that the field appearing in the Lagrangian (13) is given by , and hence . Thus, includes contributions from both the fluctuations of the dynamical photon as well as the background field generated by the monopole-instantons. As a check that our formalism gives the standard monopoles interaction, we can turn off the photon field in (13) to find that , which is the monopole-monopole Coulomb interaction. In Section 4, it will be clear how to carry out the path integral rigorously by using an abelian duality transformation.

The partition function (20) encodes all information about the system under study. 7 For example, as Polyakov did, one can completely ignore the W-bosons at zero temperature to find that there is a linear confining potential between two external charged probes. At finite temperature , we compactify the Euclidean time over a circle of radius . At low temperatures compared to the W-boson mass , one can integrate out the W-bosons. This results in an interacting gas of W-bosons and magnetic monopoles. Since at low temperatures the W-bosons have non-relativistic speeds, in the following we consider a non-relativistic version of the partition function (20). We will not try to directly start from (20) and take the non-relativistic limit. Instead, we will write down a partition function motivated by the physics of the problem. Since we have a system of W-bosons and monopole-instantons, it is tempting to write down a many-body partition function of a non-relativitic gas of physical particles (W-bosons) in the background of external field generated by an ensemble of monopole-instantons. Many-body treatment of a non-relativistic gas is a standard procedure in condensed matter that can be found in many books on the subject, see e.g. [24, 25, 26]. The new thing here, which has not been considered before, is that we add instantons to the system as background field.

2.4 The non-relativistic partition function

Let us consider a two-dimensional gas (remember that we are working in dimensions) of interacting W-bosons of mass and charges . These charged W-bosons experience logarithmic Coulomb interactions. In addition, let us consider this gas in the background of monopole-instantons which act as external time-dependent sources. The classical Hamiltonian of the system reads

(22)

where is the two-dimensional position of the W-boson, while is its Euclidean time. The expressions for the monopole-instanton field, and , are given by (15) and (17). The first term in (22) is the rest mass of the W-boson gas. The second term is the kinetic term of the non-relativistic W-bosons written in terms of the kinetic momentum . The third term describes the interaction between the W-bosons and the zeroth-component of the monopole-instantons field . The factor is acquired because of working in the Euclidean space. In fact, this term is the Aharonov-Bohm coupling between the electrically charged W-bosons and the magnetically charged monopole-instantons. Finally, the last term is the mutual Coulomb interaction between two W-bosons.

The second-quantized version of the Hamiltonian (22) can be obtained by replacing , and introducing the density operators and for the positively and negatively charged W-bosons, respectively:

(23)

The fields and are the annihilation and creation operators for the gauge bosons . They satisfy the equal time commutation relations

(24)

The equal-time commutators of all other fields vanish. We also introduce the monopole density operator

(25)

Then, the field-theoretical version of the Hamiltonian (22) reads

(26)

and is the conserved W-boson number operator

(27)

The term can be thought of as a chemical potential added to the Hamiltonian, where is the W-boson rest mass. The second-quantized version of the Lagrangian can be obtained from the Hamiltonian (26) by using the standard procedure and keeping in mind that we are working in the Euclidean space:

(28)

The finite temperature non-relativistic grand partition function can be obtained in two steps. First, we regard the annihilation and creation operators, and , as classical complex fields, and , and perform the path integral over the various fields. Then, as we did in the case of relativistic partition function, we perform a sum over an arbitrary number of monopole-instantons located at positions , taking their Coulomb interaction into account. The finite temperature effects can be taken automatically into account by compactifying the Euclidean time over a circle of circumference , where is the inverse temperature, . Then, we demand that the fields satisfy periodic boundary conditions over the circle. 8 Thus, the partition function reads

(29)

where is the monopole fugacity given by (21), and the subscript , for example in , indicates that the fields must satisfy the periodic boundary condition . The full non-relativistic Lagrangian is the sum of the Lagrangian (28) and the monopole-monopole interaction term, taking into account the periodicity of the different quantities over the thermal circle:

(30)

The propagator as well as the periodic monopole-instantons field are obtained by summing an infinite number of image charges along the compact dimension

(31)

where the superscript indicates the periodicity of the quantity. Notice that we also have to take into account the spin degeneracy factor for the W-bosons. The partition function (29) is one of the main results of the present work. Let us note that the steps of going from (22) to (29) is a standard procedure in many-body physics. However, the inclusion of instantons as background fields is new, and to the best of our knowledge, has not been incorporated before in many-body treatments. At low temperatures , and by neglecting any relativistic effects, the relativistic (20) and the non-relativistic (29) partition functions contain the same information, and in principle one can use either of them to extract the physics.

At this point, one can split the Lagrangian (30) into two parts: a free Lagrangian and interacting part such that , where

and the rest of is defined to be . Then, performing perturbation analysis, one can expand the partition function (29) as

(33)

The expansion (33) makes sense only if there is a small expansion parameter. In case of W-W interaction, the small parameter is taken to be the charge . However, for the Aharonov-Bohm term we have . In this case, the true expansion parameter is the small monopole density which is a prerequisite for the validity of the monopole-instanton dilute gas approximation.

In the next section, we use the partition function (33) to calculate the string tension between two external electric probes. In the absence of monopole-instantons the potential between the two probes is logarithmic. However, as Polyakov showed long time ago [5], including the instantons in the background creates a mass gap in the system, which in turn changes the logarithmic behavior into a linear confining potential between the probs. One expects the change from a logarithmic to linear behavior to happen at distances . However, it was never shown explicitly how this happens. Using the above formalism, we show this smooth transition takes place, as expected, at distance of order of the inverse mass gap.

3 The potential between two external electric probes at

3.1 An effective partition function and the Polyakov loop correlator

In this section, we use the perturbative expansion of the partition function (33) to calculate the potential between two external electric charges located in the background of the monopole-instanton gas, which is the same as calculating the Polyakov loop (electric) correlator. We will perform our analysis at zero temperature, or in other words, at infinite compactification radius . However, we retain all expressions as a function of such that the limit should be understood implicitly. At temperatures lower than the W-boson mass, the contribution coming from the W-bosons is accompanied by a Boltzamann suppression factor . Thus, the dynamics of the W-bosons is completely suppressed at , and one can neglect the second, third, fourth, and last terms in the Lagrangian (30). Then, we are left only with the first term, the monopole-monopole interaction. Now, let us introduce two external probes with electric charges