# Super Yang-Mills Plasma

###### Abstract

The super Yang-Mills plasma is studied in the regime of weak coupling. Collective excitations and collisional processes are discussed. Since the Keldysh-Schwinger approach is used, the collective excitations in both equilibrium and non-equilibrium plasma are under consideration. The dispersion equations of gluon, fermion, and scalar fields are written down and the self-energies, which enter the equations, are computed in the Hard Loop Approximation. The self-energies are discussed in the context of effective action which is also given. The gluon modes and fermion ones appear to be the same as those in the QCD plasma of gluons and massless quarks. The scalar modes are as of free relativistic massive particle. The binary collisional processes, which occur at the lowest nontrivial order of the coupling constant, are reviewed and then the transport properties of the plasma are discussed. The super Yang-Mills plasma is finally concluded to be very similar to the QCD plasma of gluons and light quarks. The differences mostly reflect different numbers of degrees of freedom in the two systems.

###### pacs:

52.27.Ny, 11.30.Pb, 03.70.+k## I Introduction

Supersymmetric models are considered as possible extensions of the Standard Model, see e.g. Signer:2009dx , and supersymmetry is then assumed to be a symmetry of Nature at a sufficiently high energy scale. Experiments at the Large Hadron Collider may soon show whether this is the case. This paper is devoted to a plasma system with the dynamics governed by the supersymmetric Yang-Mills theory Brink:1976bc ; Gliozzi:1976qd . In the models with an extended () supersymmetry, the left- and right-handed fermions interact in the same way, in conflict with the Standard Model where the left- and right-handed matter particles are coupled differently. Consequently, the super Yang-Mills is not treated as a serious candidate for a theory to describe the world of elementary particles. Nevertheless, the theory attracts a lot of attention because of its unique features. The super Yang-Mills appears to be finite and thus it is conformally invariant not only at the classical but at the quantum level as well.

A great interest in the super Yang-Mills theory was stimulated by a discovery of the AdS/CFT duality of the five-dimensional gravity in the anti de Sitter geometry and the conformal field theories Maldacena:1997re , for a review see Aharony:1999ti and the lecture notes Klebanov:2000me as an introduction. The duality offered a unique tool to study strongly coupled field theories. Since the gravitational constant and the coupling constant of dual conformal field theory are inversely proportional to each other, some problems of strongly coupled field theories can be solved via weakly coupled gravity. In this way some intriguing features of strongly coupled systems driven by the super Yang-Mills dynamics were revealed, see the reviews Son:2007vk ; Janik:2010we , but relevance of the results for non-supersymmetric systems, which are of our actual interest, remains an open issue. In particular, one asks how properties of the super Yang-Mills plasma (SYMP) are related to those of quark-gluon plasma (QGP) studied experimentally in relativistic heavy-ion collisions. While such a comparison is, in general, a difficult problem, some comparative analyses have been done in the domain of weak coupling where perturbative methods are applicable CaronHuot:2006te ; Huot:2006ys ; CaronHuot:2008uh ; Blaizot:2006tk ; Chesler:2006gr ; Chesler:2009yg .

We undertook a task of systematic comparison of supersymmetric perturbative plasmas to their non-supersymmetric counterparts. We started with the SUSY QED, analyzing first collective excitations of ultrarelativistic plasma which, in general, is out of equilibrium Czajka:2010zh . We computed the one-loop retarded self-energies of photons, photinos, electrons and selectrons in the Hard Loop Approximation using the Keldysh-Schwinger formalism. The self-energies, which we also analyzed in the context of effective action, enter the dispersion equations of photons, photinos, electrons and selectrons, respectively. The collective modes of SUSY QED plasma appear to be essentially the same as those in ultrarelativistic electromagnetic plasma of photons, electrons and positrons. In particular, a spectrum of photino modes coincides with that of quasi-electrons. Therefore, independently whether photon modes are stable or unstable, there are no unstable photino excitations. The supersymmetry, which is obviously broken in the plasma medium, does not induce any instability in the photino sector.

In the subsequent paper Czajka:2011zn we discussed collisional characteristics of SUSY QED plasma. For this purpose we computed cross sections of all elementary processes which occur at the lowest non-trivial order of . We found that some processes, e.g. the Compton scattering on selectrons, are independent of momentum transfer. The processes are qualitatively different from those of usual electromagnetic interactions dominated by small momentum transfers. Further on we discussed collisional characteristics of equilibrium SUSY QED plasma, observing that parameters of ultrarelativistic plasmas are strongly constrained by dimensional arguments, as the temperature is the only dimensional quantity of equilibrium system. Then, transport coefficients like viscosity are proportional to appropriate powers of temperature and the coefficients characterizing different plasmas can differ only by numerical factors. So, we derived the energy loss and momentum broadening of a particle traversing the equilibrium plasma, which depend not only on the plasma temperature but on the energy of test particle as well. We found that the two quantities have very similar structure (in the limit of high energy of the test particle) even for very different elementary cross sections. Our findings presented in Czajka:2010zh ; Czajka:2011zn show that the plasmas of SUSY QED and of QED are surprisingly similar to each other. In this paper we discuss properties of the super Yang-Mills plasma, analyzing both collective excitations and collisional characteristics of the system.

Our main aim is to confront the weakly coupled plasma driven by super Yang-Mills with the perturbative quark-gluon plasma governed by QCD. We do not attempt to compare our results to those obtained in strong coupling regime using either the AdS/CFT duality or lattice QCD. Some plasma characteristics we discuss, e.g. the energy loss, are computed in both strongly and weakly coupled systems but it is rather unclear how to study collective excitations representing colored quasiparticles in the setting of AdS/CFT duality or lattice QCD. The paper Bak:2007fk demonstrates that even the definition of Debye screening mass, which has a very simple meaning in perturbative plasmas, is far not straightforward in strongly interacting systems. For these reasons we escape from discussing our results in the context of strong coupling.

Our paper is organized as follows. In the next section, we discuss the Lagrangian of super Yang-Mills and the field content of the system under consideration. The vertexes of super Yang-Mills are collected in Appendix A. In Sec. III basic characteristics of SYMP such as energy density and Debye mass are discussed and compared to those of QGP. Then, we move to plasma collective excitations. The general dispersion equations of gauge bosons, fermions and scalars are written down in Sec. IV and the self-energies, which enter the equations, are obtained in the subsequent section. We apply here the Keldysh-Schwinger approach which allows one to study equilibrium and non-equilibrium systems. The free Green’s functions of Keldysh-Schwinger formalism are given in Appendix B. Since we are interested in collective modes, the self-energies are obtained in the long wavelength limit corresponding to the Hard Loop Approximation. The effective action of the Hard Loop Approach is derived in Sec. VI and possible structures of self-energies are considered in this context. In Sec. VII we present a qualitative discussion of collective modes in SYMP. Sec. VIII is devoted to collisional characteristics of the plasma - elementary processes and transport coefficients are briefly discussed here. Finally, we conclude our study in Sec. IX.

As we have intended to make our paper complete and self-contained, there is inevitably some repetition of the content of our previous publications Czajka:2010zh ; Czajka:2011zn , mostly in Secs. VII, VIII. Throughout the paper we use the natural system of units with ; our choice of the signature of the metric tensor is .

## Ii Super Yang-Mills Theory

We start our considerations with a discussion of the Lagrangian of super Yang-Mills theory Brink:1976bc ; Gliozzi:1976qd . We follow here the presentation given in Yamada:2006rx .

The gauge group is assumed to be and every field of the super Yang-Mills theory belongs to its adjoint representation. The field content of the theory, which is summarized in Table 1, is the following. There are gauge bosons (gluons) described by the vector field with . There are four Majorana fermions represented by the Weyl spinors with which can be combined in the Dirac bispinors as

(1) |

where with denoting Hermitian conjugation. To numerate the Majorana fermions we use the indices and the corresponding bispinor is denoted as . Finally, there are six real scalar fields which are assembled in the multiplet . The components of are either denoted as for scalars, and for pseudoscalars, with or as with .

Field’s symbol | Type of the field | Range of the field’s index | Spin | Number of degrees of freedom |
---|---|---|---|---|

vector | 1 | |||

real (pseudo-)scalar | 0 | |||

Majorana spinor | 1/2 |

The Lagrangian density of super Yang-Mills theory can be written as

where and the covariant derivatives equal

(3) |

and their explicit form can be chosen as

(4) | |||||

(5) |

where the Pauli matrices read

(6) |

As seen, the matrices are antiHermitian: , . The vertexes of super Yang-Mills, which can be inferred from the Lagrangian (II), are collected in Appendix A. The vertexes are used in perturbative calculations presented in the subsequent sections.

The Lagrangian (II) is sometimes CaronHuot:2008uh ; Chesler:2006gr ; Chesler:2009yg extended by adding a fundamental hypermultiplet to mimic a behavior of quarks in QCD plasma. The hypermultiplet is typically massive to study heavy flavors but it can be massless as well. We do not consider any extension of the Lagrangian (II) but at the end of Sec. VI we briefly comment on a possible structure of self-energies of fields belonging to the fundamental hypermultiplet.

## Iii Basic plasma characteristics

We start our discussion of SYMP with basic characteristics of the equilibrium plasma. Specifically, we consider the energy and particle densities, Debye mass and plasma parameter of SYMP comparing the quantities to those of QGP. For the beginning, however, a few comments are in order.

In QGP there are several conserved charges: baryon number, electric and color charges, strangeness. The net baryon number and electric charge are typically non-zero in QGP produced in relativistic heavy-ion collisions while the total strangeness and color charge vanish. Actually, the color charge is usually assumed to vanish not only globally but locally as well. It certainly makes sense as the whitening of QGP appears to be the relaxation process of the shortest time scale Manuel:2004gk . In SYMP, there are conserved charges carried by fermions and scalars associated with the global symmetry. One of these charges can be identified with the electric charge to couple super Yang-Mills to electromagnetic field CaronHuot:2006te . In the forthcoming the average charges of SYMP are assumed to vanish and so are the associated chemical potentials. The constituents of SYMP carry color charges but we further assume that the plasma is globally and locally colorless.

Since there are conserved supercharges in supersymmetric theories, it seems reasonable to consider a statistical supersymmetric system with a non-zero expectation value of the supercharge. However, it is not obvious how to deal with a partition function customary defined as where is the inverse temperature, is the Hamiltonian, is the supercharge operator and is the associated chemical potential. The problem is caused by a fermionic character of the supercharge . If is simply a number, as, say, the baryon chemical potential, the partition function even of non-interacting system does not factorize into a product of partition functions of single momentum modes because the supercharges of different modes do not commute with each other. The supercharge is not an extensive quantity Kapusta:1984cp . There were proposed two ways to resolve the problem. Either the chemical potential remains a number but the supercharge is modified by multiplying it by an additional fermionic field Kapusta:1984cp ; Mrowczynski:1986cu or the chemical potential by itself is a fermionic field Kovtun:2003vj . Then, and are both bosonic and the partition function can be computed in a standard way. The two formulations, however, are not equivalent to each other. According to the former one Kapusta:1984cp ; Mrowczynski:1986cu , properties of a supercharged system vary with an expectation value of the supercharge, within the latter one Kovtun:2003vj , the partition function appears to be effectively independent of . Because of the ambiguity, we further consider SYMP where the expectation values of all supercharges vanish both globally and locally.

In view of the above discussion, SYMP is comparable to QGP where the conserved charges are all zero and so are the associated chemical potentials. We adopt the assumption whenever the two plasma systems are compared to each other.

When the chemical potentials are absent, the temperature is the only dimensional parameter, which characterizes the equilibrium plasma, and all plasma parameters are expressed through the appropriate powers of . Taking into account that the right numbers of bosonic and fermionic degrees of freedom in SYMP and QGP, the energy densities of equilibrium non-interacting plasmas equal

(7) |

where the upper expression is for SYMP and the lower one for QGP with light quark flavors. The quark is light when its mass is much smaller than the plasma temperature. For , the energy density of SYMP is approximately 2.5 times bigger than that of QGP at the same temperature. The same holds for the pressure which, obviously, equals .

The particle densities in SYMP and QGP are found to be

(8) |

where is the Riemann zeta function. For we have at the same temperature.

As we show in Sec. V.1, the gluon polarization tensor has exactly the same structure in SYMP and QGP, and consequently the Debye mass in SYMP is defined in the same way as in QGP. The masses in both plasmas equal

(9) |

where, as previously, the upper case is for SYMP and the lower one for QGP. For , the ratio of Debye masses squared is 2.4 at the same value of . The Debye mass determines not only the screening length but it also gives the plasma frequency which is the minimal frequency of longitudinal and transverse plasma oscillations corresponding to the zero wave vector. The plasma frequency is also called the gluon thermal mass.

Another important quantity characterizing the equilibrium plasma is the so-called plasma parameter which equals the inverse number of particles in the sphere of radius of the screening length. When is decreasing, the behavior of plasma is more and more collective while inter-particle collisions are less and less important. For , we have

(10) |

As seen, the dynamics of QGP is more collective than that of SYMP.

The differences of and for SYMP and QGP merely reflect the difference in numbers of degrees of freedom in the two plasma systems. In case of and it also matters that (anti-)quarks in QGP and fermions in SYMP belong to different representations - fundamental and adjoint, respectively - of the gauge group.

## Iv Dispersion equations

Dispersion equations determine dispersion relations of quasi-particle excitations. Below we write down the dispersion equations of quasi-gluons, quasi-fermions, and quasi-scalars.

### iv.1 Gluons

Since the equation of motion of the gluon field can be written in the form

(11) |

where color indices are dropped, is the retarded polarization tensor and is the four-momentum, the general gluon dispersion equation is

(12) |

Strictly speaking, one should consider the equation of motion not of the gluon field but of the gluon propagator. Then, Eq. (12) determines the poles of the propagator. Due to the transversality of (), which is required by the gauge covariance, not all components of are independent from each other, and consequently the dispersion equation (12) can be much simplified by expressing the polarization tensor through the dielectric tensor which is the not matrix.

### iv.2 Fermions

The fermion field obeys the equation

(13) |

where any indices are neglected and is the retarded fermion self-energy. The dispersion equation thus is

(14) |

Further on we assume that the spinor structure of is

(15) |

Then, substituting the expression (15) into Eq. (14) and computing the determinant as explained in Appendix 1 of Mrowczynski:1992hq , we get

(16) |

### iv.3 Scalars

The scalar field obeys the Klein-Gordon equation

(17) |

where is the retarded self-energy of scalar field and any indices are dropped. The dispersion equation is

(18) |

As seen, the whole dynamical information about plasma medium is contained in the self-energies which are computed perturbatively in the next section.

## V Self-energies

We compute here the self-energies which enter the dispersion equations (12, 14, 18). The vertexes of super Yang-Mills, which are used in our perturbative calculations, are listed in Appendix A. The plasma is assumed to be homogeneous (translationally invariant), locally colorless but the momentum distribution is, in general, different from equilibrium one. Therefore, we adopt the Keldysh-Schwinger or real-time formalism which allows one to describe both equilibrium and non-equilibrium many-body systems. The free Green’s functions, which are labeled with the indices , are collected in Appendix B. What concerns the Keldysh-Schwinger formalism we follow the conventions explained in Mrowczynski:1992hq . The computation is performed within the Hard Loop Approach, see the reviews Thoma:1995ju ; Blaizot:2001nr ; Kraemmer:2003gd , which was generalized to anisotropic systems in Mrowczynski:2000ed .

### v.1 Polarization tensor

The gluon polarization tensor can be defined by means of the Dyson-Schwinger equation

(19) |

where and is the interacting and free gluon propagator, respectively. The lowest order contributions to gluon polarization tensor are given by six diagrams shown in Fig. 1. The curly, plain, doted and dashed lines denote, respectively, gluon, fermion, ghost, and scalar fields.

Using the vertexes given in Appendix A, the contribution to the contour polarization tensor of Keldysh-Schwinger formalism, which comes from the fermion loop corresponding to the graph in Fig. 1a, is immediately written down in the coordinate space as

(20) |

where the trace is taken over spinor indices. The factor due to the fermion loop is included and the relation is used here.

We are interested in the retarded polarization tensor which is expressed through as

(21) |

The polarization tensors are found from the contour tensor (20) by locating the argument on the upper (lower) and on the lower (upper) branch of the contour. Then, one gets

(22) |

As already mentioned, the system under study is assumed to be translationally invariant. Then, the two-point functions as effectively depend on and only through . Therefore, we put and we write as and as . Then, Eq. (22) is

(23) |

Since

(24) |

the retarded polarization tensor is found as

(25) |

which in the momentum space reads

(26) |

Further on the index is dropped and is denoted as , as only the retarded polarization tensor is discussed. Substituting the functions given by Eqs. (87, 89, 88) into the formula (26), one finds

where with , the traces of gamma matrices are computed and it is taken into account that . We also note that after performing the integration over , the momentum was changed into in the negative energy contribution.

In the Hard Loop Approximation, when , we have

(28) | |||||

(29) |

We note that and for . With the formulas (28, 29), Eq. (V.1) gives

(30) |

which has the well-known structure of the polarization tensor of gauge bosons in ultrarelativistic QED and QCD plasmas. As seen, is symmetric with respect to Lorentz indices and transverse , as required by the gauge invariance. In the vacuum limit, when the fermion distribution function vanishes, the polarization tensor (30) is still nonzero (actually infinite). As we will see, the vacuum contribution to the complete polarization tensor exactly vanishes due to the supersymmetry.

In analogy to the fermion-loop expression (26), one finds the gluon-loop contribution to the retarded polarization tensor shown in Fig. 1b as

(31) | |||||

where the gluon Green’s functions and are given by Eqs. (75, 78), the combinatorial factor is included and

(32) |

with

(33) |

Within the Hard Loop Approximation the tensor (32) is computed as

(34) |

where we have taken into account that .

Substituting the expressions (34) into Eq. (31), using the explicit form of the functions and given by Eqs. (75, 78), and applying the Hard Loop Approximation (28, 29) we get

(35) |

The gluon-tadpole contribution to the retarded polarization tensor, which shown in Fig. 1c, equals

(36) |

where the combinatorial factor is included and equals

(37) |

With the explicit form of the function given by Eq. (77), the formula (36) provides

(38) |

The ghost-loop contribution to the retarded polarization tensor, which is shown in Fig. 1d, equals

(39) |

where the factor is included as we deal with the fermion loop. Using the explicit form of the functions and given by Eqs. (83, 86), the formula (39) is manipulated to

(40) |

which holds in the Hard Loop Approximation.

As already mentioned, the quark-loop contribution to the polarization tensor is symmetric and transverse with respect to Lorentz indices. The same holds for the sum of gluon-loop, gluon-tadpole and ghost-loop contributions which gives the gluon polarization tensor in pure gluodynamics (QCD with no quarks). The sum of the three contributions equals

(41) |

To our best knowledge this is the first computation of the QCD polarization tensor in Hard Loop Approximation performed in the Keldysh-Schwinger (real time) formalism which explicitly demonstrates the transversality of the tensor. In Refs. Weldon:1982aq ; Mrowczynski:2000ed , where the equilibrium and non-equilibrium anisotropic plasmas were considered, respectively, the transversality of was actually assumed. In case of imaginary time formalism, the computation of the gluon polarization tensor in Hard Loop Approximation is the textbook material lebellac ; Kapusta-Gale .

The contribution to the polarization tensor coming from the scalar loop depicted in Fig. 1e is given by

(42) |

which changes into

(43) |

when the functions and given by Eqs. (92, 95) are used and the Hard Loop Approximation is adopted.

The contribution to the polarization tensor coming from the scalar tadpole depicted in Fig. 1f is

(44) |

where the combinatorial factor is included. With the function given by Eq. (94) we have

(45) |

We get the complete contribution from a scalar field to the polarization tensor by summing up the scalar loop and scalar tadpole. Thus, one finds

(46) |

which has the structure corresponding to the scalar QED. Then, it is not a surprise that the polarization tensor (46) is symmetric and transverse.

After summing up all contributions, we get the final expression of gluon polarization tensor

(47) |

where

(48) |

is the effective distribution function of plasma constituents. We observe that the coefficients in front of the distributions functions , , equal the numbers of degrees of freedom (except colors) of, respectively, gauge bosons, fermions and scalars, cf. Table 1. This is obviously a manifestation of supersymmetry. Another effect of the supersymmetry is vanishing of the tensor (47) in the vacuum limit when . Needles to say, the polarization tensor (47) is symmetric and transverse in Lorentz indices and thus it is gauge independent.

In case of QCD plasma, one gets the polarization tensor of the form (47) after the vacuum contribution is subtracted. For the QGP with the number of massless flavors, the effective distribution function equals

(49) |

where , are the distribution functions of quarks and antiquarks which contribute differently to the polarization tensor than fermions of the super Yang-Mills. This happens because (anti-)quarks of QCD belong to the fundamental representation of while the fermions belong to the adjoint representation.

### v.2 Fermion self-energy

The fermion self-energy can be defined by means of the Dyson-Schwinger equation

(50) |

where and is the interacting and free propagator, respectively. The lowest order contributions to fermion self-energy are given by diagrams shown in Fig. 2. The curly, plain, and dashed lines denote, respectively, gluon, fermion, and scalar fields.

The contribution to the fermion self-energy corresponding to the graph depicted in Fig. 2a is given by

(51) |

With the functions , and , given by Eqs. (75, 78, 87, 90), one obtains

(52) |

where the traces over gamma matrices are computed and the Hard Loop Approximation is applied. Eq. (52) has the well-known form of electron self-energy in QED.

Since there are scalar and pseudoscalar fields and , there are two contributions to the fermion self-energy corresponding to the graphs depicted in Figs. 2b, 2c. The first one corresponding to the field equals

(53) |

Due to the relations (3), one finds that . Using the result and substituting the functions , and , given by Eqs. (87, 90, 92, 95) into Eq. (53), one obtains the following result

(54) |

which holds in the Hard Loop Approximation.

The contribution due to the pseudoscalar field is

(55) |

Because , , and , we again obtain the result (54).

### v.3 Scalar self-energy

The scalar self-energy can be defined by means of the Dyson-Schwinger equation

(57) |

where and is the scalar interacting and free propagator, respectively. The lowest order contributions to the scalar self-energy are given by the diagrams shown in Fig. 3. The curly, plain, and dashed lines denote, respectively, gluon, fermion, and scalar fields.

Since there are scalar () and pseudoscalar () fields, we have to consider separately the self-energies of and . However, one observes that only the coupling of scalars to fermions differs for and . The self-interaction and the coupling to the gauge field are the same. Therefore, only the fermion-loop contribution to the scalar self-energy, which is shown in the diagram Fig. 3a, needs to be computed separately for the and fields.

In case of the scalar field, the diagram Fig. 3a provides

(58) |

where the symmetry factor and the extra minus sign due to the fermionic character of the loop are included. With the explicit form of the functions given by Eqs. (87, 90) and the identity which follows from the relations (3), one finds

(59) |

The result holds in the Hard Loop Approximation. For the pseudoscalar we obtain the same expression because ,