Universality of Hard-Loop Action
UUniversality of Hard-Loop Action
Alina Czajka
Institute of Physics, Jan Kochanowski University, Kielce, Poland
Stanis(cid:32)law Mr´owczy´nski
Institute of Physics, Jan Kochanowski University, Kielce, Poland andNational Centre for Nuclear Research, Warsaw, Poland (Dated: December 17, 2014)The effective actions of gauge bosons, fermions and scalars, which are obtained within the hard-loop approximation, are shown to have unique forms for a whole class of gauge theories includingQED, scalar QED, super QED, pure Yang-Mills, QCD, super Yang-Mills. The universality occursirrespective of a field content of each theory and of variety of specific interactions. Consequently,the long-wavelength or semiclassical features of plasma systems governed by these theories such ascollective excitations are almost identical. An origin of the universality, which holds within thelimits of applicability of the hard-loop approach, is discussed.
I. INTRODUCTION
The hard-loop approach is a practical tool to describe plasma systems governed by QED or QCD in a gaugeinvariant way which is free of infrared divergences, see the reviews [1–4]. Initially the approach was developed withinthe thermal field theory [5, 6] but it was soon realized that it can be formulated in terms of quasiclassical kinetictheory [7, 8]. The plasma systems under consideration were assumed to be in thermodynamical equilibrium but themethods can be naturally generalized to plasmas out of equilibrium [9, 10].An elegant and concise formulation of the hard-loop approach is achieved by introducing an effective action derivedfor equilibrium and non-equilibrium systems in [11–13] and [9, 14], respectively. The action is a key quantity thatencodes an infinite set of hard-loop n -point functions. A whole gamut of long-wavelength characteristics of a plasmasystem is carried by the functions. In particular, the two-point functions or self-energies provide response functions likepermeabilities or susceptibilities which control various screening lengths. The self-energies also determine a spectrumof collective excitations (quasiparticles) that is a fundamental characteristic of any many-body system.One wonders how much a given plasma characteristic is different for different plasma systems. It has been known fora long time that the self-energies of gauge bosons in the long-wavelength limit are of the same structure for QED andQCD plasmas [15]. Consequently, the collective excitations and many other characteristics are the same, or almostthe same, in the two plasma systems [16]. However, it should be remembered that these systems are so similar inthe domain of validity of the hard-loop approach that is when the momentum scale of collective degrees of freedom isneither too long nor too short. We return to this problem at the end of Sec. III.Comparing systematically supersymmetric plasmas to their non-supersymmetric counterparts, we have considered[17–19] a whole class of gauge theories including Abelian cases: QED, scalar QED, and N = 1 super QED andnonAbelian ones: pure Yang-Mills, QCD, and N = 4 super Yang-Mills. We have observed that the self-energies ofgauge bosons, fermions and scalars, which are computed in the hard-loop approximation, have unique structures forall considered theories irrespective of a field content and of variety of specific interactions. Consequently, the hard-loopeffective actions are essentially the same and so are long-wavelength characteristics of plasma systems governed by thegauge theories of interest. Although our findings are partially presented in [17–19], we have decided to collect all ourresults in this paper and to systematically elaborate on the problem. We explain an origin of the universality, that is,how it happens that the microscopically different systems are very similar to each other in the long-wavelength limit.Physical consequences of the universality and its limitations are also discussed.Our paper is organized as follows. In the next section, we briefly present the gauge theories taken into consideration.The differences and similarities of the theories are underlined. Sec. III is devoted to the self-energies of gauge bosons,fermions and scalars which are computed in the hard-loop approximation. Validity of the approximation is alsoexplained here. Knowing the self-energies, the effective action of the hard-loop approach is derived in Sec. IV. Anorigin of the universality of the hard-loop action, its physical consequences and limitations are discussed in Sec. Vwhich concludes our study.Throughout the paper we use the natural system of units with c = (cid:126) = k B = 1; our choice of the signature of themetric tensor is (+ − −− ). a r X i v : . [ h e p - ph ] D ec II. GAUGE THEORIES UNDER CONSIDERATION
We briefly present here the gauge theories under consideration stressing differences and similarities among them.We start with QED of the commonly known Lagrangian density that is L QED = − F µν F µν + i ¯Ψ D/ Ψ , (1)where the strength tensor F µν is expressed through the electromagnetic four-potential A µ as F µν ≡ ∂ µ A ν − ∂ ν A µ ,Ψ is the Dirac bispinor electron field, D/ ≡ γ µ D µ and the covariant derivative equals D µ ≡ ∂ µ − ieA µ . Since weare interested in ultrarelativistic plasmas, where the plasma constituents are treated as massless, the mass termis neglected in Eq. (1) and in all other cases under study. As well known, the Lagrangian (1) describes a systemof electrons, positrons and photons governed by a long-range electromagnetic interaction represented by the term e ¯Ψ γ µ Ψ A µ .Replacing the electron bispinor Ψ with the scalar complex field Φ, we get the scalar electrodynamics of spinlesscharges and the Lagrangian reads L scalar QED = − F µν F µν − ( D µ Φ) ∗ D µ Φ . (2)Except for the interaction terms e ( ∂ µ Φ ∗ )Φ A µ and e Φ ∗ ( ∂ µ Φ) A µ , there is a four-boson coupling e Φ ∗ Φ A µ A µ . Sucha contact interaction is qualitatively different than that caused by a massless particle exchange. In absence ofother interactions, it gives the scattering which is isotropic in the center-of-mass frame of colliding particles withcharacteristic energy and momentum transfers which are much bigger than those in one photon-exchange processes.A peculiar combination of QED and scalar QED is N = 1 super QED, see e.g. [20], with the Lagrangian of the form L super QED = L QED + i ∂/ Λ + ( D µ Φ L ) ∗ ( D µ Φ L ) + ( D ∗ µ Φ R )( D µ Φ ∗ R ) (3)+ √ e (cid:0) ¯Ψ P R ΛΦ L − ¯Ψ P L ΛΦ ∗ R + Φ ∗ L ¯Λ P L Ψ − Φ R ¯Λ P R Ψ (cid:1) − e (cid:0) Φ ∗ L Φ L − Φ ∗ R Φ R (cid:1) , where Λ is the Majorana bispinor photino field, Φ L and Φ R represent the scalar left and right selectrons; the projectors P L and P R are defined in a standard way P L ≡ (1 − γ ) and P R ≡ (1 + γ ). The supersymmetric extension of QEDdescribes a mixture of photons, Majorana and Dirac fermions, and scalars of two types with a variety of interactions.Except for the long-range one-photon exchanges, we have four-boson couplings and the Yukawa interactions of non-electromagnetic nature. The complete list of elementary processes, which is given in [18], is thus very long and itmakes the supersymmetric plasma very different at the microscopic level from the usual electromagnetic ones.The first nonAbelian plasma under study is that governed by the pure Yang-Mills theory with the SU( N c ) gaugegroup. The Lagrangian of gluodynamics is L YM = − F µνa F aµν , (4)where a, b = 1 , , . . . N c − F µνa is expressed by the four-potential A µa as F µνa ≡ ∂ µ A νa − ∂ ν A µa + gf abc A µb A νc with g being the coupling constant and f abc the structure constant of the SU( N c )group. Due to the self-interaction of Yang-Mills fields, there is the three- and four-gluon coupling.Enriching the pure gluodynamics with (massless) quarks of N f flavors, which belong to the fundamental represen-tation of the SU( N c ) gauge group, we get QCD with the Lagrangian L QCD = L YM + i ¯Ψ i D/ Ψ i , (5)where i = 1 , , . . . N f and the covariant derivative equals D µ ≡ ∂ µ − igτ a A aµ with τ a being the generator of fundamentalrepresentation of the SU( N c ) group. Except for the three- and four-gluon couplings, gluons also interact with thecolor quark current.Finally, the Lagrangian of N = 4 super Yang-Mills theory, see e.g. [21], can be written as L super YM = L YM + i ai ( D/ Ψ i ) a + 12 ( D µ Φ A ) a ( D µ Φ A ) a (6) − g f abe f cde Φ aA Φ bB Φ cA Φ dB − i g f abc (cid:16) ¯Ψ ai α pij X bp Ψ cj + i ¯Ψ ai β pij γ Y bp Ψ cj (cid:17) , where instead of quarks we have four Majorana fermions represented by Ψ ai with i, j = 1 , , , X , Y , X , Y , X , Y ). The components of Φ are either denoted as X p for scalars, and Y p for pseudoscalars, with p, q = 1 , , A with A, B = 1 , , . . .
6. The 4 × α p , β p satisfy the relations { α p , α q } = − δ pq , { β p , β q } = − δ pq , [ α p , β q ] = 0 . (7)In the super Yang-Mills theory all fields belong to the adjoint representation of the SU( N c ) gauge group and thecovariant derivative is D abµ ≡ ∂ µ δ ab + gf abc A cµ . As in QCD there are the three- and four-gluon couplings and thegluon interaction with the color fermion current. Additionally there are the four-boson couplings g Φ A Φ A A µ A µ and g Φ A Φ B Φ A Φ B . There is also the Yukawa interaction of fermions with scalars. The complete list of elementaryinteractions, which is given in [22], is again rather long and it makes the super Yang-Mills plasma quite different atthe microscopic level from the gluodynamic or QCD plasmas. III. SELF-ENERGIES
Our objective is to derive the effective action of all considered theories in the hard-loop approximation. The action S can be found via the respective self-energies which are the second functional derivatives of S with respect to thegiven fields. Thus, the self-energies of gauge boson, fermion and scalar fields equalΠ µν ( x, y ) = δ SδA µ ( x ) δA ν ( y ) , (8)Σ( x, y ) = δ Sδ ¯Ψ( x ) δ Ψ( y ) , (9) P ( x, y ) = δ Sδ Φ ∗ ( x ) δ Φ( y ) , (10)where the field indices, which are different for different theories under consideration, are suppressed. The action willbe obtained in the subsequent section by integrating the formulas (8)-(10) over the respective fields.We compute the self-energies, which enter Eqs. (8)-(10), diagrammatically. The plasma systems under study areassumed to be homogeneous in coordinate space (translationally invariant), locally colorless and unpolarized, but themomentum distribution may be arbitrary. Therefore, we use the Keldysh-Schwinger or real-time formalism, explainedin e.g. [23], which allows one to describe many-body systems both in and out of equilibrium.In the Tables I, III, and V we present the diagrams of the lowest order (one loop) contributions to the self-energiesof gauge bosons, fermions, and scalars, respectively, for all studied theories. Needless to say that the coupling constant g (or e ) is assumed to be small. Since the Feynman gauge is used, the ghost loop contributes to the gluon polarizationtensor. The curly, plain, dotted, and dashed lines denote, respectively, the gauge, fermion, ghost, and scalar fields.As seen in Table I, both the number of diagrams contributing to the polarization tensor and their forms are differentfor each theory. We have the fermion, scalar and gluon loops and the scalar and gluon tadpoles which differentlydepend on the external momentum. Accordingly, there is no surprise that the polarization tensors Π µν ( k ) are quitedifferent for each theory. However, when the external momentum k is much smaller than the internal momentum p ,which flows along the loop and is carried by a plasma constituent, that is when the hard-loop approximation ( k (cid:28) p )is applied, we get a very striking result: the (retarded) polarization tensors of all theories are of the same formΠ µν ( k ) = C Π (cid:90) d p (2 π ) f Π ( p ) E p k p µ p ν − ( k µ p ν + p µ k ν − g µν ( k · p ))( k · p )( k · p + i + ) , (11)where C Π is the factor and f Π ( p ) the effective distribution function of plasma constituents which are both given inTable II for each plasma system. f e ( p ) and ¯ f e ( p ) denote the electron and, respectively, positron distribution functions.The meaning of other functions can be easily guessed. We only add that f ˜ γ ( p ) is the distribution function of photinos.All functions are normalized in such a way that ρ f = (cid:90) d p (2 π ) f f ( p ) (12)is density of particles f of a given spin and color, if any. Particles of the same type but different spin and/or colorare assumed to have the same momentum distribution. The left and right selectrons in N = 1 super QED have the TABLE I: The diagrams of the lowest order contributions to the polarization tensors.Plasma system Lowest order diagramsQEDscalar QED N = 1 super QEDYang-MillsQCD N = 4 super Yang-Mills same momentum distribution as well. It is also assumed that quarks of all flavors, similarly as all fermions and allscalars in N = 4 super Yang-Mills plasma, have the same momentum distribution. In case of non-supersymmetricplasmas, there is subtracted from the formula (11) the (infinite) vacuum contribution which otherwise survives when f Π ( p ) is sent to zero. The subtraction is not needed for the supersymmetric theories where the vacuum effect cancelsout. The polarization tensor (11), which is chosen to obey the retarded initial condition, is symmetric in Lorentzindices, Π µν ( k ) = Π νµ ( k ), and transverse, k µ Π µν ( k ) = 0, and thus it is gauge independent. We note that thetransversality of Π µν ( k ) is not an assumption but it automatically results from the calculations, the details of whichare given in [17, 19, 24] for the electromagnetic theories, N = 4 super Yang-Mills, and QCD, respectively. In case ofnonAbelian theories, the transversality of Π µν ( k ) requires to include the Faddeev-Popov ghosts when the calculationsare performed in a covariant gauge. The problem how to include the ghosts in the Keldysh-Schwinger formalism isdiscussed in [24].One wonders how the universality of the polarization tensor emerges. This is not the case that every one-loopcontribution behaves in the same way in the long-wavelength limit. Just the opposite, the fermion loops contributedifferently than boson ones, and the tadpoles are different than the loops. However, every subset of diagrams which is,as a sum of the diagrams, gauge independent, has the same long-wavelength limit. For example, in the N = 4 super TABLE II: The factors entering the polarization tensors.Plasma system C Π f Π ( p )QED e f e ( p ) + 2 ¯ f e ( p )scalar QED e f s ( p ) + ¯ f s ( p ) N = 1 super QED e f e ( p ) + 2 ¯ f e ( p ) + 2 f s ( p ) + 2 ¯ f s ( p )Yang-Mills g N c δ ab f g ( p )QCD g N c δ ab f g ( p ) + N f N c (cid:0) f q ( p ) + ¯ f q ( p ) (cid:1) N = 4 super Yang-Mills g N c δ ab f g ( p ) + 8 f f ( p ) + 6 f s ( p ) TABLE III: The diagrams of the lowest order contributions to the fermion self-energies.Plasma system Lowest order diagramsQEDelectron in N = 1 super QEDphotino in N = 1 super QEDQCD N = 4 super Yang-Mills Yang-Mills theory we have three such subsets. The first one is simply the fermion loop, the second one is the sum ofthe scalar loop and scalar tadpole, and the third gauge independent subset is the sum of the gluon loop, the gluontadpole and the ghost loop. We also note that the universality holds within the domain of validity of the hard-loopapproximation which is explained at the end of this section after all self-energies of interest are given. A physicalorigin of the universality is discussed in Sec. V.In Table III there are listed the lowest order contributions to the fermion self-energies of every theory. In case ofthe N = 1 super QED, there are the Dirac fermions and Majorana fermions which have to be treated differently. Asin case of the polarization tensor, the fermion self-energies Σ( k ) are quite different for each theory. However, whenthe external momentum k is much smaller than the internal momentum p that is when the hard-loop approximationis applied, the (retarded) self-energies of all theories are of the same formΣ( k ) = C Σ (cid:90) d p (2 π ) f Σ ( p ) E p p/k · p + i + , (13)where C Σ and f Σ ( p ) are both given in Table IV for each plasma system. The indices m, n = 1 , , . . . N c label quarkcolors in the fundamental representation of SU( N c ) group.Table V shows the diagrams of the lowest order contributions to the scalar self-energy of three theories wherescalars occur. As in case of the polarization tensors and fermion self-energies, the self-energy of scalars P ( k ) are quitedifferent for each theory. However, within the hard-loop approximation we obtain the amazingly repetitive result -the scalar self-energies of all theories have the same form P ( k ) = − C P (cid:90) d p (2 π ) f P ( p ) E p , (14)where C P and f P ( p ) are both given in Table VI for each plasma system. As seen, the self-energy (14) is real, negativeand it is independent of the wave vector k .The universal expressions of the self-energies (11), (13), and (14) have been obtained in the hard-loop approximationthat is when the external momentum k is much smaller than the internal momentum p which is carried by a plasmaconstituent. However, it appears that the self-energies (11), (13), and (14) are valid when the external momentum TABLE IV: The factors entering the fermion self-energies.Plasma system C Σ f Σ ( p )QED e f γ ( p ) + f e ( p ) + ¯ f e ( p )electron in N = 1 super QED e f γ ( p ) + f e ( p ) + ¯ f e ( p ) + 2 f ˜ γ ( p ) + f s ( p ) + ¯ f s ( p )photino in N = 1 super QED e f e ( p ) + ¯ f e ( p ) + f s ( p ) + ¯ f s ( p )QCD g N c − N c δ mn δ ij f g ( p ) + N f (cid:0) f q ( p ) + ¯ f q ( p ) (cid:1) N = 4 super Yang-Mills g N c δ ab δ ij f g ( p ) + 8 f f ( p ) + 6 f s ( p ) TABLE V: The diagrams of the lowest order contributions to the scalar self-energies.Plasma system Lowest order diagramsscalar QED N = 1 super QED N = 4 super Yang-Mills k is not too small. It is most easily seen in case of the fermion self-energy (13) which diverges as k →
0. When wedeal with an equilibrium (isotropic) plasma of the temperature T , the characteristic momentum of (massless) plasmaconstituents is of the order T . One observes that if the external momentum k is of the order g T , which is the so-called magnetic or ultrasoft scale, the self-energy (13) is not perturbatively small as it is of the order O ( g ). Therefore, theexpression (13) is meaningless for k ≤ g T . Since k must be much smaller than p ∼ T , one arrives to the well-knownconclusion that the self-energy (13) is valid at the soft scale that is when k is of the order gT . Analyzing higherorder corrections to the self-energies (11), (13), (14), one shows that they are indeed valid for k ∼ gT and they breakdown at the magnetic scale because of the infrared problem of gauge theories, see e.g. [25] or the review [4]. Whenthe momentum distribution of plasma particles is anisotropic, instead of the temperature T , we have a characteristicfour-momentum P µ of plasma constituents and the hard-loop approximation requires that P µ (cid:29) k µ which shouldbe understood as a set of four conditions for each component of the four-momentum k µ . Validity of the self-energies(11), (13), and (14) is then limited to k µ ∼ g P µ . IV. EFFECTIVE ACTION
Having the self-energies Π µν ( k ) , Σ( k ), and P ( k ) given by Eqs. (11), (13), and (14), respectively, we can reconstructthe effective action. Integrating the formulas (8)-(10) over the respective fields, we obtain the Lagrangian densities L A ( x ) = 12 (cid:90) d y A µ ( x )Π µν ( x − y ) A ν ( y ) , (15) L Ψ2 ( x ) = (cid:90) d y ¯Ψ( x )Σ( x − y )Ψ( y ) , (16) L Φ2 ( x ) = (cid:90) d y Φ ∗ ( x ) P ( x − y )Φ( y ) . (17)In case of N = 4 super Yang-Mills, where the scalar fields are real, there is an extra factor 1/2 in the r.h.s of Eq. (17).The subscript ‘2’ indicates that the above effective actions generate only two-point functions. We omit the fieldindices in Eqs. (15)-(17) to keep the expressions applicable to all considered theories. The action is obviously relatedto the Lagrangian density as S = (cid:82) d x L . Using the explicit expressions of the self-energies (11), (13), and (14), the TABLE VI: The factors entering the scalar self-energies.Plasma system C P f P ( p )scalar QED e f γ ( p ) + f s ( p ) + ¯ f s ( p ) N = 1 super QED e f γ ( p ) + f e ( p ) + ¯ f e ( p ) + 2 f ˜ γ ( p ) + f s ( p ) + ¯ f s ( p ) N = 4 super Yang-Mills g N c δ ab δ AB f g ( p ) + 8 f f ( p ) + 6 f s ( p ) Lagrangians (15)-(17) can be manipulated, as first shown in [13], to the forms L A ( x ) = C Π (cid:90) d p (2 π ) f Π ( p ) E p F µν ( x ) p ν p ρ ( p · ∂ ) F µρ ( x ) , (18) L Ψ2 ( x ) = C Σ (cid:90) d p (2 π ) f Σ ( p ) E p ¯Ψ( x ) p · γp · ∂ Ψ( x ) , (19) L Φ2 ( x ) = − C P (cid:90) d p (2 π ) f P ( p ) E p Φ ∗ ( x )Φ( x ) , (20)where the operator inverse to p · ∂ acts as 1 p · ∂ Ψ( x ) ≡ i (cid:90) d k (2 π ) e ik · x p · k Ψ( k ) . (21)The operator ( p · ∂ ) − is defined analogously.The n − point functions with n >
2, which are generated by the actions (18)-(20), identically vanish, as the actionsare quadratic in fields. We also observe that the action of scalars (20) is gauge invariant for every theory which includesthe scalar field. Moreover, the gauge boson action (18) is invariant as well but only in the Abelian theories. Thefermion action is gauge dependent in all theories under consideration. Therefore, the fermion action and, in general,the gauge boson action need to be modified to comply with the principle of gauge invariance. This is achieved bysimply replacing the usual derivative ∂ µ by the covariant derivative D µ in Eqs. (18) and (19). Thus, we obtain L A HL ( x ) = C Π (cid:90) d p (2 π ) f Π ( p ) E p F µν ( x ) p ν p ρ ( p · D ) F µρ ( x ) , (22) L ΨHL ( x ) = C Σ (cid:90) d p (2 π ) f Σ ( p ) E p ¯Ψ( x ) p · γp · D Ψ( x ) , (23) L ΦHL ( x ) = − C P (cid:90) d p (2 π ) f P ( p ) E p Φ ∗ ( x )Φ( x ) . (24)The forms of covariant derivatives present in Eqs. (22) and (23) depend on the theory under consideration. In theelectromagnetic theories, the derivative in the gauge boson action (22) is, as already mentioned, the usual derivativewhile that in the fermion action (23) is D µ = ∂ µ − ieA µ . The operator ( p · D ) − acts as1 p · D Ψ( x ) ≡ p · ∂ ∞ (cid:88) n =0 (cid:16) − iep · A ( x ) 1 p · ∂ (cid:17) n Ψ( x ) . (25)In the N = 4 super Yang-Mills the covariant derivatives in Eqs. (22) and (23) are both in the adjoint representationof SU( N c ) gauge group. The formula (25) should be then appropriately modified. In QCD, the covariant derivativein Eq. (22) is in the adjoint representation but that in Eq. (23) is in the fundamental one. As already mentioned,there is an extra factor 1/2 in the r.h.s of Eq. (24) in case of N = 4 super Yang-Mills.The hard-loop actions (22), (23), and (24) are all of the universal form for a whole class of gauge theories. However,the case of Abelian fields differs from that of nonAbelian ones. In the electromagnetic theories the gauge boson andscalar actions are quadratic in fields. Therefore, the n − point functions generated by these actions vanish for n > V. DISCUSSION
We have shown that the hard-loop self-energies of gauge, fermion, and scalar fields are of the universal structuresand so are the effective actions of QED, scalar QED, N = 1 super QED, Yang-Mills, QCD, and N = 4 super Yang-Mills. One asks why the universality occurs physically. Taking into account a diversity of the theories - various fieldcontent and microscopic interactions - the uniqueness of the hard-loop effective action is rather surprising.To better understand the problem in physical terms, let us consider the QED plasma of spin 1/2 electrons andpositrons and the scalar QED plasma of spin 0 particles and antiparticles. The universality of hard-loop action meansthat neither effects of quantum statistics of plasma constituents are observable nor the differences in elementaryinteractions which govern the dynamics of the two systems. Both facts can be understood as follows. The hard-loop approximation requires that the momentum at which a plasma is probed, that is the wavevector k , is muchsmaller than the typical momentum of a plasma constituent p . Therefore, the length scale, at which the plasma isprobed, 1 /k , is much greater than the characteristic de Broglie wavelength of plasma particle, 1 /p . The hard-loopapproximation thus corresponds to the classical limit where fermions and bosons of the same masses and charges arenot distinguishable. The fact that the differences in elementary interactions are not seen results from the very natureof gauge theories - the gauge symmetry fully controls the interaction. And the hard-loop effective actions obey thegauge symmetry.The universality of hard-loop actions has far-reaching physical consequences: the characteristics of all plasmasystems under consideration, which occur at the soft scale, are qualitatively the same. In particular, spectra ofcollective excitations of gauge, fermion, and scalar fields are the same. Therefore, if the electromagnetic plasma witha given momentum distribution is, say, unstable, the quark-gluon plasma with this momentum distribution is unstableas well. We conclude that in spite of all differences, the plasma systems under consideration are very similar to eachother at the soft scale. However, the hard-loop approach breaks down for the momenta at and below the magneticsale. Then, systems governed by different theories can behave very differently. In particular, the QED plasma is verydifferent from the QCD one, as in the latter case effects of confinement apparently appear at the magnetic scale.Recently, there have been undertaken several efforts to extend methods of the hard-loop approach to the ultrasoftscale [26–30]. These efforts explicitly show limitations of the universality we have elaborated on here. Acknowledgments
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