London model of dual color superconductor

. The Meissner e ff ect for the chromoelectric field ⃗ E a is a property of the non-perturbative QCD vacuum medium assumed to explain the observed confinement of color. The color dielectric function ϵ of such a medium should vanish. By Lorentz invariance ϵµ = 1 i.e., its color magnetic permeability µ should diverge. The assumption based on analogy is well motivated: Ordinary superconductor is the physical medium with µ = 0, confining the opposite magnetic charges (would they exist). For the first successful phenomenological description of both the Meissner e ff ect and superconductivity within Maxwell equations Fritz London guessed two equations for the superconductivity current. We adapt his arguments to QCD and come with two analogous manifestly non-Abelian London-like equations. One equation describes the dual Meissner e ff ect. The analogy is, however, not perfect: We can only speculate that there is some sort of chromo-magnetic superfluidity in QCD phenomena to which the second equation might be ascribed. Moreover, from more advanced Ginzburg-Landau (GL) theory of superconductivity it follows that the Meissner e ff ect is a consequence of spontaneous breaking of the underlying global U (1) symmetry. This is certainly not the case of the dual Meissner e ff ect. Fortunately, the derived London-like equations suggest a strongly color-paramagnetic behavior. We interpret it as a guide to the phenomenological Ginzburg-Landau-like man-ifestly gauge-invariant low-momentum QCD with the confining vacuum which behaves as a perfect color paramagnet ( µ = + ∞ ).

Introduction.Our trust in QCD relies mainly on its distinguished perturbative property of asymptotic freedom [1].It uniquely describes the experimental facts in hadron physics caused by the short-distance interactions of the colored quarks and the colored gluons [2].Another experimental fact, also expected to be described by QCD, the permanent confinement of the colored quarks and the colored gluons inside the colorless hadronic jails, stays unexplained [2].Yet it is being repeatedly confirmed by the extensive lattice QCD computations.The color confinement is generally attributed to the peculiar properties of the non-perturbative QCD vacuum medium.The intuitive bag-model picture [3] is that the non-perturbative QCD vacuum is a medium which does not allow for the penetration of the chromo-electric field, of which the colored quarks and the colored gluons are the sources: Its chromo-dielectric function ǫ vanishes.
As in the Lorentz-invariant theories the vacuum must look the same in all Lorentz frames i.e., the confining QCD vacuum behaves simultaneously as a perfect color paramagnet: its chromo-magnetic permeability should diverge: µ → ∞.Realizations of this general idea started with [4].Particularly attractive are those which employ the physical analogy with the "dual" phenomenon, the confinement of the magnetic field in superconductors [5].
In perturbation theory the relation ( 1) is well understood [6]: The asymptotic freedom or anti-screening of the perturbative QCD vacuum medium (ǫ < 1) results from a loop computation, and it is difficult to understand in physical terms.The complementary relation µ > 1 results from the summation over the Landau levels of the color-charged particles in an external chromo-magnetic field, and it is easy to understand: a weak color paramagnetism of the perturbative QCD vacuum is due to the fact that the color-charged QCD gluons carry the spin one [7].
The knowledge of the perturbative QCD Lagrangian is ultimately sufficient for the computation of both the short-and the long-distance quark and gluon properties.The lattice computations support such a point of view.It is, however, quite conceivable [8] that for the intuitive, physical understanding of the QCD quantum ground state, unlike in the infrared-free theories, the classical QCD Lagrangian is practically useless.An effective QCD Lagrangian seems much more efficient.We believe that in this respect the lessons from the long-lasting development of understanding superconductivity are highly relevant, and we try to develop here the simplest one.
The phenomenon of superconductivity of electrons in metals at temperatures close to the absolute zero was discovered in 1911 by Camerling-Ones.In 1933 Meissner had found that the superconductors are not only the perfect conductors but also the perfect diamagnets.In 1935 H.London and F. London [9] came to the idea to replace the Ohm's electric current for the electric E and magnetic B fields by a supercurrent form j s .The reason is clear: For superconductors the conductivity σ in the Ohm's current is infinite.They suggested for the superconducting current j s the equations It is utmost important that these famous London equations are in accord with the Bianchi identities (second pair of the Maxwell equations) div B ≡ 0 ( 6) which follow from the definition of the gauge-invariant field tensor F µν .The equation ( 4) is responsible for superconductivity [10].The equation ( 5) is responsible for the Meissner effect: If we apply rot to (3) we get (for ∂ ∂t E = 0) the equation which says that the magnetic field penetrates into the body of superconductor only up to the London penetration length λ = 1/ √ κ.In the book from 1950 [10] its author justifies the equations (4) and ( 5) with a great charm: "The equations at which we have arrived are distinguished by simplicity and symmetry in such a way that we could hardly avoid writing them down."F. London continues: "However, it is quite conceivable that future developments may necessitate substantial modifications of this theory".
Indeed, the modifications did follow, and it is fascinating that the prescient Fritz London envisaged all of them [10]: The first mandatory question was: Where the equation which clearly underlies the London equations might come from?Being not gauge-invariant Londons never used this equation for the description of the properties of superconductors.F. London, however, ingeniously speculated [10] that such a current could be obtained from the non-relativistic quantum-mechanical current of many electrons if superconductivity is a quantum phenomenon on macroscopic scale characterized by a kind of solidification or 'condensation of the average momentum distribution'.
Ginzburg and Landau (GL) realized that it could be the electric current of the charged bosons provided the boson field can be replaced by a condensate, and developed in 1950 their famous phenomenological theory of superconductivity [11].We notice that both in the London and the GL theories the very notion of the fermionic electrons as the carriers of the electric super-current is entirely missing.
Finally, the second mandatory question was: How the (doubly charged) scalar order parameter of Ginzburg and Landau can emerge in an interacting many-electron system?The paradigm-changing answer was found by Bardeen, Cooper and Schrieffer (BCS) in 1957 [12]: It is a Hartree-Fock-type mean field in the momentumcondensed ground state of a many-electron quantum system with spontaneously broken fermion number.
Why the crude BCS approximation is so successful numerically was, however, a mystery for years.It was clarified only in 1992 by Polchinski [13].
In the following we present what we think is the London theory of the Meissner effect for the chromo-electric field.It requires specification of the non-perturbative form of the color-gluon current, and we briefly discuss its natural physical consequences.
The QCD London equations.We attempt here to follow the logic of the London theory, and develop the analogous phenomenological description of the experimental fact that the chromo-electric field becomes effectively short-range at strong coupling.The equations of motion corresponding to the perturbative QCD Lagrangian Here the chromo-electric field E a and the chromomagnetic field B a are defined as particular components of the covariant color gluon field tensor In the gauge A 0 a = 0 (which, however, is not free of subtleties [2]) they are defined as Knowing that all analogies somehow falter we point out the distinct properties of QCD: 1.In QCD the colored quarks seem entirely irrelevant for the ultimate structure of the confining QCD vacuum medium, and are not considered here.For practical applications, e.g. in hadron spectroscopy the quarks are, however, indispensable as the sources of the chromoelectric field.Their particular colorless and spin configurations ultimately determine the shapes, spins and masses of hadronic bags.
2. The QCD Maxwell equations are non-linear, and cannot be expressed entirely in terms of the chromoelectric and chromo-magnetic fields.Both their left-and right-hand sides are gauge-dependent.
3. The Bianchi identities which follow from the definitions of E a and B a (second pair of the QCD Maxwell equations) have the non-trivial right-hand sides: The pseudo-scalar chromo-magnetic density ρ (M) a was a sign for the existence of the chromo-magnetic monopoles [14].Their condensate, as an analog of the condensate of the Cooper pairs in superconductors, is in the heart of the underlying theories of the dual color superconductors as the models of color confinement [5].
It is our understanding that the pseudo-vector chromomagnetic current j (M) a is another, more general indication of the relevance of non-trivial chromo-magnetic configurations in the confining QCD vacuum medium.In any case its rot will be present in the time derivative of the QCD London current.
4. The non-relativistic non-perturbative phenomena of superconductivity are well described by theories at weak coupling.In sharp contrast, the non-perturbative phenomena in QCD take place only at large distances where their description requires the strong-coupling tools.
We assume that there is a gauge and an appropriate, yet unknown, manifestly gauge-invariant effective QCD Lagrangian at strong coupling which yields, upon appropriate condensation, the QCD Maxwell equations in the form div E a = 0 (16) rot The intention is to find the explicit form of J a which will yield the Meissner effect for the chromo-electric field.It is natural to expect that the new current will be relevant also for the description of other QCD phenomena at strong coupling.In particular, we guess it should describe some sort of super-fluidity of the strongly coupled (predominantly gluonic) QCD matter.First we apply the operation rot to the Eq.( 15), use the QCD equations of motion ( 16) and ( 17) modified for the strong coupling, and get This equation suggests to postulate the first QCD London equation in the form for it leads to the equation for the Meissner effect of the chromo-electric field Here λ = 1/µ is the London penetration length of the chromo-electric field.It should be of the order of the typical hadron size.Second, we apply rot to (19), and after a simple manipulation we obtain the second QCD London equation It is readily verified that the new formulas ( 19) and ( 21) for the chromo-electric and chromo-magnetic fields, respectively, identically fulfil the second pair of the QCD Maxwell equations ( 14) and (15).
In order to have the parallel with the ordinary superconductivity complete we present the QCD London equations also for the case of the static chromo-electric field ( ∂ ∂t E a = 0): It is easy to check that the equation implies the finite penetration length for the static chromo-electric field: The second London equation easily follows: Within our assumptions the gauge dependence of the QCD London current seems unavoidable.With some hesitation we write and leave a discussion of this rather suspicious concept to the concluding section.
It is perhaps worth of noticing that in this case the signs of the "Abelian" pieces of ∂ ∂t J a and rot J a proportional to E a and B a , respectively, are identical to their London counterparts.We take the parallel with the arguments of F. London seriously and discuss also the flow properties of J a having in mind the London QCD equations (22) and (24).
The colored gluon super-fluid.Clearly, the 'derivation' of the QCD London equations depends upon the strong assumptions formulated above which can be verified only a posteriori i.e., once the yet unknown gauge-invariant GL-like theory is found.Phenomenologically the primary outcome, the Meissner effect for the chromo-electric field, is welcome: it is in accord with the successful MIT bag model.
It is then natural to ask whether, similarly to the ordinary superconductivity, the QCD London current describes some super-flow of the strongly interacting QCD matter."We could hardly avoid" a speculation that it might be responsible for the observed almost perfect fluidity of the strongly interacting colored gluon plasma [15] (with an innocent role of the colored quarks).To demonstrate this convincingly deserves extra work.Here we restrict ourselves merely to two qualitative remarks which hopefully support such a speculation.
First, it is gratifying that the suggestion respects the limited value of analogies.It is related to the point 4 above: Both the London electrodynamics of ordinary conductivity and of superconductivity dealing with flow and super-flow are the macroscopic descriptions.In contrast, the perturbative QCD Maxwell equations ( 9) a (10) are by definition appropriate for the short-distance color-gluon phenomena.Consequently, the current j a can hardly be compared with a macroscopic Ohm-like flow.The QCD Maxwell equations ( 16) and ( 17) are, however, the effective ones, intended for the description of the large-distance color-gluon phenomena.Hence, the longdistance flow property associated with the current J a should have the reasonable macroscopic physical sense.By assumption, the "new" definitions of the chromoelectric and chromo-magnetic fields E a and B a in terms of the London QCD currents ( 22) and ( 24), respectively, represent the natural physical degrees of freedom.It is of course utmost important that they fulfil the universally valid identities ( 14) and (15).
Second, association of J a with the macroscopic flow of a strongly coupled color-gluon super-fluid definitely amounts to a non-trivial modification of the velocity field due to the fact that the time derivative of the London current contains the rot of the genuinely non-Abelian pseudo-vector magnetic current j M a .At the London phenomenological level such a description incorporates unspecified chromo-magnetic degrees of freedom in the strongly interacting gluon plasma.It is encouraging that in more explicit particular realizations of such a 'magnetic scenario' the plasma contains, besides the colored gluons, also the chromo-magnetic monopoles, dyons or the magnetic strings [16].In the condensed form these topological objects are suggested for the explanation of the color confinement.The reliability of such a picture is supported by the lattice QCD simulations [17].

Conclusion and outlook.
Within the assumptions formulated above the present London model provides a natural phenomenological basis for the whole class of models of the confining QCD vacuum medium viewed as a dual color superconductor: (1) It yields the Meissner effect for the chromo-electric field, the necessary ingredient for the hadronic bag formation.(2) By definition, the QCD vacuum medium, a perfect color paramagnet, is full of non-perturbative chromo-magnetic configurations.This is phenomenologically incorporated in the fixed chromomagnetic admixture in the London QCD current.(3) The model hopefully describes also a sort of super-fluidity of the strongly interacting quark-gluon droplets [18].It is an unintended, though quite natural, fortunate bonus.(4) In the macroscopic flow of the strongly interacting quark-gluon plasma the unique chromo-magnetic component of the London QCD gluon current should have the specific experimental manifestation.
At the same time the present London-type approach suggests that some physical significance of the gauge potentials for the observable large-distance phenomena in QCD is unavoidable.The implementation of the gauge invariance in general can be a rather subtle issue: (i) In the Abelian electrodynamics the general expectation is that the physical quantities should be gauge invariant i.e., the functions of the gauge invariant fields E and B. Yet, there is the famous quantum loophole, the experimentally observed Aharonov-Bohm effect [19].(ii) Even more to the point, the subtlety of the gauge invariance of the London theory of superconductivity mentioned in the Introduction is nicely discussed in the footnote of [20].(iii) Finally and most important, the new phenomena related to the non-zero mean value A aµ A µ a are expected to emerge in the non-Abelian gauge theory (QCD) at strong coupling [20,21].Gubarev and Zakharov in their insightful paper [21] argue that the topological defects responsible for confinement (like the chromo-magnetic monopoles or the magnetic strings) should manifest themselves by the gauge-dependent vacuum condensate A aµ A µ a .Below we briefly illustrate how such a condensate might justify the form of our London QCD current (25).
To the best of our knowledge the generally accepted effective theory of the strongly interacting QCD (gluon) matter does not exist yet.It should, for example, provide the field-theoretic description of the phenomenological MIT bag model.In the superconductivity language the formation of the bags created by the magnetic monopoleanti-monopole pair in the Abelian GL superconductivity field was elaborated by Ball and Caticha [22].We believe that the general ideas of F. London about macroscopic quantum nature of the future molecular theory of superconductivity apply also in the case of the strongly interacting QCD matter.
Pagels and Tomboulis [8] have suggested the nonpolynomial effective theory in terms of F µν a .It results in the particular dia-electric picture of the QCD vacuum.Another option mentioned by these authors is to consider the algebraically independent polynomials constructed from the QCD field tensor F µν a .There are nine of them for the gauge group SU (2) [23].As far as we know, for the physically distinct SU (3) even their list is not known.
We believe that the starting point for such a program, following Steven Weinberg [24], should be in principle infinite series of gauge-invariant terms formed by the gauge-covariant F µν a and its covariant derivatives.As an illustration consider L ef f = − 1 4 F µν a F aµν + ... It is important that the higher derivatives in an effective Lagrangian are harmless in the sense that they do not cause the Ostrogradsky instabilities [25].Quantized is only the mandatory lowest-order term which defines the QCD degrees of freedom in terms of which we want to describe the system; the higher orders are treated as interactions.
It is suggestive, following Ginzburg and Landau to replace in the most rigid terms of L ef f the products of the two gauge vector potentials A i a A j b by the corresponding Gubarev-Zakharov (GZ) vacuum condensate (Hartree-Fock-like mean field) i.e., by a constant In ordinary GL (or better the Abelian Higgs) model this step corresponds to replacing in the kinetic term (D µ Φ)D µ Φ of the gauge-invariant Higgs Lagrangian the complex scalar field Φ by its gauge-dependent vacuum expectation value Φ = v.From the resulting London Lagrangian L London = −e 2 v 2 A A we easily get the (London) current δL London /δ A = " Relying upon the usefulness of A µ a A aµ [21] we suggest to proceed in L ef f analogously: First, consider in the mandatory, ordinary QCD term of L ef f only its non-derivative part.In the gauge A 0 a = 0 it reads as Performing in this term one HF contraction (26) in all possible positions we arrive at L ef f = −3g 2 Λ 2 A a A a .It yields the first term −µ 2 A a of the London QCD current (25) for µ 2 = 6g 2 Λ 2 .It is perhaps interesting to note that even it is formally identical with the ordinary London superconductivity current, the first term of the London QCD current is of the entirely different, genuinely non-Abelian origin.
Second, the straightforward idea is to obtain the second term of the London QCD current (25) by similar procedure from appropriate dim 6 terms in which we keep one derivative of the gauge potentials.The nonderivative dim 6 terms give rise to a correction to the first term using two contractions.Finding in such a way the term 1  2 gf abc rot( A b × A c ) will be the first step in our future work.
For us the most flagrant use of a gauge-non-invariant vacuum condensate is the glorious BCS [12]: Immediately after its advent Yoichiro Nambu asked himself [26]: How can one trust the BCS with its gauge non-invariant twofermion condensate Φ ∼ ψ ↑ ψ ↓ for discussing the electromagnetic properties of superconductors like the Meissner effect if its only excitations are the Bogoliubov [27]-Valatin [28] fermionic quasi-particles not carrying the definite electric charge ?He ingeniously saved the electric charge conservation and the gauge invariance of BCS by adding to it [29] the now famous collective exitations, the Nambu-Goldstone bosons.
It is conceivable that the use of the GZ gauge-noninvariant but color-symmetry-conserving (!) condensate together with appropriate account of the fixed topological defects (monopoles, strings, ..., ?) responsible for confinement will result in a novel implementation of gauge invariance.The London QCD current (25) seems indispensable for fixing the form of an intuitively transparent effective strong-coupling QCD.
I am grateful to Tom Brauner, Petr Beneš and Adam Smetana for their interest in this work, and for valuable suggestions.