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Intermediate energy spectrum of five colour QCD at one loop

I consider the monopole condensate of five colour QCD. The naive lowest energy state is unobtainable at one-loop for five or more colours due to simple geometry. The consequent adjustment of the vacuum condensate generates a hierarchy of confinement scales in a natural Higgs-free manner. QCD and QED-like forces emerge naturally, acting upon matter fields that may be interpreted as down quarks, up quarks and electrons.


Introduction
It is already known [1,2,3], that SU (N ) QCD can lower the energy of its vacuum with a monopole background field along the Abelian directions, where the Abelian components are equal in magnitude but orthogonal in real space [2,3]. This orthogonality, while of no special consequence in SU (3) QCD in three space dimensions, does have consequences when the number of Abelian directions is greater than three. As noted originally by Flyvbjerg [2], SU (N ≥ 5) QCD cannot realise its true minimum because four orthogonal vectors cannot fit in three dimensions. I shall call a system kept from reaching its true lowest energy state by a lack of spatial dimensions dimensionally frustrated.
The research in this chapter seeks to identify the monopole condensate of five-colour QCD, or at least a good candidate for it, and examine the consequences. It assumes the dual superconductor model of confinement [4,5,6,7,8] in which chromomagnetic monopole-antimonopole pairs play the dual role to Cooper pairs, restricting the electric component of the chromodynamic field to flux tubes. This model is by no means proven but the case for it is very strong. I shall handle the monopole degrees of freedom with the Cho-Faddeev-Niemi-Shabanov decomposition [7,9,10,11] which specifies the internal directions corresponding to the Abelian generators in a gauge-covariant way, automatically introducing the monopole field in the process. It is explained in detail in section 2.
At extremely high energies where the effects of confinement are not significant, the dynamics are simply those of SU (5) QCD in the far ultraviolet. I shall show however that the dimensionally frustrated condensate is anisotropic and this causes some colours to be confined more tightly than others. Even more interesting, white combinations do not necessarily need to contain all five colours as one would expect. The first three colours, labelled red, blue and green, form unconfined combinations among themselves, as do the additional two colours, which I have called ultraviolet and infrared. (Actually the confining effect is not quite zero but is much smaller than other effects and is therefore ignored.) This is but one way in which the SU (3) symmetry naturally breaks off from the rest of the symmetry group.
Examination of the gluon dynamics in section 4 finds that those corresponding to one particular root vector are confined more tightly than the rest. At intermediate energies these gluons drop out of the dynamics, causing the coupling constants of those that remain to scale differently and again leading to the separation of SU (3). The emergence of an unconfined Abelian gauge field that can be identified with the photon is demonstrated in section 5. While these sections are chiefly reviews of work already completed [12], section 6 contains new material concerning the matter field representations and identifies the neutral, or white, colour combinations as well as the natural emergence of both up and down quarks and the electron, all with the correct relative electric and colour charges.
While this is strictly speaking an examination of a one-loop effect in SU (5) QCD, the prospect of grand unification does arise. The conclusion of this chapter is that its effective theory does not include weak nuclear decay but that it could well describe a unification of QCD with QED. The value of such a unification is also discussed along with the predictions of this theory in section 7, before ending with a summary in section 8.

The Cho-Faddeev-Niemi-Shabanov decomposition
My treatment of the monopole condensate rests on the Cho-Faddeev-Niemi-Shabanov decomposition [7,9,10,11]. I use the following notation: The Lie group SU (N ) has N 2 − 1 generators λ (j) , of which N − 1 are Abelian generators Λ (i) . For simplicity, I specify the gauge transformed Abelian directions (Cartan generators) witĥ In the same way, I replace the standard raising and lowering operators E±α for the root vectors α with the gauge transformed ones where E±α refers to the gauge transformed operator throughout the rest of this chapter. Gluon fluctuations in theni directions are described by c (i) µ . The gauge field of the covariant derivative which leaves theni invariant is In general this is where summation is implied over i. Bµ can be a attributed to non-Abelian monopoles, as indicated by theni describing the homotopy group π2 The monopole field strength has only Abelian components, ie.
where H (i) µν has the eigenvalue H (i) . Since I am only concerned with magnetic backgrounds, H (i) is considered the magnitude of a background magnetic field H (i) . The field strength of the Abelian components c (i) µ also lies in the Abelian directions as expected and is shown by where The Lagrangian of the Abelian and monopole components is The dynamical degrees of freedom (DOF) perpendicular toni are denoted by Xµ, so if Aµ is the gluon field then where Because Xµ is orthogonal to all Abelian directions it can be expressed as a linear combination of the raising and lowering operators E±α, which leads to the definition so is the monopole field strength tensor felt by X (α) µ . I also define the background magnetic field µν 's non-zero eigenvalue. Since both Bµ, Xµ contain off-diagonal degrees of freedom, it is worth clarifying that Xµ contains the quantum fluctuations taking place on a generally non-trivial background whose topology is contained in the monopole field Bµ.

The Vacuum State of five-colour QCD
The one-loop effective energy of five-colour QCD is given by [2,3] it follows that which means that the chromomagnetic field components must be equal in magnitude but mutually orthogonal in the lowest energy state. However three dimensional space can only accomodate three mutually orthogonal vectors. Since the number of Cartan components, ie. components corresponding to Abelian generators, is always N − 1 in SU (N ) it follows that QCD with more than four colours cannot achieve such an arrangement.
One could substitute the Cartan basis H (i) but this leads to intractable equations that cannot be solved analytically. It is reasonable to expect that the lowest attainable energy state is only slightly different from (16) and that this difference is due to the failure of mutual orthogonality. I therefore propose the ansatz that all Cartan components are equal in magnitude to what they would be in the absence of dimensional frustration, and that their relative orientations in real space are chosen so as to minimise the energy. In practice this means that three of the four are mutually orthogonal and the remaining one is a linear combination of those three. This remainder will increase the effective energy through its scalar products with the mutually orthogonal vectors but not all scalar products contribute equally. This follows from the form of the root vectors in eq. (18). This means that the orientation of the remaining real space vector in relation to the mutually orthogonal ones impacts the effective energy.
A little thought reveals that the lowest energy state should have only one scalar product contribute to it. The problem of finding the lowest available energy state therefore reduces to finding the scalar product that contributes to it the least. The six candidates are As can be seen from while the strongest is and the weakest are Remember that the negative signs are affected by H (3) , H (4) being antiparallel.
Assuming the dual superconductor model of confinement [4,5,6,7,8], it follows that different valence gluons and even different quarks (in the fundamental representation) will be confined with different strengths and therefore at different length scales. Those that feel the background will be confined the most strongly, those that feel the backgrounds of the form H will be confined least strongly. The remainder will be confined with intermediate strength.
At highest energy then, we have the full dynamics of SU (5) QCD. Moving down to some intermediate energy however, finds that the dynamics associated with the root vector 0, 0, − are confined out of the dynamics. The remaining gluons interact among themselves. Moving to lower energy scales I find that those dynamics are all removed in their turn except for those corresponding to the root vectors ?, ?, 1 √ 24 , almost leaving an SU (2) gauge field interaction. I say 'almost' because I shall later demonstrate that the form of the monopole condensate is sufficiently different from the SU (2) condensate to alter the dynamics, producing three confined U (1) gauge fields, two of which are contained within SU (3), a further unconfined U (1) gauge field that may be identified with the photon, and three copies of the valence gluons of SU (2). At lowest energies only the unconfined gauge field remains. In this way a hierarchy of confinement scales and effective dynamics emerges naturally, without the introduction of any ad. hoc. mechanisms like the Higgs field.

Intermediate Energy Dynamics
In constructing the hierarchical picture above, I began with SU (5) and finished with U (1) but had no apparent gauge group governing the dynamics at the intermediate energy scale. The dynamics of this energy scale will prove to be quite interesting.
To facilitate the discussion I introduce a notation inspired by the Dynkin diagram of SU (5). The root vectors implicitly specified in eq. (18) are all linear combinations of a few basis vectors, which according to Lie algebra representation theory can be chosen for convenience. I take the basis vectors which I shall each represent by respectively. The remaining root vectors are sums of these basis vectors.
In this notation their representation contains an 'O' if the corresponding basis vector is included and 'X' if it is not. For example the root vector is represented by OOXX = OXXX + XOXX.
When convenient, a ' ?' is used to indicate that either 'O' or 'X' might be substituted.
In addition to its brevity, this notation has the nice feature of making obvious which root vectors can be combined to form other root vectors because there are no root vectors with an 'X' with 'O's on either side. There is no OXXO for example.
, the valence gluon corresponding to XXXO, out of the dynamics directly affects only those remaining valence gluons that couple to it, those of root vectors of the form ??O?. The remaining gluons, corresponding to the root vectors OOXX, XOXX and OXXX (collectively given by ??XX), may still undergo the full set of interactions available to them at higher energies. It is easy to see that these are the root vectors that comprise the group SU (3), to which the other valence gluons couple forming two six dimensional representations. Subsequent discussion shall extend the X,O,? notation to include the valence gluons corresponding to a root vector. Whether it is the gluon or the root vector that is meant will be clear from context.
Consider the beta function, or to be less imprecise, the scaling of the various gluon couplings. I shall now demonstrate that the loss to confinement of the root vector XXXO causes unequal corrections to the running of the couplings for different gluons. Since this is only an introductory paper the following analysis is only performed to one-loop.
The gluons ??XX, corresponding to the above-mentioned SU (3), retain their original set of interactions. Performing the standard perturbative calculation [2] therefore yields the standard result for SU (5) QCD. The remaining gluons do not. The absence of the maximally confined XXXO restricts their three-point vertices to those of SU (4), since all root vectors are now of the form ???X. The same is not true of the four-point interactions, but the exceptions do not contribute to the scaling of the coupling constant at one-loop [20]. We have then that the SU (3) subgroup's coupling scales differently from the rest of the unconfined gluons when the maximally confined valence gluons XXXO drop out.
The beta function is proportional to the number of colours in pure QCD at one-loop, so as the length scale increases, the coupling among gluons within the SU (3) subgroup initially grows faster than the couplings involving the other gluons. As noted above, the SU (3) couplings will initially scale as in the five-colour theory, while the remainder scale as though there were only four colours. This specific behaviour must soon change due to both non-perturbative contributions and because the non-SU (3) gluons have a weaker coupling. A detailed understanding requires a nonperturbative analysis well beyond the scope of this chapter. Indeed, the application of one-loop perturbation theory at anything other than the far ultraviolet is questionable in itself. The point remains that the SU (3) subgroup ??XX separates from the remaining gluons by its stronger coupling strength.
The symmetry reduction that takes place in this model is suggestive of boson mass generation but there appears to be no obvious specific mechanism. Kondo et. al. have argued for the spontaneous generation of mass through various non-trivial mechanisms [18,21,3]. This is consistent with the well-studied correlation between confinement and chiral symmetry breaking (see [22,23,24,25] and references therein).

The emergence of QED
Neglecting off-diagonal gluons, the equality H (3) = −H (4) allows the change in variables Substituting eqs (28) into the Abelian dynamics (9) finds that the antisymmetric combination Eµ couples to the background H(n3 −n4), but the symmetric combination Aµ does not. Again by the dual superconductor model the former is confined (along with c µn2 ) while the latter is not. Since the electromagnetic field is long range it is natural to interpret Aµ as the photon.
The rotation from c µn3 , c µn4 to Eµ, Aµ in interactions with valence gluons is only meaningful if the gluon in question couples either to both of c

Matter field representations
I have shown how the coupling of the gluons to the monopole background determines their confinement strength and subsequent phenomenology. I now consider the matter fields and focus in particular on the fundamental representation of SU (5). The confinement of the fundamental representation is determined by the maximal stability group [26,27], which for SU (5) is U (4) ≈ SU (4) ⊗ U (1), where for any given element (· · · ψ · · ·) T , the SU (4) acts only on the remaining orthogonal elements while the U (1) causes it inconsequential phase changes. This latter U (1) describes the monopole condensate contributing to the confinement and is given by the corresponding weight of the fundamental representation. As a concrete example, consider the fundamental element (0 0 0 0 ψ) T . The SU (4) of its maximal stability group are the matrices where Ti are the standard SU (4) matrices T1 . . . T15, while the U (1) is generated by T24 = 1 √ 20 diag (1 1 1 1 − 4). Therefore the only component of the chromomonopole condensate that contributes to the confinement of (0 0 0 0 ψ) T is that generated by T24. This is what would be expected based on the weights of the fundamental representation, and indeed the main result of [27] is that the weight of the representation determines which components of the monopole condensate contribute to a given particle's confinement.
The weights of the fundamental representation of SU (5) are (1 0 0 0 0) T : If all Abelian components of the chromomonopole condensate were of equal magnitude and mutually orthogonal in real space so that crossterms could be neglected then they would all be confined at equal length scales and nothing remarkable would happen. However, we already know that such is not the case.
First consider the first three lines in equation (30). They all have identical (small) dependence on H (3) , H (4) , and the same dependencies on H (1) , H (2) as in SU (3) QCD. Consider now that the last two lines show no dependence on H (1) , H (2) , and it is only natural to equate the first three elements with the quark colours of the standard model. This is supported by noting that the two additional weight entries would provide very little additional confinement because the cross terms between H (3) , H (4) have negative sign due to their antiparallelism. In fact the total contribution squared of H (3) , H (4) is As can be seen from the bracket on the right-hand-side, the cross terms almost cancel this contribution entirely. According to the näive interpretation of the dual superconductor model employed by this paper this corresponds to extremely weak confinement. Since it is inconsequential compared to the QCD confinement and might very well scale to zero at lower energies anyway (although I have not shown this!) I shall assume that this corresponds to an unconfined state. This all ties in rather nicely with the result of section 4 in which the corresponding SU (3) dynamics separate from the remaining dynamics through stronger scaling of the coupling constant.
The remaining weights have non-zero elements only in the third and fourth position. The final weight corresponds to a colour charge which I shall refer to as infrared (ir), whose confinement is slightly stronger than that of the QCD colours discussed above, while the penultimate one corresponds to the colour charge ultraviolet (uv), whose confinement is nearly twice as strong as that of the QCD colours. (I shall refer to both of these charges as the invisible colours.) This occurs because there are positively contributing cross terms between H (3) , H (4) (remember their antiparallelism). It follows that ultraviolet must be combined into some neutral combination with infrared at a very small length scale. Only combined with infrared does ultraviolet form an unconfined physical state.
Note that the third and fourth entries of the sum of the ultraviolet and infrared weights are exactly negative three times those of the QCD quark weights. We have already seen that such a combination effectively feels no confining effect from the background condensate. Remembering that the third and fourth Abelian directions provide the unconfined photon Aµ of section 5 gives the electric charge ratio between QCD quark states and white states. In conventional QCD the white states comprise both white combinations of QCD quark colours (hadrons) and truly colourless particles (leptons), and it is simply a fortunate coincidence that both have integer multiples of the electron charge. In this model, states carrying both of the invisible colours but no QCD colours are white and electrically charged, as are white combinations of the QCD colours. From the above discussion of the weights it follows that both these cases also have the same electric charge, up to a negative sign. It will therefore be natural to interpret the state with both invisible colours but no QCD colours as the electron/positron. The reader may recall that the third and fourth Abelian directions also provide the confined photon Eµ. This may confuse some. While it may provide some additional interactions at close range, Eµ is confined, not confining, so the electric charges are still freeto separate to very large distances.
Based on the proceeding discussion the fundamental representation can be shown as where the electric charge of the QCD quarks is implicit in the colour. The red, blue and green quark colours are exchanged by the corresponding SU (3) but the invisible colours are not exchanged except at extremely high energies due to the extra-strong confinement of the ultraviolet charge. At highest energies the dynamics are those of SU (5) in the extreme weakcoupling limit. Electric charge has no meaning at such energies. At intermediate energies at which the gluons XXXO have been confined out of the dynamics not only is there no available gluon to exchange the invisible colours, but the ultraviolet colour itself is confined just as QCD colours are confined at larger distances. This not only removes ultraviolet from the effective dynamics but the infrared as well, because the all-white combinations involving the infrared, apart from a meson-like bound state, require the ultraviolet. This confinement of infrared leaves no source for gluons of the form ??OX which, as discussed in section 4, have an ill-defined electric charge. They may still occur in gluon-antigluon pairs but this is no threat to electric charge.
The intermediate dynamics consist primarily of conventional QCD and QED, but there is no obvious way to include the SU (2) of weak nuclear decay or to turn a QCD quark into a lepton.
I now turn to the asymmetric representation where [1] is the fundamental representation and ⊗AS indicates an antisymmetric cross-product. The top-left-hand corner has the same interpretation as in conventional GUT theories [28]. Red/blue is effectively antigreen etc, and the U (1) (in this case electric) charge is double that of the QCD quarks in the fundamental representation. In other words the 3 × 3 block matrix in the top-left-hand corner can be associated with the anti-up quark when the fundamental representation contains the down quark. The remaining entries of [2] contain either an ultraviolet or an infrared colour charge, which confines them out of intermediate energy level dynamics. The one exception contains both invisible colour charges and is therefor a colourless state with electric charge negative three times that of the down quark, ie. a positron as discussed above. The effective dynamics of this representation, and its complex conjugate [3], are dominated by the colour and electric interactions of the up quark and electric interactions of the electron/positron.

Predictions and prospects for grand unification
This approach yields no weak interaction dynamics but there is a gluonmediated exchange between the up quarks and the electron. This is reminiscent of proton decay in conventional GUTS, in which a down quark becomes a positron and an up quark becomes an anti-up quark which forms a meson with the remaining up quark. In this case however, an up quark becomes an electron and the mediating gluon carries a charge of + 5 3 . This cannot be absorbed by anything within the proton so proton decay is forbidden. It could however be absorbed by an anti-up quark so that an extremely high energy collision between a proton and an antiproton yielding an electron-positron pair and a neutral pair of pions, by which I mean either two π 0 s or one π + and one π − . Unfortunately such a sequence can also occur through ordinary standard model interactions so this is not much of a prediction.
It was natural to hope that dimensional frustration might yield a Higgsless GUT but it makes a good start, unifying SU (3)c with U (1)EM , the forces acting on the right-handed matter fields, which do not feel the weak nuclear force. While I cannot yet claim to have done so, a physically realistic unification of SU (3)c with U (1)EM would be a unification of the forces affecting right-handed matter fields.
Assuming that nature does employ this mechanism to unify the strong nuclear and electromagnetic interactions of right-handed particles, what new phenomena could we expect to see? Obviously there would be new hadrons containing the invisible quarks. These can be divided into two types. There are mesons, we could call them invisible mesons, composed of invisible quark-antiquark pairs. Remember that such pairs can be taken not just from the fundamental representation which contains pure invisible states, but also from the [2] representation which contains states carrying both a QCD colour and an invisible colour. From QCD colour symmetry such states would be mixed, so the distinguishable invisible mesons are It is, of course, possible that these states also mix, either with each other or with standard model mesons. This requires further study and is beyond the scope of this paper. There is one more invisible meson which is listed separatedly because it is already known. Recall that the positron is the quark in the [2] representation with the colour charge uv/ir. The invisible meson associated with the positron is obviously positronium, so e + − e − ⇐⇒ uv/ir − uv/ir.
Another obvious combination is the quark pair made up of each invisible colour from the fundamental representation, uv − ir, and of course its antimatter partner. Next there are particles made up of those quarks in [2] that carry both a QCD colour and an invisible colour, ie. {rbg}/uv and {rbg}/ir. An unconfined combination needs each QCD colour in equal numbers and each of the invisible colours in equal numbers, so at least six quarks are needed.
The quark combinations in the last paragraph are neither mesonic nor baryonic and may be candidates for dark matter. There may also be fancier combinations involving more quarks or even gluons similar to the exotic states discussed in relation to conventional QCD.
Conventional GUTs predict proton decay and monopole production. Dimensional frustration predicts neither of these. It does predict an anomolous scattering however. If an electron is fired into a proton at sufficiently high energies it may turn into an up quark and emit a mediating boson that is absorbed by an up quark and turns it into an electron. The experimenter sends in an electron and sees an electron emerge so this is just a scattering experiment, but it is additional scattering to the electromagnetic interaction already observed in deep inelastic scattering experiments. The strength of the scattering for a given electron energy has not yet been calculated and requires the mediating gluon mass which is currently unknown, although it is natural to expect that it is very massive so that extremely high energy scattering experiments would be needed. It is worth noting however that while reaching ever higher energies is becoming increasingly difficult, an experiment of this kind does not, by modern standards, require sophisticated detection equipment or calculations as it is only measuring electron scattering. Again, further work is required.

Conclusion
I have studied the long known but generally ignored result that QCD with five or more colours has an altered vacuum state due to the limited dimensionality of space, a condition dubbed 'dimensional frustration'. It appears to lead to a unified theory of the strong and electromagnetic interactions, which is not the conventional approach to grand unification, but these two forces are the only ones acting on the right-handed matter fields. Identification of the physical vacuum encounters an intractable set of non-analytic equations but further analysis was enabled by a wellmotivated ansatz.
Assuming the dual superconductor model, a range of confinement scales emerged with one root vector (XXXO) being confined more strongly than all the rest, while some others are less tightly confined (??OO). Gluons remaining at intermediate energy scales exhibit unconventional dynamics because only some of them couple to the XXXO. An SU (3) subset have stronger interactions among themselves at increasing length scales, suggesting the separation of QCD dynamics at lower energy. In addition to a weakly confined U (1) and off-diagonal SU (2) generators, there also emerged a single, unconfined U (1) gauge field consistent with the photon. The theory appears to be a unification of QCD with electromagnetism. Such a unification, if consistent with experiment, is of interest to the standard model in which right-handed matter fields only couple to those two forces.
Study of the matter field representations found that the fundamental representation [1] comprises the three colours of down quark and two more so-called invisible colours, named ultraviolet and infrared. These two colour charges combined will neutralise each other, as do white combinations of the QCD colours. According to the area law of the Wilson loop [29] combined with the non-Abelian Stokes theorem [26,27], the asymmetric arrangement of the chromomonopole condensate gives the invisible colours, especially ultraviolet, an extremely short confinement scale.
The antisymmetric combination of [1] with itself, denoted [2], contains the anti-up quark and the positron, as well as various combinations of the QCD and invisible colours. (The remaining matter representations [3], [4] are the complex conjugates of [1], [2].) Again the invisible-coloured quarks have a very short confinement scale and make little contribution to the intermediate energy dynamics so that the effective theory reduces to QCD and electromagnetism.
Much work remains to be done. The masses of gauge bosons coupling the QCD colours to the invisible colours need to be calculated. The breaking off of QCD symmetry, both through gluon interaction strength and quark confinement scales, suggests a strong symmetry breaking that should render these bosons very massive.
Dimensional frustration is a natural, almost inevitable, means of generating a hierarchy in QCD with five or more colours without resorting to contrived symmetry breaking methods such as the Higgs field. Even a simplistic analysis such as this finds a rich phenomenology.
The author thanks K.-I. Kondo for helpful discussions. This work was partially supported by a fellowship from the Japan Society for the Promotion of Science (P05717), with hospitality provided by the physics department of Chiba University.