Lucini also provides a few interesting insights into ways that walking technicolor theory might remain relevant. In particular, he raises the possibility that the observed Higgs boson could actually be a composite glueball in this beyond the Standard Model theory.
After the quoted material, I make a few observations of my own.
Recent numerical calculations of the glueball spectrum in QCD, in SU(N) Yang-Mills theory in the large-N limit and in candidate theories of strongly interacting dynamics beyond the standard model (in which the lowest-lying scalar plays the role of the Higgs boson) are reviewed and their implications for our theoretical understanding of glueballs in QCD-like theories and in strongly coupled gauge theories with a (near-) conformal dynamics are discussed.
1. Introduction . . .
Among SU(N) gauge groups, phenomenologically, a special role is played by SU(3), which is the gauge group of QCD. One expects that glueball states also appear in QCD. However, to date, glueballs have eluded any experimental attempt aimed at their identification in the QCD spectrum. The fundamental reason for the lack of experimental evidence of glueballs in the spectrum can be traced back to the fact that there is no quantum number that distinguishes glueballs from isosinglet mesons in the same JPC channel. Hence, physical states could naturally be thought of as admixtures of states that in idealised conditions one would call glueballs and states that are naturally identified as isosinglet mesons. This mixing can be assumed to be the result of some off-diagonal effective Hamiltonian that couples nearby states in the spectrum with the same quantum numbers. If the mixing is small, then one could classify the states that are coupled by this Hamiltonian under gluon-rich and quark-rich. However, it is a dynamical problem to establish the strength of the mixing terms.
As for other non-perturbative features of QCD, numerical calculations with Monte Carlo methods in the theory discretised on a spacetime lattice provide invaluable insights on the nature of glueballs. Those calculations should measure the effective couplings for the production and annihilation of glueballs and of mesons (which will be related to the diagonal elements of the effective Hamiltonian describing the interactions of those states) as well as the couplings between glueballs and mesons. Although progress in this direction has been achieved, a calculation of this type is still unviable. Thus, most of current lattice calculations assume that the mixing is small and compute masses of mesons and glueballs separately, the hope being that the real interacting spectrum will not look much different from this simplified case. In addition, since glueball measurements often demand a large statistics, which in turn requires a huge computational effort, the glueball spectrum is often provided in the quenched approximation, in this case amounting to studying the problem in the SU(3) Yang-Mills theory.
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In addition to the theoretical and experimental interest for glueballs in QCD, more recently there has been an increasing interest in glueball-like states in theories that like QCD confine in the infrared and are asymptotically free in the ultraviolet, but unlike QCD have an intermediate energy region in which the physics is governed by an infrared fixed point in an enlarged parameter space. These theories, normally referred as technicolour, are said to be walking, since the coupling runs very slow in this intermediate energy range.1 Walking, and more in general infrared conformal gauge theories, can provide a dynamical mechanism of electroweak symmetry breaking. A full discussion of this mechanism goes beyond the scope of this work, and we refer the interested reader to the specialised literature.
1 Although the coupling is not a physical quantity, the concept of running can be formalised in terms of spectral observables.
Our interest here is in the fact that in this framework the lowest-lying isosinglet scalar plays a special role, since it should be identified with the recently discovered Higgs boson. Once again, the existence of such a light scalar is a dynamical problem that can be addressed with lattice calculations.
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2. Extracting glueball masses from lattice calculations . . .
[One has to keep in mind that the calculation is variational. While this is not a big problem for the lowest-lying states, the variational estimate of higher excitations is less and less accurate, since the number of operators in the variational basis decreases as we go higher in mass. These issues have been known since the early days of glueball calculations on the lattice. Various approaches have been developed over the years and glueball calculations for the ground state and a few excitations in some spin channels (e.g. the 0++ and the 2++) are already quite accurate, at least in the pure Yang-Mills case. . . .
3. From SU(3) to SU(infinity) and back to QCD . . .
For a long time, performing calculations at quark masses that are low enough for being phenomenologically relevant has been a big practical problem in lattice QCD. Recent advances in numerical algorithms and computational power allow calculations to be done at a pseudoscalar mass close to that of the physical pi. However, this still poses a computational challenge that limits the statistics. The number of configurations that can be obtained are generally sufficient for getting the meson spectrum at an acceptable accuracy level. However, numerical calculations of glueball masses require higher statistics (at least by a factor of ten) Monte Carlo simulations. The reason for the different statistical requirements to get an error of the same order of magnitude is well understood, and can be explained in terms of QCD dynamics. While the field is progressing at a fast pace, still the required computational time is highly non-trivial to obtain. Because of these high computational demands, it still makes sense to resort to the so-called quenched approximation, in which the quarks are treated as external sources. In glueball calculations of SU(3), this amounts to studying the pure Yang-Mills theory. At this stage, it is an open problem to establish whether and how these calculations could be relevant for real-world QCD. . . .
[O]ne would expect the physics in SU(3) to only differ from that of the large-N extrapolated theory by a quantity of the order of 10% in each observable. . . . The emerging picture is that, at level of accuracy of around 5-10%, glueballs are well described by their large-N limit, the correction coefficient being of order one. This is consistent with the expectation of the large-N series being convergent at N = 3. In this sense and following the fact that the corrections are of the order of magnitude one would naively expect, one can say that SU(3) is close to SU(infinity). Hence, it makes sense to look at the physics of glueballs in the simpler N = infinity theory. A calculation performed in [another paper] shows – albeit at fixed lattice spacing – that similar conclusions seem also to be valid for all glueball states currently accessible to numerical simulations. This is good news for large-N based approaches to glueballs and QCD-like strong interactions, like for instance the gauge-string duality or, more recently, topological field theory.
When dynamical fermions are considered, it can be shown that they provide a leading contribution that is 1=N-suppressed with respect to the Yang-Mills leading term. Hence, if SU(3) is close to SU(infinity) and the contribution given by quarks stays negligible for N = 3, then glueball calculations in SU(3) gauge theory can still provide a useful guidance to experiments. This provides a motivation to perform high precision quenched calculations.
Another relevant consequence of the large-N limit is that mesons and glueballs do not mix for infinite number of colours. If SU(3) is close to the large-N limit, the suppression of mixing at large-N can justify an approach to glueballs that neglects their mixing with mesons.
Very recently, high statistics numerical calculations of the glueball spectrum in a full QCD setup have been performed and extended to include more spectral states. . . . Most of the glueball states seem to be at quite high mass, in a region that will be accessible to the PANDA experiment. A current focus of lattice calculations is to reduce both the statistical error and the systematic error (e.g. by taking into account mixing with mesons and improving spin identification). In particular, lattice calculations are focusing on exotic quantum numbers (as for instance 2+- and 0-+), since these can not be explained by the simple quark model. . . .
[Ed. the figures in the paper shows central estimates of glueball masses in the 3 GeV to 5.5 GeV ranges (and high end masses within error bars of up to about 6.5 GeV) for all states calculated in this paper except for the ground states for 0++ and 2++ which have masses of more than about 1.7 GeV and less than 3 GeV. These calculations are generically heavier than the calculations from two other sources that are presented for comparison in the same referenced figure which are from about 0.7 GeV to 3 GeV for the various glueball states, with most glueball state masses much more tightly bunched together around a central value.]
4. Glueballs in near-conformal gauge theories . . .
A walking or a conformal novel interaction could provide a dynamical explanation of electroweak symmetry breaking. [Ed. i.e. walking technicolor theories.] As with other strong interactions, the lattice approach can provide essential information on the physical properties of those theories. . . .
As a fundamental step, one must be able to disentangle features of (near-) conformality from a typical confining behaviour when the fermions have a finite (albeit small) mass. One of the signatures of a conformal gauge theory is a spectrum whose mass ratios do not depend on the fermion mass, possibly presenting an inverted mass hierarchy (i.e., unlike in QCD, glueball masses can be lower than the pseudoscalar meson mass). This feature is particularly important in the light of the recent discovery of the Higgs particle, which in this framework will be identified with a scalar that is anomalously light near a point in which scale invariance gets restored. . . .
The experimental identification of glueballs is one of the most urgent open problems in QCD. Thanks to recent progress, lattice calculations are getting closer and closer to providing firm theoretical predictions for glueball masses in all possible channels. Further progress will include a quantification of the mixing with mesons, a systematic study of states with exotic quantum numbers and a more accurate reconstruction of the continuum spin from the lattice data.
More recently, glueballs (or better, 0++ isosinglet states with a non-negligible contribution coming from the pure glue dynamics) are starting to play a key role in understanding the dynamics of strongly-coupled conformal or near-conformal gauge theories and in their phenomenological signatures, with a light glueball spectrum (originally identified for one of those models) now starting to emerge as a key signature of those theories.
Whether electroweak symmetry breaking is described by a gauge theory in the near conformal-phase and the Higgs particle should be seen as a glueball of a new strong interaction is current the subject of a fervid theoretical and phenomenological investigation, with the lattice proving to be once more an invaluable quantitative tool.
From Biagio Lucini, Glueballs on the Lattice (January 7, 2014) (emphasis added, end note references omitted, paragraph breaks adjusted for readability in blog format after omissions).
A Few Observations.
The precision of QCD calculations of glueball masses varies a great deal by type of glueball. The masses of the ground states of the 0++ and 2++ glueball have been calculated very precisely (on the order of 1%). At the other extreme, the precision of QCD estimates of the mass of the 0+-, 0--, and 2-- ground states are roughly +/- 20%. Other masses of other kinds of glueballs and excited states of the more precisely known glueball masses have intermediate precision.
If most glueball states are indeed rather heavy, this is good news for people who are trying to experimentally observe glueballs, because there are fewer mesons in the vicinity of those masses which would have to be distinguished from glueballs.
I had previously been unaware of the gap in theoretical knowledge concerning the extent to which glueballs and mesons mix in Standard Model QCD. The fact that results in N=3 are well approximated across the board by N=infinity models, and the glueballs and mesons have no mixing in N=infinity models provides real hope that the mixing of glueballs and mesons in the real world is minimal. This is because the prospects for experimentally identifying glueballs in the near to medium term future are much better if this mixing is small.
It is also notable that lattice QCD calculations are generally being done in practice these days with a pure Yang Mills SU(3) approximation anyway, rather than full QCD. This implies the recent computational innovation called the Amplituhedron can be used to simplify these calculations without any loss of accuracy relative to the way that these QCD calculations are currently being done. So, progress on the theoretical calculation side of the process of identifying and characterizing glueballs may be closer than it has seemed until recently.
Higgs Bosons As Technicolor Glueballs?
Lucini also deserve credit for his observation that the nature of the glueball mass spectrum provides a means of experimentally testing the theoretical predictions of technicolor, and more further, to provide an experimental test of QCD at a fairly gross scale.
Experimental tests of the QCD part of the Standard Model are particularly valuable. Difficulties in making numerically precise calculations of QCD observables, and the fact that experimental measurements have been made of most QCD observables already and at greater precision than QCD theory allows, means that QCD has mostly tested "post-dictions" and that QCD is one of the less rigorously tested parts of the Standard Model. It has been accepted widely by physicists and not even received much theoretically consideration of alternatives, despite the absence of highly precise experimental confirmation of its fine details, as much as anything because no one has come up with any very good alternatives and because there have been few experimental hints to motivate alternatives to it.
Technicolor theories, like the one discussed in this paper, have largely fallen out of favor. If the Higgs boson had not been discovered, this would have been one of the only alternatives to explain how the rest of the Standard Model works. But, now that the Higgs boson has been discovered, Occam's Razor tends to favor the Standard Model approach, in which the Higgs boson is simply one more fundamental particle with experimentally established properties. This is because the Technicolor approach, while eliminating the need for the Higgs boson to be a new fundamental particle, does so at the expense of adding complexity to some of the underlying guts of Standard Model symmetries and equations, particularly as they evolve at higher energies.
While technicolor theories were mostly proposed as ways to deal with the possibility of the non-detection of the Higgs boson, however, as Lucini's paper notes, there are a subclass of technicolor theories in which a composite technicolor boson, akin to the glueball, can look like a Higgs boson even though it is not the simple fundamental particle Higgs boson of the Standard Model.
The notion that the Higgs boson could be a composite glueball-like body does have some attractions. In general, bosons can occupy the same space at the same time, unlike fermions. So, a composite boson could seem point-like. There are already multiple tempting hints that make a composite-like Higgs boson attactive:
(1) the exceedingly tight coincidence between the Higgs boson mass and the sum of the masses of the four electroweak bosons (W+, W-, Z and the photon), divided by the square root of four (i.e. the square root of the number of electroweak bosons by analogy to a similar formula used for some purposes for linear combinations of electromagnetically neutral mesons like kaons),
(2) the fact that the sum of the electric charges of the four electroweak bosons (i.e. zero) is the same as the Higgs boson electric charge,
(3) the fact that the sum of the spins of the four electroweak bosons could add up to the zero of the Higgs boson (if the W+ and W- have oppositely aligned spins and so do the Z and the photon), and
(4) the fact that only particles that interact with the electroweak bosons have masses generated by the Higgs mechanism.
Of course, all of those hints, which point to the possibility that the Higgs boson could be in some way a composite particle of the four electroweak bosons not gluons or some other beyond the Standard Model second strong force. So, these hints of the Higgs boson as an electroweak boson composite simultaneously tend to disfavover the notion that the Higgs boson has origins as a glueball of the QCD strong force or some strong force-like interaction of some new beyond the Standard Model force. Still, it is an interesting theoretical possibility to muse upon, and it may well be appropriate to do experiments that could distinguish a Standard Model Higgs boson from a technicolor composite Higgs boson.