Energy scales

PHYSICS REPORTS (Review Section of Physics Letters) 104, Nos. 2-4 (1984) 107-111. North-Holland, Amsterdam

Energy Scales Steven WEINBERG* This talk centered on the problems raised by the variety of energy scales in fundamental physics. Most obvious is the mass scale of the "known" quarks and leptons, extending over some 3 orders of magnitude, and the larger mass scale of the W and Z. These all arise from SU(2) x U(1) breaking, but we are still not sure whether the mechanism for this breakdown is the appearance of vacuum expectation values of elementary scalars, or the binding of composite Goldstone bosons by a new extra-strong "technicolor" force. In the former case the mass scale of SU(2) x U(1) breaking is set by the scalar mass in the Lagrangian, or perhaps by the renormalization group-invariant scale parameter of the ~14 coupling; in the latter, by the RNG-invariant scale parameter A Tc ~ 300 GeV of the technicolor force. But in either case, new elements must be introduced to understand the quark and lepton masses: either a whole host of very small Yukawa couplings of quarks and leptons to elementary scalars, or else yet another extra-strong force, "extended technicolor", with A ETC > 20 TeV. Another class of energy parameters are the RNG-invariant scale parameters of the various gauge couplings: the Ac ~-200 MeV of quantum chromodynamics, and perhaps the much larger scales ATc and A ETC referred to above. Although they are rarely mentioned, there are also the scale parameters of electroweak SU(2) x U(1), where the couplings gl and g2 become of order unity (or would do so, if we ignored SU(2)x U(1) breaking). For gl this is enormous, much larger than 1019GeV; for g2 it is tiny, much smaller than me. One popular way to understand the enormous disparities in the various A's is to suppose that all of the different gauge groups, SU(3) x SU(2) x U(1) x (perhaps) TC x ETC x . . . are tied together by some sort of grand unification, which dictates that the gauge couplings (suitably normalized) become equal at a common scale MGuT. The individual gauge couplings vary slowly, like 1 / X / ~ , and with different proportionality coefficients, so if the common coupling is small at Maux the different A's where the individual COUl61ingsbecome of order unity will be very different from each other and from MauT. For instance, if we suppose that SU(3)x SU(2)x U(1) is part of a simple group G that has an irreducible representation containing only known quarks and leptons plus possible neutrals (e.g. SU(5), SO(10), SU(4) 4, etc.), and that G is broken in one step at Mcu~r to SU(3) x SU(2) x U(1), then A OCD/MGUTis predicted to be roughly exp(-~r/lla), so that MGuT~ 1015 GeV, and the ratio of the SU(2) and U(1) couplings at 100 GeV is successfully predicted. The large size of MatJx has well known implications for possible baryon nonconservation at the edge of observability. It also hints at a connection with gravitation. Perhaps (as in "induced gravity" theories) the coefficient (167rG) -1 = (1.7 x 1018GeV) -2 of the Einstein-Hilbert Lagrangian comes out of the same dynamics that is responsible for the spontaneous breakdown of the grand gauge group. There may also be very light scalars obeying symmetries which only allow them to have very weak non-renormalizable interactions. But there are problems. There is the hierarchy problem: why is the SU(2)x U(1) breaking scale * Research supported in part by the Robert A. Welch Foundation.

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MEW = 300 GeV so different from MGUT? A related problem: why are MEW and Aoco so much closer to each other than either is to MGUT? There is also a less frequently discussed problem: why is the characteristic scale A GUT, at which the grand unified gauge coupling would become of order unity if the grand gauge group were unbroken, so different from MGUT? Our answer to the first two problems depends of course on what we think is the mechanism for SU(2) x U(1) breaking. These problems could find a rather natural solution in technicolor models: MEW is here just another A-parameter, Afc, SO it is very different from MGUT for the same reason as are AocD, Asu(2), and Au(1), and ATC may be relatively close to AocD if the QCD and technicolor gauge groups are not too different. It is in theories with elementary scalars that these problems are so difficult. It was widely hoped that supersymmetry would help to solve the hierarchy problem by providing suitable elementary scalars of low mass. In order for this to be possible, some vestige of supersymmetry must survive down to energies of order 300 GeV. Unfortunately, the idea that supersymmetry is simply unbroken at energies above 300 GeV seems to lead to inescapable phenomenological difficulties. Much attention has been given recently to the possibility that supersymmetry is spontaneously broken at very high energy, but only in some isolated sector of fields that do not interact much with the fields of known particles. In one popular variant of this idea it is gravity that carries the message that supersymmetry is broken to the known particles. These models are not (yet) in disagreement with experiment, and predict all sorts of superpartner particles that might be detected even with present accelerators. Unfortunately one generally has to introduce a new energy scale, the scale of supersymmetry breaking, intermediate between the Planck or G U T scale and the scale of observed phenomena, so this is not yet a satisfying solution of the hierarchy problem. The third problem was to understand why the RNG-invariant scale AGUT is SO different from the GUT-breaking scale MGUT. This is the same as asking why the G U T coupling constant measured at energy MGUT is substantially less than unity. Such problems will have to wait until we have some idea of what it is that determines the gauge couplings in general.

Discussion

R. GATFO: I have a question concerning the hidden sector. Is there an additional symmetry or selection rule or else that defines it as hidden? Technically: in the initial superpotential in principle nothing forbids coupling the additional gauge singlets to the other scalars. One would not like to introduce a sort of philosophical principle saying that something is hidden and something else is open. S. WEINBERG: It is possible to invent symmetries which enforce the structure, but they are not very natural. They may be technically natural but they are really quite contrived. I think this is done "faute de mieux". It seems to me attractive that supersymmetry is broken and somehow or other the news of the breakdown of supersymmetry does not appear at low energy except in certain limited ways. This is one way that that can happen, and it is not the most appealing way. It leaves you with a great sense of mystery, I agree. This does not solve the hierarchy problem. R. HOFSTADTER: I would like to ask whether the/x ~ ey branching ratio, if determined, will give information on the scales of energy that you have discussed. S. WEINBERG: Your question reminds us of another way in which experimental physics at accessible energies can give us a handle on a higher scale. There is a theorem that if you take a minimal model with just

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the known quarks and leptons and just the known electroweak and strong interactions, # does not decay into ey. It is not a selection rule we invented which may be only approximately true. It is analogous to the conservation of baryon number, or in the earlier days to the conservation of strangeness. As in all those cases, if you invent new kinds of particles (Higgs bosons or new gauge interactions) then # - ey can occur and the rate gives a handle on the relevant energy scale. I believe the non-observation of/x - ey is one of the reasons why the technicolor scale (if it is there) must be above a certain value, say 10 TeV. But not observing this decay, we know that there is certain physics that does not occur below that energy. The ETC picture was very attractive but is already in trouble because it predicts things like this that do not happen. There are ways of tinkering the ETC picture, but they are not as attractive as the original picture. So I cannot give you a number for the rate to look for but it is important to look for rare processes. Two processes we should keep looking for are # ~ e y and K ~ # e .

R. HOFSTADTER: The present branching ratio limit for # ~ e7 is 2 x 10 the range around 10-~2 will have been explored.

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but in less than a year

E. WIT-FEN: I would like to draw attention to what I regard as the biggest disappointment about the inverted hierarchy model. In such a model one would hope that, fundamentally, there would be only one mass scale including gravity as well as other interactions. If the fundamental mass scale is much less than the Planck mass, then the effective Planck mass should be renormalized upwards- along with the grand unified mass - in the inverted hierarchy process. This could quite plausibly happen if the theory contains a Brans-Dicke scalar that acquires an anomalously large expectation value in the inverted hierarchy process; its expectation value could be the effective Planck mass. The disappointment is that this does not work. When one actually couples the inverted hierarchy model to supergravity, one finds that rather than the scalar field driving the Planck mass up to a big value, the supergravity interactions prevent the inverted hierarchy mechanism from working at all.

F. WILCZEK: I would like to endorse and extend Professor Weinberg's remarks concerning the possibilities for interesting physics associated with very light scalar particles. One interesting possibility is that broken family symmetries, such as the symmetry between muon and electron or between strange and down quarks, might arise by spontaneous symmetry breaking at a large scale from an exact symmetry. If these symmetries were not gauged, one would expect massless Nambu-Goldstone bosons to arise. It appears that the decay mode K + ~ ~ + + f (f = massless neutral scalar), is the most promising place to look for such things and that symmetry breaking scales up to 10~2-1013GeV might be probed this way. Second, if axions are associated with a symmetry breaking scale 10 ~2GeV, as has been argued on cosmological grounds, then their Compton wavelength is about 1 cm. The coupling is severely suppressed by the fact that it is P and T odd but nevertheless could just possibly be comparable to gravity below these scales. One can also dream about an analog to the axiom call it a cosmion, which would play a role in cancelling the cosmological constant similar to the role of the axion in cancelling the 0-parameter. This would be a very light particle coupled to the trace of the energy momentum tensor. I have not been very successful in numerous attempts to build concrete models of this sort but I think this is another important motivation for Cavendish-type experiments at small distances.

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An important advantage of postulating an extended family symmetry is that such a symmetry could legitimize the Peccei-Quinn "quasi-symmetry", making it an automatic, so to speak accidental, consequence of a real symmetry plus renormalizability. Models of this kind are readily constructed. It is not inconceivable that supersymmetry might play a similar role in legitimizing scale invariance.

S. WEINBERG: I agree with Wilczek's remarks, and in fact I have been working on similar ideas, in part with John Preskill. First, with regard to possible long-range forces produced by axions, the problem seems to be that axions are CP-odd, so they do not couple appreciably to the sort of CP-even operators like 47q, which can add up coherently for macroscopic samples of matter. It is true that CP is not exactly conserved, but as far as I can see true coupling of axions to CP even operators like fO induced by CP violations is too weak to allow axion-exchange to compete with gravitation. One can perhaps evade this problem by preparing a macroscopic system in a state which is not an eigenstate of T. For instance, a pear-shaped rotating nucleus will have a non-vanishing value for E . B (and its color equivalent) so that it couples without derivatives to soft axions. All you need do is prepare macroscopic samples of pear-shaped nuclei, all pointing the same way and all rotating the same way. The axion-exchange forces between two such samples would be much stronger than the gravitational force. It's not so easy. In order to avoid this CP problem one can ask whether scalars could couple to the densities of other topological invariants, such as a well-known linear combination of R 2, R,~R '*~,R , ~ , R ,v~p, which are CP even. Second, I have like Wilczek been attracted by the idea that the cosmological constant problem would be solved through the relaxation of the vacuum expectation value of some sort of very light scalar. It seems to me that the analogy with the invisible axion of Dine, Fischler, and Srednicki is very instructive. They proposed a Peccei-Quinn symmetry spontaneously broken at very high energies, above 10 9 GeV, but it is not possible to understand how this solves the strong CP problem by calculations at these high energies. After all, above 109 GeV the color instantons which are the hub of the problem make an absolutely negligible contribution to the vacuum energy. What really solves the strong CP problem is the appearance in the low-energy effective Lagrangian of a very light scalar, the Goldstone boson of PQ-symmetry breaking, whose vacuum expectation value relaxes to a value that just cancels the strong CP-violation. In the same way, it does no good to solve the cosmological constant problem in terms of calculations at a "fundamental" scale like 1019GeV. What we need to know is not only why quantum gravitational fluctuations or GUT fields do not produce a cosmological constant, but also why this is not produced by low-energy effects like the vacuum energy of QED and QCD. The only possible way that I can see to understand this is to suppose that the low-energy effective field theory contains a very light scalar, whose vacuum expectation value can relax to a value that cancels the gravitational effect of the vacuum energy.

E. WITTEN: I wish to respond to the remarks by Wilczek and Weinberg concerning the cosmological constant. There are certain theories in which- at least classically- the value of the physical cosmological constant (the curvature of space) is not determined by the parameters in the Lagrangian but arises as an integration constant in the course of solving the equations of motion. The only cases of this type that I know of are certain slightly exotic Kaluza-Klein theories (one needs a third rank tensor elementary field such as exists in ll-dimensional supergravity). In these theories, regardless of the numerical values of the constants in the Lagrangian, the equations admit a solution in which four-dimensional space-time is flat. However, this is only one of a one-parameter family of solutions and I do not know why nature would choose it.

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F. WILCZEK: I have two comments: (i) the cosmological term also appears as an integration constant if we work in the gauge X/g = 1 in ordinary gravity, (ii) I find it suspiciously suggestive that in many supersymmetry models the divergences of the dilatation current and of a certain chiral current occur in the same supermultiplet. If the chiral symmetry can be axionized I expect something like a cosmion would be the partner of the axion.