Theoretical High Energy Physics
University of Heidelberg

Philosophenweg 16 -- D-69120 Heidelberg -- Germany

Tel: +49-6221-54-9421, +49 721 608 22240 -- Fax: +49-6221-549-333

E-mail:R.Hofmann(at)ThPhys.Uni-Heidelberg.de; hofmann(at)particle.uni-karlsruhe.de; ralf.hofmann2(at)kit.edu

courtesy: 60 years of Yang-Mills Gauge Field Theories

Announcement: "3rd Symposium on Analysis of Quantum Field Theory", Thessaloniki, 27-30 September 2017

Book "The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications (1st edition)"

Book "The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications (2nd edition)"

Review, 1st edition: MathSciNet (Mathematical Reviews)
Review, 1st edition: Zentralblatt Math
Review, 1st edition: Contemporary Physics
Review, 2nd edition: MathSciNet (Mathematical Reviews)

Perspectives article on CMB anomalies in Nature Physics

Research interests:


Former, present students and their thesis topics:


Lecture, summer semester 2017:

Talks given recently:

University of California at Santa Barbara
29th Johns Hopkins workshop on Theoretical Physics, Budapest
Free University of Brussels
7th conference Continuous advances in QCD, Minneapolis
Spinoza Instituut, University of Utrecht
Outstanding questions for the cosmological Standard Model, Imperial College, London
Kolloquium ueber Theoretische Physik, Universitaet Karlsruhe
Symmetry in nonlinear mathematical physics, Kyiv, Ukraine
Quantum Field Theory under External Conditions 2007, Leipzig, Germany
Delta Meeting 2007, Heidelberg, Germany
Physikalisch-Technische Bundesanstalt 2010, Berlin, Germany
I@ICNAAM 2011, Halkidiki, Greece
II@ICNAAM 2011, Halkidiki, Greece
III@ICNAAM 2011, Halkidiki, Greece
IV@ICNAAM 2011, Halkidiki, Greece
I@Winter Workshop on Non-perturbative QFT, 2011, Sophia-Antipolis, France
II@Winter Workshop on Non-perturbative QFT, 2011, Sophia-Antipolis, France
Seminar talk, INLN, 2012, Sophia-Antipolis, France
Winter Workshop on Non-perturbative QFT, 2013, Sophia-Antipolis, France
Cosmology and fundamental physics with Planck, 2013, CERN
New Frontiers in Physics, 2013, Kolymbari, Crete
Seminar on Particle Physics, University of Vienna, 2014
Seminar on Theoretical Condensed Matter Physics, KIT, 2014
Joburg Workshop on QCD and Matrices, University of the Witwatersrand, Johannesburg, 2014
Winter Workshop on Non-perturbative QFT, 2015, Sophia-Antipolis, France
60 Years of Yang-Mills gauge field theories, 2015, Nanyang Technological University, Singapore
14th Workshop on non-perturbative QCD, 2016, L' Institute d' Astrophysique de Paris, France
Winter Workshop on Non-perturbative QFT, 2017, Sophia-Antipolis, France
Winter Workshop on Non-perturbative QFT - Ingolf Bischer, 2017, Sophia-Antipolis, France
Winter Workshop on Non-perturbative QFT - Steffen Hahn, 2017, Sophia-Antipolis, France

SU(2) or SU(3) Yang-Mills thermodynamics, nonperturbatively:

Each of these two theories is defined in terms of interacting, massless gauge fields subject to an infinite spacetime resolution. In a (Euclidean) thermodynamical formulation, apart from the propagating fields, which are solutions to the field equations in the limit of zero coupling only, nonpropagating solutions (calorons) exist at any value of the coupling strength. Calorons can be interpreted as particular quantum fluctuations: They are different from those quantum fluctuations that are associated with the propagating modes in possessing no energy density and no pressure (BPS saturation) and yet the potency of originating mostly short-lived, massive matter upon deformation through inelastic scattering with the propagating modes. In a given caloron this temporarily generates a nontrivial holonomy and thus a pair of a magnetic monopole and its antimonopole. That is, the BPS saturation of the caloron configuration is lifted together with a departure from trivial holonomy. Notice that BPS saturated configurations of topological charge modulus |Q|=1, which are of nontrivial holonomy, were constructed. However, the static holonomy parameter inherent to these solutions of the Yang-Mills equations annihilates their potential contribution to the partition function in the infinite-volume limit. In a caloron, deformed away from trivial holonomy by inelastic scattering with a propagating mode, the by far typical situation is that a monopole attracts its antimonopole and vice versa thus causing the system to annihilate with the effect that the life-cycle of the monopole-antimonopole pair can start over again. Very rarely though, there is repulsion between the magnetically charged constituents of a strongly deformed caloron. This facilitates a much longer life of the monopole and its antimonopole in isolation. Thermodynamically, the overwhelming situation of attraction leads to a negative ground-state pressure. By frequent interactions with the calorons a part of the spectrum of propagating gauge fields acquires mass. As a consequence, the computation of thermodynamical quantities such as the pressure or the energy density is enabled in an extremely efficient way: The so called loop-expansion converges very rapidly. Most likely, there is only a finite number of one-particle irreducible contributions to polarizations tensors. It is important to point out that as of yet only the case of unadulterated thermodynamics can be treated exactly, Local distortions by static or time dependent sources can be accounted for in terms of adiabatic approximations due to the strongly correlating nature of the thermal ground state.

Using effective-theory methods, we are now convinced that although isolated and screened magnetic monopoles and antimonopoles do exist in the deconfining plasma their net flux at distances comparable to the magnetic screening length is too small to generate an area law for the spatial Wilson loop at large spatial contour size. This is in contrast to the results of lattice gauge theory. We strongly suspect the reason for this discrepancy to be rooted in the use of the Wilson action at finite lattice spacing a. While the latter does capture the bulk of the nonperturbative ground-state physics at high temperature (trace-anomaly for energy-momentum tensor) such subtle effects like the proper distribution and screening of isolated, long-lived magnetic charge seem to escape these lattice simulations. We would then predict that the use of a perfect lattice action, nonperturbatively adjusted to a given finite a such as to leave the partition function unchanged when changing a->a', yields the same perimeter law for the spatial Wilson loop that we observe in the effective theory.

Both theories, SU(2) and SU(3) Yang-Mills, undergo two transitions to phases with reduced gauge symmetry if the temperature is decreased. There is a very narrow, intermediate phase where all propagating gauge fields are massive. In that phase the overall pressure is negative and dominated by the physics of contracting, magnetic vortex loops. At low temperature excitations change their statistical properties: Magnetic vortex loops are now stable. If there are selfintersections within a given vortex loop then the associated state is a massive, charged or neutral spin-1/2 fermion, if there is no selfintersection then the state is associated with a massless spin-1/2 Majorana fermion. Remarkably, the ground-state pressure is precisely zero in the low-temperature phase (cosmological constant problem!). This statement is proved by integrating out all fluctuations enabled by the (slightly unconventional) Borel summability of the asymptotic series representing the pressure. As a by-product the growing violation of thermal equilibrium by turbulences is shown with increasing temperature. We believe that this result is of relevance when addressing the instability problems encountered in terrestial nuclear fusion experiments with magnetic plasma confinement. Together with Sakharov's three conditions for the generation of charge-asymmetry and the hypothesis of a Planck-scale axion field this gross violation of thermal equilibrium by the fluctuations of very abundant but instable objects may be responsible for the charge-violating emission of the solar wind, see figure for the description of the according table-top experiment in an old german physics textbook.

More applications:

Radiation history of the Universe and massive photon in the future:

Most of the photons in our world are part of the so-called cosmic microwave background radiation (CMB) which, in an ever cooling Universe, got released by the formation of neutral hydrogen atoms at about 300.000 years after the Big Bang (recombination).

From that time on the conventional view is that CMB photons, which became less and less energetic by their redshift due to the Universe's expansion, did not interact with anything except, maybe, rather recently with the free electronic charges that are provided by reionized gases induced by radiation stemming from stars (early reionization of the Universe). This conclusion is drawn from an analysis by the WMAP collaboration of the observed correlation between the fluctuations in temperature and electric-field polarization of the CMB at low multipoles.

An entirely new perspective on the issue of reionization emerges if one allows for an embedding of the U(1) gauge group, associated with the photon, into a larger gauge symmetry SU(2). Namely, a nonperturbative analysis of SU(2) thermodynamics reveals that there is only one point in temperature T(c,E) in the phase diagram of this theory where, in accord with our daily experience, the photon is precisely massless and un(anti)screened. Thus one is lead to identify the present temperature of the CMB of about 2.7 Kelvin with T(c,E). With the Universe's temperature being slightly higher than T(c,E), however, the photon experienced tiny interactions with charged and massive vector particles whose existence is an immediate consequence of the (dynamically broken) SU(2) gauge symmetry. The effect of these interactions on the fluctuations of the CMB temperature and its electric-field polarization peaks at a redshift of about z=1 and dies off rapidly for larger redshifts (or temperatures). Observationally (WMAP 5 years), the dipole and monopole subtracted angular two-point correlation of temperature is consistent with zero for angles larger than 60 degrees, and the spatial orientations of low-lying multipoles seem to be statistically correlated. There is potential for these large-angle 'anomalies' in the CMB to be resolved by the effects of SU(2) adulterated propagation of thermalized photons at temperatures of about twice the present CMB temperature.

Another interesting consequence of SU(2) gauge dynamics determining the physics of the CMB is the prediction that the CMB photon necessarily would have to become massive by an event taking place suddenly in less than 2.2 billion years because the Universe's ground state would then transform into a superconductor.

Observational evidence for the CMB being on the verge of a phase transition:

The data on CMB line temperatures at very low frequencies (Arcade2, radio-frequency surveys, one of Arcade2's papers) point towards a huge excess in comparison with the CMB baseline temperature T=2.725 K extracted at higher frequencies by FIRAS (COBE). This excess can be explained by presuming that the present CMB baseline temperature is just slightly below the critical temperature of the deconfining-preconfining transition thus generating a Meissner mass for low-frequency CMB photons. If the frequency falls below this mass then the associated mode can no longer propagate as a wave (evanescence). As a consequence, its contribution to the frequency spectrum of intensity no longer follows the Planck curve. The intensity of calibrator modes, however, is strictly Planck (or Rayleigh-Jeans) distributed. Thus, by nulling the former with the latter one actually compares `apples' with `pears' giving rise to the before-mentioned excess of line temperature. Extracting the photon mass from the data, one observes saturation of its value at the lowest frequencies. This and the fact that the asymptotic spectral index for line temperature T as a function of frequency, which theoretically is -2, starting from about -2.62 moves in the right direction when invoking lower and lower frequencies in the according fit to the data, indicates that the CMB is on the verge of a phase transition towards a superconducting ground state. Since a U(1) gauge theory for photon propagation predicts a trivial phase diagram and since there is only a single species of photons the only viable candidate gauge theory is based on the group SU(2).

Because of often-voiced confusion concerning the photon-mass bounds cited by the particle-data group we add the following clarifying sentences: All laboratory (null tests of Coulomb law or Amperes' law deviations) or observational (magnetic fields of Jupiter, Earth, or Milky Way) investigations extract their photon mass bounds, which are by at least four orders of magnitude lower than the above-discussed Meissner mass, at local energy densities in a 'nonthermal' situation. By 'nonthermal' we mean that in such situations no spatially and globally constant temperature prevails. These energy densities by many orders of magnitude exceed that of the above-discussed (modified) Planck spectrum at a few Kelvin temperature. Matching these local energy densities to thermal energy densities in some sort of local adiabatic approximation (in the sense of the inverse effect of thermoelectric or - magnetic power in condensed-matter systems), yields local temperatures that are many times the CMB baseline value T=2.725 K. But at these high temperatures no nonabelian effects ((anti)screening or a Meissner mass) take place, and the SU(2) photon propagates in agreement with experiment and observation. Notice that the concept of local thermalization effectively giving rise to macroscopic, space-dependent forces may be quite universally applicable, that is, beyond the situation of a Yang-Mills theory of finite gauge symmetry defined on a flat spacetime.

Gap in the black-body spectrum at low temperatures and low momenta:

A consequence of an SU(2) gauge symmetry being responsible for the existence of propagating photons is the prediction that the black-body spectrum exhibits a sizable gap at low momenta and temperatures. Interestingly, this spectral gap decays with the same power one half of temperature as the effective adjoint scalar field (average over noninteracting trivial-holonomy calorons) does. Physically, the gap emerges because photons are screened by the charged and massive vector modes of the SU(2) theory. The effect is entirely negligible for temperatures, say, larger than 50 Kelvin, and it is absent at the present temperature of the CMB: T(c,E)=2.73 Kelvin. The figure shows the screening function |G| as a function of photon momentum (left) and photon frequency (right) all scaled dimensionless with appropriate powers of temperature (logarithmic plot).

The various solid lines correspond to the exact SU(2) results for T=5.45 (dark grey), 8.2 (grey), 10.9 (light grey) Kelvin, the respective dashed lines are SU(2) results obtained within the approximation that the photon's momentum is on its U(1) mass shell (p squared equals zero). To the right of the deep dip G is negative (antiscreening) while it is positive to the left of the dip (screening). There is a rapid rise of G to the left of the dip indicating that the photon's (screening) mass increases strongly with a decreasing momentum. The black dashed line in the right panel indicates the endpoint in frequency for photon propagation. For low-temperature and low-frequency modifications of black-body spectral energy density see the next figure. Dashed lines indicate the SU(2) results in the approximation that the photon's momentum is on its U(1) mass shell (p squared equals zero), the grey solid lines are the exact SU(2) result, and the solid black lines refer to the conventional Planck spectrum of energy density. All quantities are dimensionless in natural units (c=hbar=k=1). In particular, Y=omega/T is the dimensionless frequency.

Below, we also show the spectral radiance in SI units for the SU(2) black body (spectral radiance and not spectral energy density is the quantity measured radiometrically) at temperatures 5.4, 8 and 12 Kelvin. Red curves are the SU(2) modified spectra and grey curves the convential Planck spectra.

There is a certain amount of astrophysical evidence that an SU(2) gauge theory could describe photon propagation: The observation of large, cold, and dilute clouds of atomic hydrogen inbetween spiral arms of our galaxy. (There is no standard explanation why the hydrogen atoms did not form molecules over a period of about 50 million years, see L. B. G. Knee and C. M. Brunt, Nature 412, 308-310 (2001)). An SU(2) gauge theory, however, explains the absence of the photons required for the mediation of the dipole force between hydrogen atoms by a large screening. The absence of these photons possibly also explains why there is a suppression of large angles in the CMB temperature-temperature correlation function: The correlation of temperature fluctuations, that is present within the standard U(1) description of photon propagation, is strongly suppressed at low redshifts because long-wavelength photons are switched off by screening. Moreover, appealing to the violation of the conventional temperature - scale factor relation at low redshift as introduced by the (nonconformal) SU(2) equation of state, it appears that the discrepancy of redshift for instantaneous reionisation, as extracted from an analysis of the TT angular CMB spectrum (z~11, Planck collaboration) and by onset of the Gunn-Peterson trough at z~6.28 (R. Becker et al., 2001), is resolved.

No dark matter at high redshifts and value of H0

Based on a modified temperature-redshift relation for the CMB, due to the assumption that it is an SU(2) rather than a U(1) photon gas, one concludes by a simple argument, resting on approximate thermal recombination (Saha equation) and matter domination during CMB decoupling, that during this epoch the role of dark matter plus baryonic matter in LambdaCDM cosmology solely is played by baryonic matter. Thanks to the observational extraction of a relation between the present cosmological expansion rate H0 and the comoving sound horizon at baryon drag rs, performed by J.L. Bernal, A. Riess et al. in 2016, such a new high-z cosmological model, however, explains the present discrepancy between the low value of H0 obtained from a fit of LambdaCDM to the TT power spectrum of the CMB on one hand and the high value of H0 observed in terms of cosmologically local signatures on the other hand (H0licow coll. (2016), A. Riess et al. (2016)).

Large-angle anomalies of the CMB, confirmed by PLANCK

The rapid build-up of a cosmologically local, spherically symmetric temperature profile at redshift around one (or a former CMB temperature of about 5.5 Kelvin), sourced by the black-body anomaly arising from thermal, generalised photon dynamics due to the above-mentioned SU(2) Yang-Mills theory of scale 0.0001 eV, would explain the soundly confirmed suppression of the TT correlation function at angles > 60 degrees (PLANCK-CMB). For a 2D analogue, imagine a trampoline whose wobbling surface represents primordial temperature fluctuations of equal strength on all length scales within the visible Universe (horizon defined by the trampoline frame). Imagine further that the center of the trampoline's surface is locally depressed by pull from a rope attached to it from below. The amplitude of this depression supposedly is a hundred times larger than the typical amplitude of a primordial wobble. This creates a 2D spherically symmetric profile whose gradient defines a preferred direction for an `observer' located at a given point of its slope. While primordial wobbles of large scales (order z=1) are smoothed by this process small-scale fluctuations are much less affected. In the CMB the gradient to the black-body anomaly induced 3D spherically symmetric temperature profile gives rise to a dynamic contribution to the CMB dipole, plausibly accounting for the discrepancy between the relativistic Doppler-shift inferred and gravitationally surveyed motion of the Local Group of galaxies, and it predicts a Cold Spot.

Solitonic fermions:

The doublets of single and one-fold selfintersecting center-vortex loops, as they emerge as stable solitons in the confining phase of an SU(2) Yang-Mills theory, may represent the lepton families of the Standard Model of Particle Physics (SM). If true then this would mean that the fundamental gauge-symmetry structure of leptons and their weak (very weak and very, very weak) interactions is a triple product of SU(2) with Yang-Mills scales matching the masses of charged leptons. The apparent structurelessness of the latter down to small distances, as inferred from high-energy collision experiments, would then be a consequence of the excitability of a tower of instable vortex loops with a larger number of selfintersections than unity. This tower mediates the undeterministic transition of initial into final states, well described by perturbative SM vertices, and at the same time protects the incoming lepton from having to reveal its internal structure due to temporary generation of a large but well localized entropy. Certain signatures in high-temperature Z pinch experiments, the production of an anomalous multiplicity of charged leptons at large impact parameter plus recent evidence for a neutral vector resonance of mass ~ 240 GeV decaying into electron-positron pairs as seen in the Tevatron Run II data, and conventionally not explicable narrow and correlated electron and positron peaks as generated in supercritical heavy-ion collisions provide experimental reasons to not dismiss the above-sketched scenario of leptons and their weak (very weak and very, very weak) interactions emerging from pure and confining SU(2) Quantum Yang-Mills dynamics.
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R.Hofmann, June 2017