Theoretical High Energy Physics
University of Heidelberg

Philosophenweg 16 -- D-69120 Heidelberg -- Germany

Tel: +49-162 1361402-- Fax: +49-6221-549-333


(courtesy: 60 Years of Yang-Mills Gauge Theory)

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:


Block Lecture, winter semester 2023/2024:

Yang-Mills thermodynamics and applications in particle physics / cosmology

29 July - 02 August 2024, 9:15 - 17:00, Philosophenweg 12, Übungsraum, 3rd floor, lecture will be given in presence

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
ICNAAM 2017, Thessaloniki, Greece
ICNFP 2018, Kolymbari, Crete, Greece

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 and at sufficiently high temperatures, 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. These calorons can be interpreted as quantum fluctuations, endowing propagating modes with a quantum dispersion. In spite of their selfduality, which entails zero pressure and energy density (BPS saturation), calorons carry a potency of originating mostly short-lived, massive, and charged matter upon deformation through inelastic scattering with propagating modes. More specifically, in a given caloron, a nontrivial holonomy is generated which implies a manifest pair of a magnetic monopole and its antimonopole. BPS saturation of the caloron configuration is lifted together with a departure from trivial holonomy. Notice that also 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 degenerate back into a pointlike monopole and an antimonopole whose charge is smeared across the entire 3-space. Once this situation is re-instated the life-cycle of the monopole-antimonopole pair can start over again. Very rarely though, there is repulsion between these 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 caloron centers a part of the spectrum of propagating gauge fields acquires mass (adjoint Higgs mechanism). 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. It is important to point out that as of yet only the case of unadulterated thermodynamics is under theoretical control. 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 can show 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.

Both theories, SU(2) and SU(3) Yang-Mills, undergo two transitions to phases with reduced gauge symmetry if 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, embedded into a condensate of magnetic monopole-antimonopole pairs. The transition between to the low-temperature phase is characterised by a stark deperture from thermal equilibrium and that fact that excitations change their statistical properties: (Center-)vortex loops with a single selfintersection and no slefintersection at all are now (quasi-)stable, vortex loops with higher intersection numbers are instable and occur with large abundancies. 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 nearly 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 demonstrated 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 condensed Planck-scale axions this gross violation of thermal equilibrium by the fluctuations of very abundant but instable objects may be responsible for lepton-number violation in the very early Universe

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 400.000 years after the Big Bang (recombination).

From that epoch onwards 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 for the free electronic charges that are provided by reionized gases induced by radiation stemming from stars (reionization of the Universe).

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).

Gap in the blackbody spectrum at low temperatures and low frequencies

A consequence of an SU(2) gauge symmetry being responsible for the existence of propagating photons is the prediction that the blackbody 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 caloron centers) 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.

Reduced dark matter at high redshifts and the 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 that during this epoch the dark-matter density parameters is reduced compared to the one of LambdaCDM cosmology. For such a cosmological model to match LambdaCDM at low redshift an epoch is required where dark matter is being created from dark energy. Fitting the cosmological parameters of the modified model to the CMB power spectra observed by the PLANCK collaboration, the discrepancies between reionisation redshifts, baryon densites, and the value of H0 are resolved in favour of cosmologically local observation.

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 and the dark Universe

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 (SMPP). 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 corresponding to 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 and the thermal nature of the blob of selfintersection which contains a frequently perturbed monopole providing the charge of the soliton. The interplay a gravitational chiral symmetry breaking at the Planck scale with these SU(2) Yang-Mills theories, based on the axial anomaly, produces ultralight axions whose condensates would represent a unification of astrophysical and cosmological dark matter. In addition, the SU(2) Yang-Mills theory describing the CMB would generate an axion condensate whose extent stretches well beyond the cosmological horizon and therefore acts as dark energy. Within such framework, a communication between particle physics and cosmology is observed: The gauge dynamics, which associates with the emergence of and the interactions between leptons, is, by virtue of an enormous hierarchy between leptonic mass scales and the Planck mass as well as gravitationally mediated chiral symmetry breaking, responsible for the emergence of the dark Universe.
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R.Hofmann, August 2023