Search

We show how chiral four-dimensional spinors can be obtained from six-dimensional riemannian geometry, if two dimensions form a noncompact internal space with finite volume.

We examine patterns where ratios of the fermion masses and the W-boson mass ( x i = m i m w ) are proportional to powers of a small parameter λ(xi=ciλPi). For a simple estimate of the uncertainty in the coefficients ci we determine the allowed values of Pi and the corresponding range of λ. Using this information we search for realistic patterns in a large class of anomaly-free SU(3)×SU(2)×U(1)×U(1) models where λ is related to a symmetry breaking scale and the Pi follow from the quantum numbers. No realistic model is found. In contrast, realistic mass patterns can be induced from an anomalous U(1) symmetry.

Einstein's equations with a cosmological constant in arbitrary dimensions admit solutions with unobservable internal space and flat four-dimensional space. Internal space is non-compact, has finite volume and admits a compact group of symmetries. No fine tuning of parameters is needed to obtain a vanishing four-dimensional cosmological constant.

The number of chiral fermions may change in the course of spontaneous symmetry breaking. We discuss solutions of a six-dimensional Einstein-Yang-Mills theory based on SO(12). In the resulting effective four-dimensional theory they can be interpreted as spontaneous breaking of a gauge group SO(10) to H = SU (3) C × SU (2) L × U (1) R × U (10) B−L . For all solutions, the fermions which are chiral with respect to H form standard generations. However, the number of generations for the solutions with broken SO(10) may be different compared to the symmetric solutions. All solutions considered here exhibit a local generation group SU(2)G × U(1)G. For the solutions with broken SO(10) symmetry, the leptons and quarks within one generation transform differently with respect to SU(2)G × U(1)G. Spontaneous symmetry breaking also modifies the SO(10) relations among Yukawa couplings. All this has important consequences for possible fermion mass relations obtained from higher-dimensional theories.

We argue that the vamshmg of the cosmological constant obtains as a result of the dynamics of a new field-the cosmon-which is the Goldstone boson of dilatation lnvanance, assumed to be broken spontaneously near the Planck scale The presence of the cosmon, coupled to the fact that scale lnvarlance is anomalous at the quantum level, drives the cosmological constant to zero, provided that the energy-momentum trace is purely anomalous. Furthermore, these quantum anomalies give the cosmon a small mass, giving rise to an intermediate range force (2~< 10 4 m). We can eshmate the effect of the cosmon force between macroscopic matter distributions The dominant component of the force is attractive, couples to mass and should be weaker than gravity (a ~ 10-2 10 3). There is, however, also a repulsive baryon-number-dependent component of calculable strength ( ~ ~a) and an even smaller contribution proportional to Z-N. Einstein's equations admit Minkowski space as a solution only if the trace of the energy-momentum tensor T ~" in vacuum vanishes (( T~ ) is the cosmological constant). In the standard model of the strong and electroweak interactions, however, there exist various nonvanishing contributions to this trace. For instance, chiral symmetry breaking in QCD gives a contribution to (Ta a) proportional to (mulaU 2

We discuss cosmologies where the Planck length is not a fundamental constant but rather evolves with time. The dynamics which should be responsible for today's tiny value of this length scale are governed by the effective potential of a Brans-Dicke type theory. Qualitative properties of this potential depend on the short distance behaviour of the unifying fundamental theory. We discuss criteria for the asymptotic behaviour of realistic cosmologies and show that the role of a possible cosmological constant is quite different from the case of standard cosmology.

Gravity in 13 dimensions with ground state M 4 × S 9 exhibits a local SO(10) gauge group. We discuss the dimensional reduction for this model. The model is enlarged to 17 dimensions to include a generation group for four fermion generations.

We discuss the speculation that the sharp positron lines and correlated e/sup +/e/sup -/ and ..gamma gamma.. signals seen in heavy-ion collisions may be evidence for a new phase of QED. We examine several characteristics of the data which argue for this interpretation and point out further experimental observations which would favor this hypothesis. However, we detail also theoretical difficulties and experimental contradictions which considerably weaken the basis for this speculation. In particular, we argue that for the formation of a new phase or a solitonlike structure in QED it is necessary that nonlinear effects in electrodynamics become important. Even though Z..cap alpha.. is large, these effects always entail a suppression factor of ..cap alpha.., which is difficult to overcome.

The relation m/sub t//m/sub d/roughly-equalm/sub b//m/sub s/, which predicts a top-quark mass close to 40 GeV, is derived in the context of SO(10) and (and E/sub 6/) grand-unified theories with suitable intermediate scales.

Some consequences of an SO(10) gauge theory which breaks down to SU(3)c × U(1)e.m. via SU(4)c × SU(2)L × SU(2)R are presented. These include: (i) A proton lifetime estimate of (1–20) × 1031 yr. (ii) Mass relations involving quarks and charged leptons. The explanation of the violation of the asymptotic SU(5) relation me = md is linked to the new intermediate mass scale in SO(10). The top quark mass is estimated to be 20 ± 2 GeV at the toponium mass. (iii) The possibility of a fourth generation (ντ′, τ′, b′, t′) with mτ′ ≅ 17–35 GeV, m b ′ ⋍ 42–118 GeV and m t ′ ⋍ 150–230 GeV . (iv) Heaviest neutrino mass in the 1–10 eV range. In addition, the masses of the superheavy neutrinos are of order 5 × 1012 − 5 × 1013 GeV. (v) Neutron-antineutron oscillations with τ n n > 10 38 yr.

Higher dimensional theories predict ratios between Yukawa couplings and gauge couplings in the resulting four-dimensional gauge theory. After spontaneous symmetry breaking of weak interactions, these relations are converted into relations between particle masses. We calculate explicitly the Yukawa couplings resulting from a six-dimensional SO(12) gauge theory. We argue that the fermion masses follow the pattern mt, ≫ mb, mc mτ ≫ ms, mμ ≫ md, mu, me.

We discuss possibilities of obtaining chiral four-dimensional fermions from dimensional reduction of pure higher dimensional gravity. We explore a modification of riemannian geometry where the Lorentz rotations are treated in close analogy to usual gauge theories. The metric is not the product of two vielbeins and the vielbein may not be invertible everywhere. The bundle structure of Lorentz transformations is distinguished from the bundle structure of tangent space rotations and the gravitational index theorems have to be modified for this case. We also investigate noncompact internal spaces with finite volume in the context of riemannian geometry. Chiral fermions are obtained in the latter case. As a byproduct of this work, we find that for the usual torsion theories the Dirac operator is not the relevant mass operator for dimensional reduction of fermions.

We consider fermions in theories of higher dimensional gravity where the four-dimensional gauge group is embedded in the invariance group of d dimensional (d>4) Lorentz and general co-ordinate transformations. It is a necessary condition for obtaining massless chiral fermions from dimensional reduction that the d dimensional spinor does not admit a mass term consistent with Lorentz and general co-ordinate transformations. This is the case for a Weyl spinor for d = 6 8 mod 8, a Majorana spinor for d = 9 mod 8 or a Majorana-Weyl spinor for d = 2 mod 8.

Models are considered which have no small Yukawa couplings unrelated to symmetry. This situation is generic in higher dimensional unification where Yukawa couplings are predicted to have a strength similar to the gauge couplings. Generations have then to be differentiated by symmetry properties and the structure of fermion mass matrices is given in terms of quantum numbers alone. Possible symmetries leading to realistic mass matrices are scanned.

Certain embeddings of the low energy SU(3) C × SU(2) L × U(1) Y gauge group within the higher dimensional symmetry lead to exotic chiral generations in addition to a number of standard generations. An example is studied, leading to an anomaly free chiral exotic generation consisting of quarks with electric charge + 1 6 and − 5 6 , the corresponding antiquarks and addition integer charged leptons. The most striking predictions from such a compactification would be the existence of stable half-integer charged hadrons with a mass of the order of the Fermi scale. Detection of such particles would be an impressive hint for the existence of more than four dimensions.

Within the context of a class of higher dimensional models of gravity, we investigate the heating of the universe following an inflationary phase. High temperatures, typically on the order of 10 17 GeV can be achieved. This allows for a subsequent production of baryon asymmetry and, if existing, superheavy cosmic strings.

The number of four-dimensional chiral fermions obtained from dimensional reduction of models with spinor matter fields coupled to pure gravity in d > 4 dimensions is linked to topological properties of the internal d − 4 dimensional space. This gives important restrictions on possible ground states of such models consistent with a realistic four-dimensional unified theory. Connections with spontaneous symmetry breaking and Yukawa couplings of fermions in unified theories are discussed.

We find an absolute minimum of an SO(10) symmetric potential with SU(3) × U(1) invariance. By fixing the higher scales MR(〈126〉) ⪡ Mx(〈54〉), the model is consistent with the experimental knowledge about matter stability and the value of sin2 θw. We determine the spectrum of scalar particles and show that their tree-diagram contributions to nucleon decay amplitudes are proportional to 1/Mx2.

Six-dimensional Einstein-Maxwell theory admits a continuous spectrum of classical solutions, where the four-dimensional cosmological constant is a free integration constant. It is shown how the existence of this continuous spectrum is related to the non-compactness of internal space. The need for a fine tuning of parameters to obtain the zero four-dimensional cosmological constant arises only if one requires compact internal space. For non-compact internal space one can envisage dynamical mechanisms leading to a vanishing or extremely small value of the four-dimensional cosmological constant. Similar ideas apply to the gauge hierarchy problem. For non-compact internal spaces we find solutions where the scale of spontaneous symmetry breaking is a free integration constant.

A scenario describing how the four-dimensional Friedmann cosmology may evolve from a model of gravity in 4 + D dimensions which admits spontaneous compactification of D dimensions is presented. In particular, the possibility that the transition to the standard hot big bang model involves an inflationary phase is explored.

A systematic discussion of the structure of fermion mass matrices in terms of quantum numbers is presented. Small ratios between fermion masses and small mixing angles are related to a fine structure of scales around the unification scale. We argue that in higher dimensional models all small fermion masses should be explained from symmetry considerations since no free small Yukawa couplings are available. This leads to a scanning procedure selecting higher dimensional models consistent with realistic fermion mass patterns. We present a six-dimensional model admitting “compactifications” with only SU(3)c×SU(2)L×U(1)γ gauge symmetry, a vanishing cosmological constant and three generations of quarks and leptons. The field equations have solutions with a gauge hierarchy for weak symmetry breaking for a large range of model parameters without the need of fine tuning. The weak scale MW is a free integration constant and the mechanism determining its order of magnitude is not yet identified. These solutions have a good chance to be classically stable. For one particular solution the largest fermion mass is the top quark mass which is of the same order as MW. At the next level the fermion masses mb, mτand mc are supressed by a small ratio of symmetry breaking scales γ. For the mixing between the second and third generation one finds θ23≈mb/mt≈γ. The relation mb(M) = mτ(M)(1+Oγ)) is predicted. Corrections of order γ2 quark and the muon with the relation ms(M) = 13 mμ(M). This reproduces the qualitative order of magnitude ms/mb ≈ mc/mt. Unfortunately this particular solution fails by predicting maximal Cabibbo mixing and θ13 ≈ γ. The model can be interpreted as a subgroup analysis for E8 × E8 superstrings. We also give a systematic discussion of higher dimensional scalar fields in non-trivial representations of the gauge group. We describe the higher dimensional Higgs effect which can lead to a stabilization of the ground state.

Yukawa couplings or the quartic scalar coupling may become strong at an energy scale below the unification mass. This could lead to a natural explanation of a large gauge hierarchy. The new strong interactions require the scalar mass to be larger than 173 GeV.

The couplings of a light scalar to ordinary matter induce non-vanishing fields φ inside and around stars. In turn, the masses and couplings of ordinary particles depend on φ in general. Thus, fundamental coupling constants can be altered by dense matter nearby. We estimate the resulting effect on the cooling of neutron stars, the neutrino burst from supernova 1987 A and the period of the pulsar recently discovered in its remnant, and obtain bounds on the mass and couplings of scalar particles. We also comment on cosmological and other constraints on such scalar particles, and discuss in particular bounds on the mass of the dilaton in string theories.

We discuss a six-dimensional SO(12) gauge theory which can be obtained from pure gravity in 18 dimensions coupled to a Majorana-Weyl spinor, if the ground state is characterized by a noncompact internal space without boundary with small finite volume. The six-dimensional SO(12) theory spontaneously compactifies to a four-dimensional SO(10) theory with local generation group SU(2)G × U(1)G. We obtain an even number of chiral fermion generations transforming as ( 16, k, ± 1 2 ) under SO(10) × SU(2)G × U(1)G. Adding a scalar field to the six-dimensional theory provides us with fields carrying all the quantum numbers needed for a realistic spontaneous symmetry breakdown to SU(3)c × U(1)e.m.

We investigate an inflationary scenario leading to a power spectrum for the scale dependence of primordial density fluctuations, Δϱ/ϱ ∼ l−ν. The model has no very small dimensionless parameters and Δϱ/ϱ is of order unity for scales lE corresponding to the horizon near the end of inflation. The small primordial density fluctuations on galactic scales lG are explained by the huge ratio lG/lE, and are therefore a direct consequence of the many e-foldings of the scale factor during inflation. Possible observable consequences for the structure formation in the universe are shortly addressed.

We estimate the mass of the fractionally charged gauge bosons contained in the SU(4) colour subgroup of O(10), with lepton number regarded as the fourth colour. By comparing the electro-weak couplings, a lower bound of 3 × 109 GeV is obtained. This is increased to 2 × 1010 GeV if strong interactions are included in the considerations. The proton lifetime is briefly discussed.

We investigate within the standard model the possibility that nonperturbative QCD effects determine the Fermi scale and electroweak symmetry breaking is a consequence of chiral symmetry breaking. In this scenario the ratio between the Fermi scale and the quark condensate 〈 ψ ψ〉 0 1 3 comes out inversely proportional to the Yukawa coupling of the strange quark, consistent with observation. The Higgs particle mass is predicted in the range of 200 keV.

We investigate unified models where all small quantities in the fermion mass matrices are given in terms of one small ratio of symmetry breaking scales. We describe for a U(1) generation symmetry how the size of masses and mixings is determined, including possible contributions from heavy mirror quarks and leptons. This can be used for computerized model scanning. We search for realistic mass patterns in anomaly-free SU(5) × U(1)G models and find several examples. Interesting patterns for neutrino masses can be obtained.

A systematic study of neutrino masses in models with local B - L symmetry is presented. The observed SU(4) c violation in fermion masses, which is necessary to explain why m e is not equal m d , is related to the scale of B - L violation. An alternative approach uses renormalization group methods to determine this scale. The heaviest neutrino mass is predicted to be 0.1–50 eV in the case of four fermion generations. Two different generation patterns for neutrino masses are found, one predicting large mixing between ν e and ν μ (and eventually ν τ ) and the other predicting leptonic mixing angles of the same order as quark mixing angles.

We argue that an inflationary phase in the very early universe is related to the transition from a higher dimensional to a four-dimensional universe. We present details of a previously considered model which gives sufficient inflation without fine tuning of parameters.

The authors discuss the gauge hierarchy problem in theories that possess an additional discrete symmetry. As examples, they consider two models based on the gauge groups U(1)/sub L/*U(1)/sub R/ and SU(2) /sub L/*SU(2)/sub R/*U(1), with parity transformation as the discrete symmetry. By employing a minimal choice of Higgs multiplets there is no freedom, in the semi-classical approximation, to arrange for an arbitrary hierarchy. Either one has a left-right symmetric phase (no hierarchy) or a totally asymmetric phase (infinite hierarchy). It is shown that radiative corrections a la Coleman-Weinberg (1973) do not smooth out the transition region separating the two phases. A finite gauge hierarchy is not realized. (12 refs).

It is shown how an inflationary phase can be realized in the context of a higher-dimensional model of gravity. Density fluctuations which are approximately scale invariant and have magnitude ≈10 −4 −10 −5 , are also obtained.

There are speculations that grand unified theories (GUT's) may arise from higher-dimensional models of gravity in which the extra dimensions are spontaneously compactified. The GUT predictions for proton decay and sin/sup 2/theta/sub w/ can be significantly modified when nonrenormalizable interactions, scaled by inverse powers of the compactification scale M/sub c/, are added to the standard GUT Lagrangian. For example, the decay rate for p..-->..e/sup +/..pi../sup 0/ in minimal SU(5) can be lowered by one (two) orders of magnitude if M/sub c/ is on the order of 10/sup 17/ GeV. However, sin/sup 2/theta/sub W/ then decreases by 0.005 (0.01). The SO(10) model is also discussed.

We analyze various experimental indications and counterindications for deviations from Newtonian gravity under the assumption that a new interaction is mediated by a scalar particle.

We present a model with continuous or discrete abelian generation group G, which predicts all orders of magnitude for fermion masses ad mixings correctly as a function of only one small parameter φSM ≈ 110. Here φS is the scale of G symmetry breaking and M the fundamental mass scale of the theory. No small Yukawa couplings or special choices of the scalar potential are needed. We perform a systematic computerized scan for similar models with abelian generation group and we find a few other examples. However, for a wide range of charges we find no anomaly free continuous symmetry and also none without mixed SU(3) × SU(2) × U(1), U(1)G anomalies consistent with realistic fermion masses if G is broken by one field with definite charge. We also scan a class of models with generation symmetry derived from a higher dimensional framework.

We discuss the properties of codimension-two branes and compare them to codimension-one branes. In particular, we show that for deficit angle branes the brane energy momentum tensor is uniquely related to integration constants in the bulk solution. We investigate chiral fermions whose wave functions are concentrated on the brane, while all their properties in the effective four-dimensional world can be inferred from the tail of the wave function in the bulk, thereby realizing a holographic principle. We propose holographic branes for which the knowledge of the bulk geometry is sufficient for the computation of all relevant properties of the observable particles, independently of the often unknown detailed physics of the branes., 13 pages, references added

We consider large classes of chiral extensions of the Standard Model, including new quark generations that do not involve additional neutrinos as well as lepton generations without quarks. An analysis of renormalization flows of Yukawa and quartic scalar couplings reveals that additional quarks are not compatible with a scenario of grand unification without violating the strong bounds from direct and Higgs searches at colliders. Constraints from direct searches, electroweak precision observables, and Higgs physics, together with the assumption that additional new physics beyond the extended chiral field content should enter significantly above the TeV scale, allows us to make predictions for searches at the LHC.

Neutrinos interacting with the quintessence field can trigger the accelerated expansion of the Universe. In such models with a growing neutrino mass the homogeneous cosmological solution is often unstable to perturbations. We present static, spherically symmetric solutions of the Einstein equations in the same models. They describe astophysical objects composed of neutrinos, held together by gravity and the attractive force mediated by the quintessence field. We discuss their characteristics as a function of the present neutrino mass. We suggest that these objects are the likely outcome of the growth of cosmological perturbations., Comment: 9 pages, 4 figures, references and discussion of formation added

We discuss the dimensional reduction for Weyl, Majorana, or Majorana-Weyl spinors coupled to pure d -dimensional ( d ⪢ 4) gravity. The only case where a realistic four-dimensional low-energy spectrum for the fermions may be obtained, is for a Majorana-Weyl spinor in d = 2 mod 8 dimensions. Chiral massless fermions are not excluded in this case. The minimal number of dimensions for the construction of a realistic theory out of pure gravity is d = 18.

We investigate discrete symmetries in theories of higher-dimensional ( d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d − 4 dimensional internal space. If the action of pure d -dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d -dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor. A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong- CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.

As we have learned from previous talks of this conference1 by Perry and Sorkin, monopoles occur naturally in Kaluza-Klein theories. Kaluza-Klein theories are unified theories of gravitation and gauge interactions. In these theories, monopoles can be understood as geometrical objects. We want to know if such monopoles could be observed and therefore have to investigate how many of them are produced in the very early universe, and how many should be left in our present universe. This necessitates a study of the cosmology derived from Kaluza-Klein theories. The work on Kaluza-Klein cosmology reported here was done in collaboration with Q. Shafi.

Higher-dimensional gravity theories based on dimensionally continued Euler forms lead to a degenerate spectrum. It contains fewer excitations than the one of a genetic theory with arbitrary coefficients of the higher derivative terms. This can be traced back to a residual topological invariance of the action. The symmetry of the theory is enhanced. As a consequence the stability group of the vacuum can be larger than in the generic case. At the four-dimensional level this can even lead to infinite dimensional (Kac-Moody type) gauge algebras. A simple six-dimensional example leads to infinite towers of massless gravitons and scalars, but no spin-one bosons (gauge bosons) are present. We describe how modes decouple when the action approaches the Euler form. Concerning classical stability the Euler form corresponds to a singular point in the space of six-dimensional gravity actions.

We describe the renormalization group evolution of the scalar mass term in unified theories. There is no fine-tuning problem in renormalization group improved perturbation theory for scalar fields if an appropriate set of short distance parameters is used. The gauge hierarchy problem has two possible issues: it may either be solved by the renormalization group approach itself with a strong scale dependence of the scalar mass, or it is reduced to the question “why is the mass of the scalar doublet normalized at the Planck scale much smaller than the Planck mass?” In the second case, a possible solution of the gauge hierarchy problem refers to a unification with gravity.

We investigate the conditions under which Kaluza-Klein theories with non-compact internal spaces admit a discrete spectrum.

The time development of equal-time correlation functions in quantum mechanics and quantum field theory is described by an exact evolution equation for generating functionals. This permits a comparison between classical and quantum evolution in non-equilibrium systems., Comment: 7 pages, LaTex

We explore the possibility that QCD may undergo a phase transition as a function of the strange quark mass. This would hint towards models with ``spontaneous color symmetry breaking'' in the vacuum. For two light quark flavors we classify possible colored quark-antiquark, diquark and gluon condensates that are compatible with a spectrum of integer charged states and conserved isospin and baryon number. The ``quark mass phase transition'' would be linked to an unusual realization of baryon number in QCD$_2$ and could be tested in lattice simulations. We emphasize, however, that at the present stage the Higgs picture of the vacuum cannot predict a quark mass phase transition - a smooth crossover remains as a realistic alternative. Implications of the Higgs picture for the high density phase transition in QCD$_2$ suggest that this transition is characterized by the spontaneous breaking of isospin for nuclear and quark matter., published version, new section on heavy quark potential,29 pages,LaTex

We review the formalism of the effective average action in quantum field theory which corresponds to a coarse grained free energy in statistical mechanics. The associated exact renormalization group equation and possible nonperturbative approximations for its solution are discussed. This is applied to QCD where one observes the consecutive emergence of mesonic bound states and spontaneous chiral symmetry breaking as the coarse graining scale is lowered. We finally present a study of the chiral phase transition in two flavor QCD. A precision estimate of the universal critical equation of state for the three-dimensional 0( 4) Heisenberg model is presented. We explicitly connect the 0( 4) universal behavior near the critical temperature and zero quark mass with the physics at zero temperature and a realistic pion mass. For realistic quark masses the pion correlation length near Tc turns out to be smaller than its zero temperature value. Quantum chromodynamics (QCD) describes qualitatively different physics at different length scales. At short distances the relevant degrees of freedom are quarks and gluons which can be treated perturbatively. At long distances we observe hadrons, and an essential part of the dynamics can be encoded in the masses and interactions of mesons. Any attempt to deal with this situation analytically and to predict the meson properties from the short distance physics (as functions of the strong gauge coupling o: 8 and the current quark masses mq) has to bridge the gap between two qualitatively different effective descriptions. Two basic problems have to be mastered for an extrapolation from short distance QCD to mesonic length scales: • The effective couplings change with scale. This does not only concern the running gauge coupling, but also the coefficients of non-renormalizable opera­ tors as, for example, four quark operators. Typically, these non-renormalizable terms become important in the momentum range where 0:8 is strong and deviate substantially from their perturbative values. Consider the four-point function which results after integrating out the gluons. For heavy quarks it contains the information about the shape of the heavy quark potential whereas for light quarks the complicated spectrum of light mesons and chiral symmetry breaking are encoded in it. At distance scales around 1 fm one expects that the effec­ tive action resembles very little the form of the classical QCD action which is relevant at short distances. • Not only the couplings, but even the relevant variables or degrees of freedom are different for long distance and short distance QCD. It seems forbiddingly

We compute the free energy of helium-4 near the lambda transition based on an exact renormalization-group equation. An approximate solution permits the determination of universal and nonuniversal thermodynamic properties starting from the microphysics of the two-particle interactions. The method does not suffer from infrared divergences. The critical chemical potential agrees with experiment. This supports a specific formulation of the functional integral that we have proposed recently. Our results for the equation of state reproduce the observed qualitative behavior. Despite certain quantitative shortcomings of our approximation, this demonstrates that ab initio calculations for collective phenomena become possible by modern renormalization-group methods., 9 pages, 6 figures, revtex updated version, journal reference