# Condensates in quantum chromodynamics and the cosmological constant

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Edited by* Roger D. Blandford, Stanford University, Menlo Park, CA, and approved October 20, 2010 (received for review September 18, 2009)

## Abstract

Casher and Susskind [Casher A, Susskind L (1974) *Phys Rev* 9:436–460] have noted that in the light-front description, spontaneous chiral symmetry breaking is a property of hadronic wavefunctions and not of the vacuum. Here we show from several physical perspectives that, because of color confinement, quark and gluon condensates in quantum chromodynamics (QCD) are associated with the internal dynamics of hadrons. We discuss condensates using condensed matter analogues, the Anti de Sitter/conformal field theory correspondence, and the Bethe–Salpeter–Dyson–Schwinger approach for bound states. Our analysis is in agreement with the Casher and Susskind model and the explicit demonstration of “in-hadron” condensates by Roberts and coworkers [Maris P, Roberts CD, Tandy PC (1998) *Phys Lett B* 420:267–273], using the Bethe–Salpeter–Dyson–Schwinger formalism for QCD-bound states. These results imply that QCD condensates give zero contribution to the cosmological constant, because all of the gravitational effects of the in-hadron condensates are already included in the normal contribution from hadron masses.

Hadronic condensates play an important role in quantum chromodynamics. Two important examples are and , where *q* is a light quark (i.e., a quark with current quark mass small compared with the quantum chromodynamics (QCD) scale Λ_{QCD}), , *a*, *b*, *c* are color indices, and *N*_{c} = 3. (For most of the paper we focus on QCD at zero temperature and chemical potential, *T* = *μ* = 0.) For QCD with *N*_{f} light quarks, the condensate spontaneously breaks the global chiral symmetry SU(*N*_{f})_{L} × SU(*N*_{f})_{R}, where [SU(*N*) denotes the group of special unitary *N* × *N* matrices] down to the diagonal, vectorial subgroup SU(*N*_{F})_{diag}, where *N*_{f} = 2 (or *N*_{f} = 3 because *s* is a moderately light quark). Thus in the usual description, one identifies and , where Λ_{QCD} ≃ 300 MeV. These condensates are conventionally considered to be properties of the QCD vacuum and hence are constant throughout space-time. A consequence of the existence of such vacuum condensates is contributions to the cosmological constant from these condensates that are 10^{45} times larger than the observed value. If this disagreement were really true, it would be an extraordinary conflict between the experiment and the standard model.

A very different perspective on hadronic condensates was first presented in a seminal paper by Casher and Susskind (1) published in 1974, see also ref. 2. These authors argued that “spontaneous symmetry breaking must be attributed to the properties of the hadron’s wavefunction and not to the vacuum” (1). The Casher–Susskind argument is based on Weinberg’s infinite momentum frame (3) Hamiltonian formalism of hadronic physics, which is equivalent to light-front (LF) quantization and Dirac’s front form (4) rather than the usual instant form. Casher and Susskind also presented a specific model in which spontaneous chiral symmetry breaking occurs within the confines of the hadron wavefunction due to a phase change supported by the infinite number of quark and quark pairs in the LF wavefunction. In fact, the Regge behavior of hadronic structure functions requires that LF Fock states of hadrons have Fock states with an infinite number of quark and gluon partons (5–7). Thus, in contrast to formal discussions in statistical mechanics, infinite volume is not required for a phase transition in relativistic quantum field theories.

Spontaneous chiral symmetry breaking in QCD is often analyzed by means of an approximate solution of the Dyson–Schwinger equation for a massless quark propagator; if the running coupling exceeds a value of order 1, this equation yields a nonzero dynamical (constituent) quark mass Σ (8–12). Because in the path integral, Σ is formally a source for the operator , one associates Σ ≠ 0 with a nonzero quark condensate. However, the Dyson–Schwinger equation, by itself, does not incorporate confinement and the resultant property that quarks and gluons have maximum wavelengths (13); further, it does not actually determine where this condensate has spatial support or imply that it is a space-time constant.

In contrast, let us consider a meson consisting of a light quark *q* bound to a heavy antiquark, such as a *B* meson. One can analyze the propagation of the light quark *q* in the background field of the heavy quark. Solving the Dyson–Schwinger equation for the light quark, one obtains a nonzero dynamical mass and thus a nonzero value of the condensate . But this quantity is not a true vacuum expectation value; instead, it is the matrix element of the operator in the background field of the quark; i.e., one obtains an in-hadron condensate.

The concept of in-hadron condensates was in fact established in a series of pioneering papers by Roberts and coworkers (14–16) using the Bethe–Salpeter–Dyson–Schwinger analysis for bound states in QCD in conjunction with the Banks–Casher relation , where *ρ*(*λ*) denotes the density of eigenvalues ± *iλ* of the (antihermitian) euclidean Dirac operator (17). These authors reproduced the usual features of spontaneous chiral symmetry breaking using hadronic matrix elements of the Bethe–Salpeter eigensolution. For example, as shown by Maris et al. (14), the Gell-Mann–Oakes–Renner relation (18) for a pseudoscalar hadron in the Bethe–Salpeter analysis is , where is the sum of current-quark masses and *f*_{H} is the meson decay constant: [1]The essential quantity is the hadronic matrix element: [2]which takes the place of the usual vacuum expectation value. Here *T*_{H} is a flavor projection operator, *Z*_{2}(Λ) and *Z*_{4}(Λ) are renormalization constants, *S*(*p*) is the dressed quark propagator, and , where FT is the Fourier transform, is the Bethe–Salpeter bound-state vertex amplitude. The notation in the Bethe–Salpeter analysis thus refers to a hadronic matrix element, not a vacuum expectation value. The Bethe–Salpeter analysis of Roberts and coworkers (14) reproduces the essential features of spontaneous chiral symmetry breaking, including as well as a finite value for *f*_{π} at *m*_{q} → 0.

One can recast the Bethe–Salpeter formalism into the LF Fock state picture by time-ordering the coupled Bethe–Salpeter equation in *τ* = *t* + *z*/*c* or by integrating over *dk*^{-} where *k*^{-} = *k*^{0} - *k*^{3} and using the Wick analysis. This procedure generates a set of equations which couple the infinite set of Fock states at fixed *τ*. Thus the Casher–Susskind and Bethe–Salpeter descriptions of spontaneous chiral symmetry breaking and in-hadron condensates are complementary.

In this paper we show from several physical perspectives that, because of color confinement, quark and gluon QCD condensates can be regarded as being associated with the dynamics of hadron wavefunctions, rather than the vacuum itself. Thus we analyze the condensates and 〈*G*_{μν}*G*^{μν}〉 and address the question of where they have spatial (and temporal) support. We argue, in agreement with the original work of Casher and Susskind (1), that these condensates have spatial support restricted to the interiors of hadrons, as a consequence of the fact that they are due to quark and gluon interactions, and these particles are confined within hadrons. Higher-order in-hadron condensates such as , , etc. are also present, and our discussion implicitly applies to these too. [The fact that QCD experimentally conserves *P* and *T* shows that *P*- and *T*-noninvariant condensates such as , where are negligible; explaining this fact is part of the strong *CP* problem, where *C*, *P*, and *T* denote charge conjugation, parity, and time reversal transformations.] Our analysis includes consideration of condensed matter analogues, the Anti de Sitter/conformal field theory (AdS/CFT) correspondence, and the Bethe–Salpeter–Dyson–Schwinger approach for bound states. Our analysis highlights the difference between chiral models where mesons are treated as elementary fields and QCD in which all hadrons are composite systems. We note that an important consequence of the in-hadron nature of QCD condensates is that QCD gives *zero* contribution to the cosmological constant, because all of the gravitational effects of the in-hadron condensate are already included in the normal contribution from hadron masses.

We emphasize the subtlety in characterizing the formal quantity in the usual instant form, where is a product of quantum field operators, by recalling that one can render this quantity automatically zero by normal-ordering because one has to divide *S*-matrix elements by vacuum expectation values. It should be noted that perturbative contributions to the vacuum in the instant form are not frame independent as can be seen by computing any bubble diagram—e.g., in *ϕ*^{3} theory. As shown by Weinberg (3), these contributions are suppressed by powers of for an observer moving at high momentum *P*. However, such contributions are removed by normal-ordering, thus restoring Lorentz invariance of the instant form vacuum. Such subtleties are especially delicate in a confining theory, because the vacuum state in such a theory is not defined relative to the fields in the Lagrangian, quarks, and gluons, but instead relative to the actual physical, color-singlet, states.

In the front form, the analysis is simpler, because the physical vacuum is automatically trivial, up to zero modes. There are no perturbative bubble diagrams in the LF formalism, so the front-form vacuum is Lorentz-invariant from the start. The LF method provides a completely consistent formalism for quantum field theory. For example, it is straightforward to calculate the coupling of gravitons to physical particles using the LF formalism; in particular, one can prove that the anomalous gravitational magnetic moment vanishes, Fock state by Fock state (19), in agreement with the equivalence principle (20). Furthermore, the LF method reproduces quantum corrections to the gravitational form factors computed in perturbation theory (21).

## A Condensed Matter Analogy

A formulation of quantum field theory using a euclidean path integral (vacuum-to-vacuum amplitude), *Z*, provides a precise meaning for as [3]where *J* is a source for . The path integral for QCD, integrated over quark fields and gauge links using the gauge-invariant lattice discretization exhibits a formal analogy with the partition function for a statistical mechanical system. In this correspondence, a condensate such as or 〈*G*_{μν}*G*^{μν}〉 is analogous to an ensemble average in statistical mechanics. To develop a physical picture of the QCD condensates, we pursue this analogy. In a superconductor, the electron–phonon interaction produces a pairing of two electrons with opposite spins and 3-momenta at the Fermi surface, and for *T* < *T*_{c}, an associated nonzero Cooper pair condensate 〈*ee*〉_{T} (22, 23). (Here 〈…〉_{T} means thermal average.) Because this condensate has a phase, the phenomenological Ginzburg–Landau free energy function, [4]uses a complex scalar field Φ to represent it. The formal treatment of a phase transition such as that in a superconductor begins with a partition function calculated for a finite *d*-dimensional lattice, and then takes the thermodynamic (infinite volume) limit. The nonanalytic behavior associated with the superconducting phase transition only occurs in this infinite volume limit; for *T* < *T*_{c}, the (infinite volume) system develops a nonzero value of the order parameter, namely 〈Φ〉_{T}, in the phenomenological Ginzburg–Landau model, or 〈*ee*〉_{T}, in the microscopic Bardeen–Cooper–Schrieffer theory. In the formal statistical mechanics context, the minimization of the |∇Φ|^{2} term implies that the order parameter is a constant throughout the infinite spatial volume.

However, the infinite-volume limit is an idealization; in reality, superconductivity is experimentally observed to occur in finite samples of material, such as Sn, Nb, etc., and the condensate clearly has spatial support only in the volume of these samples. This observation is evident from either of two basic properties of a superconducting substance, namely, (*i*) zero-resistance flow of electric current, and (*ii*) the Meissner effect that [5]for a magnetic field **B**(*z*) and a distance *z* inside the superconducting sample, where the London penetration depth *λ*_{L} is given by , where *n* = electron concentration; both of these properties clearly hold only within the sample. The same statement applies to other phase transitions such as liquid gas or ferromagnetic; again, in the formal statistical mechanics framework, the phase transition and associated symmetry breaking by a nonzero order parameter at low *T* occur only in the thermodynamic limit, but experimentally, one observes the phase transition to occur effectively in a finite volume of matter, and the order parameter (e.g., magnetization *M*) has support only in this finite volume, rather than the infinite volume considered in the formal treatment. Similarly, the Goldstone modes that result from the spontaneous breaking of a continuous symmetry (e.g., spin waves in a Heisenberg ferromagnet) are experimentally observed in finite-volume samples. There is, of course, no conflict between the experimental measurements and the abstract theorems. The key point is that these samples are large enough for the infinite volume limit to be a useful idealization. Finite-size scaling methods that make this connection precise are standard tools in studies of phase transitions and critical phenomena (24–26).

There is another important distinction between condensed matter physics and relativistic quantum field theories. The proton eigenstate in QCD is a summation over Fock states [6]where *x*_{i} denotes the fraction of the total proton momentum carried by the parton *i*, *k*_{⊥,i} denotes the transverse momentum, *λ*_{i} denotes the helicity, and the summation extends over states with an unlimited number of gluons and sea quarks and antiquarks. In fact, the Regge behavior, , of hadronic structure functions at small *x* requires that the hadronic wavefunction has Fock states |*n*〉 with an infinite number of quark and gluon partons. [Here, in standard notation, *x* = -*q*^{2}/(2*M*_{N}*ν*), where *ν* denotes the energy transfer, *β*_{R} denotes the amplitude with which a Regge trajectory contributes to the scattering, and *α*_{R} denotes the intercept of this trajectory.] This relation applies in the Regge region, with *t* = -*q*^{2} fixed, i.e., small *x*. For example, Mueller (5) has shown that the Balitsky–Fadin–Kuraev–Lipatov behavior of the structure functions at *x* → 0 is a result of the infinite range of gluonic Fock states. The relation between Fock states of different *n* is given by an infinite tower of ladder operators (6). In the analysis by Casher and Susskind (1), spontaneous chiral symmetry breaking occurs within the confines of the hadron wavefunction due to a phase change supported by the infinite number of quark and quark pairs in the LF wavefunction. Thus, as noted above, unlike the usual discussion in condensed matter physics, infinite volume is not required for a phase transition in relativistic quantum field theories.

## A Picture of QCD Condensates

The condensed matter physics discussion above helps to motivate our analysis for QCD. The spatial support for QCD condensates should be where the particles are whose interactions give rise to them, just as the spatial support of a magnetization *M* is inside, not outside, of a piece of iron. The physical origin of the condensate in QCD can be argued to be due to the reversal of helicity (chirality) of a massless quark as it moves outward and reverses its three-momentum at the boundary of a hadron due to confinement (27). This argument suggests that the condensate has support only within the spatial extent where the quark is confined; i.e., the physical size of a hadron. Another way to motivate this observation is to note that in the LF Fock state picture of hadron wavefunctions (1, 28–30), a valence quark can flip its chirality when it interacts or interchanges with the sea quarks of multiquark Fock states, thus providing a dynamical origin for the quark running mass. In this description, the and 〈*G*_{μν}*G*^{μν}〉 condensates are effective quantities which originate from and gluon contributions to the higher Fock state LF wavefunctions of the hadron and hence are localized within the hadron. There is a natural relation with the nucleon sigma term, , where here the nucleon states are normalized as 〈*N*(*p*^{′})|*N*(*p*)〉 = (2*π*)^{3}*δ*^{3}(**p** - **p**^{′}). As discussed in the Introduction, the vacuum condensate appearing in the Gell-Mann–Oakes–Renner relation (18) [7]is replaced by the in-hadron condensate, as defined in Eq. **2**.

## Chiral Symmetry Breaking in the AdS/CFT Model

The AdS/CFT correspondence between string theory in AdS space and CFTs in physical space-time has been used to obtain an analytic, semiclassical model for strongly coupled QCD which has scale invariance and dimensional counting at short distances and color confinement at large distances (31–34). Color confinement can be imposed by introducing hard-wall boundary conditions at *z* = 1/Λ_{QCD} (*z* = AdS fifth dimension) or by modification of the AdS metric. This AdS/QCD model gives a good representation of the mass spectrum of light-quark mesons and baryons as well as the hadronic wavefunctions (31–34). One can also study the propagation of a scalar field *X*(*z*) as a model for the dynamical running quark mass (31–34). The AdS solution has the form (35, 36) [8]where *a*_{1} is proportional to the current-quark mass. The coefficient *a*_{2} scales as and is the analogue of ; however, because the quark is a color nonsinglet, the propagation of *X*(*z*), and thus the domain of the quark condensate, is limited to the region of color confinement. The AdS/QCD picture of effective confined condensates is in agreement with results from chiral bag models (37–39), which modify the original MIT bag (where MIT denotes the Massachusetts Institute of Technology) by coupling a pion field to the surface of the bag in a chirally invariant manner. Because the effect of *a*_{2} depends on *z*, the AdS picture is inconsistent with the usual picture of a constant condensate.

## Empirical Determinations of the Gluon Condensate

The renormalization invariant quantity 〈(*α*_{s}/*π*)*G*_{μν}*G*^{μν}〉, where [9]can be determined empirically by analyzing vacuum-to-vacuum current correlators constrained by data for *e*^{+}*e*^{-} → charmonium and hadronic *τ* decays (40–47). [Here we use units where *ℏ* = *c* = 1, and our flat-space metric is *η*_{μν} = diag(1,-1,-1,-1).] Some recent values (in GeV^{4}) include 0.006 ± 0.012 (43), 0.009 ± 0.007 (44, 45), and -0.015 ± 0.008 (46, 47). These values show significant scatter and even differences in sign. These are consistent with the picture in which the vacuum gluon condensate vanishes; it is confined within hadrons, rather than extending throughout all of space, as would be true of a vacuum condensate.

## Some Other Features of QCD Condensates

In the picture discussed here, QCD condensates would be considered as contributing to the masses of the hadrons where they are located. This observation is clear, because, e.g., a proton subjected to a constant electric field will accelerate and, because the condensates move with it, they comprise part of its mass. Similarly, when a hadron decays to a nonhadronic final state, such as *π*^{0} → *γγ*, the condensates in this hadron contribute their energy to the final-state photons. Thus, over long times, the dominant regions of support for these condensates would be within nucleons, because the proton is effectively stable (with lifetime *τ*_{p}≫*τ*_{univ} ≃ 1.4 × 10^{10} y), and the neutron can be stable when bound in a nucleus. In a process like *e*^{+}*e*^{-} → hadrons, the formation of the condensates occurs on the same time scale as hadronization. In accord with the Heisenberg uncertainty principle, these QCD condensates also affect virtual processes occurring over times *t* ≲ 1/Λ_{QCD}.

Moreover, in our picture, condensates in different hadrons may be chirally rotated with respect to each other, somewhat analogous to disoriented chiral condensates in heavy-ion collisions (48–50). This picture of condensates can, in principle, be verified by careful lattice gauge theory measurements. Note that the lattice measurements that have inferred nonzero values of were performed in finite volumes, although these were usually considered as approximations to the infinite volume limit. [For an early review of lattice measurements, see ref. 51; recent reviews are given in the annual Symposia on Lattice Field Theory.] In *SI Text* we discuss an application of these ideas to other asymptotically free gauge theories.

## The Case of an Infrared-Free Gauge Theory

Our discussion is only intended to apply to asymptotically free gauge theories. However, we offer some remarks on the situation for a particular infrared-free theory here, namely a U(1) gauge theory with gauge coupling *e* and some set of fermions *ψ*_{i} with charges *q*_{i}. Here there are several important differences with respect to an asymptotically free non-Abelian gauge theory. First, whereas the chiral limit of QCD, i.e., quarks with zero current-quark masses, is well-defined because of quark confinement, a U(1) theory with massless charged particles is unstable, owing to the well-known fact that these would give rise to a divergent Bethe-Heitler pair production cross-section. It is therefore necessary to break the chiral symmetry explicitly with bare fermion mass terms *m*_{i}. If the running coupling *α*_{1} = *e*^{2}/(4*π*) at a given energy scale *μ* were sufficiently large, *α*_{1}(*μ*) ≳ *O*(1), an approximate solution to the Dyson–Schwinger equation for the propagator of a fermion *ψ*_{i} with *m*_{i} ≪ *μ* would suggest that this fermion gains a nonzero dynamical mass Σ_{i} (8–12) and hence, presumably, there would be an associated condensate (no sum on *i*). However, in analyzing S*χ*SB, it is important to minimize the effects of explicit chiral symmetry breaking due to the bare masses *m*_{i}. The infrared-free nature of this theory means that for any given value of *α*_{1} at a scale *μ*, as one decreases *m*_{i}/*μ* to reduce explicit breaking of chiral symmetry, *α*_{1}(*m*_{i}) also decreases, approaching zero as *m*_{i}/*μ* → 0. Because *α*_{1}(*m*_{i}) should be the relevant coupling to use in the Dyson–Schwinger equation, it may in fact be impossible to realize a situation in this theory in which one has small explicit breaking of chiral symmetry and a large enough value of *α*_{1}(*m*_{i}) to induce spontaneous chiral symmetry breaking. A full analysis would require knowledge of the bound-state spectrum of the hypothetical strongly coupled U(1) theory, but this spectrum is not reliably known.

## Finite-Temperature QCD

So far, we have discussed QCD and other theories at zero temperature (and chemical potential or equivalently here, baryon density). For QCD in thermal equilibrium at a finite-temperature *T*, as *T* increases above the deconfinement temperature *T*_{dec}, both the hadrons and the associated condensates eventually disappear, although experiments at European Organization for Nuclear Research and Brookhaven National Laboratory-Relativistic Heavy Ion Collider show that the situation for *T* ≳ *T*_{dec} is more complicated than a weakly coupled quark–gluon plasma. The picture of the QCD condensates here is especially close to experiment, because, although finite-temperature QCD makes use of the formal thermodynamic, infinite volume limit, actual heavy-ion experiments and resultant transitions from confined to deconfined quarks and gluons take place in the finite-volume and time interval provided by colliding heavy ions. Indeed, one of the models that has been used to analyze such experiments involves the notion of a color glass condensate (58–60).

## QCD and the Cosmological Constant

One of the most challenging problems in physics is that of the cosmological constant Λ; recent reviews include refs. 61–67. This quantity enters in the Einstein gravitational field equations as (68–72) [10]where *R*_{μν}, *R*, *g*_{μν}, *T*_{μν}, and *G*_{N} are the Ricci curvature tensor, the scalar curvature, the metric tensor, the stress-energy tensor, and Newton’s constant. One defines [11]and [12]where [13]and are the Hubble constant in the present era, with *a*(*t*) being the Friedmann–Robertson–Walker scale parameter (68–72). The field equations imply and , where *ρ* = total mass/energy density, *p* = pressure, and *k* is the curvature parameter; equivalently, 1 = Ω_{m} + Ω_{γ} + Ω_{Λ} + Ω_{k}, where , , and . Long before the current period of precision cosmology, it was known that Ω_{Λ} could not be larger than O(1). In the context of quantum field theory, this empirical fact was very difficult to understand, because estimates of the contributions to *ρ*_{Λ} from (*i*) vacuum condensates of quark and gluon fields in QCD and the vacuum expectation value of the Higgs field hypothesized in the standard model (SM) to be responsible for electroweak symmetry breaking, and from (*ii*) zero-point energies of quantum fields appear to be too large by many orders of magnitude. Observations of supernovae showed the accelerated expansion of the universe and are consistent with the hypothesis that this accelerated expansion is due to a cosmological constant, Ω_{Λ} ≃ 0.76 (73–79). The supernovae data (73–79), together with measurements of the cosmic microwave background radiation, galaxy clusters, and other inputs, e.g., primordial element abundances, have led to a consistent determination of the cosmological parameters (80–83). These include *H*_{0} = 73 ± 3 km/(s - Mpc), *ρ*_{c} = 0.56 × 10^{-5} GeV/cm^{3} = (2.6 × 10^{-3} eV)^{4}, total Ω_{m} ≃ 0.24 with baryon term Ω_{b} ≃ 0.042, so that the dark matter term is Ω_{dm} ≃ 0.20. In the equation of state *p* = *wρ* for the “dark energy,” *w* is consistent with being equal to -1, the value if the accelerated expansion is due to a cosmological constant. [Other suggestions for the source of the accelerated expansion include modifications of general relativity and time-dependent *w*(*t*), as reviewed in refs. 61–67.]

Here, using our observations concerning QCD condensates, we propose a solution to problem (*i*) of the contributions by these condensates to *ρ*_{Λ}, which, in the conventional approach, are much too large. The QCD condensates form at times of order 10^{-5} s in the early universe, as the temperature *T* decreases below the confinement–deconfinement temperature *T*_{dec} ≃ 200 MeV. As noted above, for *T* ≪ *T*_{dec}, in the conventional quantum field theory view, these condensates are considered to be constants throughout space. If one accepts this conventional view, then these condensates would contribute , so that (*δ*Ω_{Λ})_{QCD} ≃ 10^{45}. On the contrary, if one accepts the argument that these condensates [and also higher-order constants such as and ] have spatial support within hadrons, not extending throughout all of space, then one makes considerable progress in solving the above problem, because the effect of these condensates on gravity is already included in the baryon term Ω_{b} in Ω_{m} and, as such, they do not contribute to Ω_{Λ}.

Another excessive type-(i) contribution to *ρ*_{Λ} is conventionally viewed as arising from the vacuum expectation value of the SM Higgs field, , giving and hence (*δ*Ω_{Λ})_{EW} ∼ 10^{56} for the electroweak (EW) theory. Similar numbers are obtained from Higgs vacuum expectation values in supersymmetric extensions of the SM [recalling that the supersymmetry breaking scale is expected to be the tera-electron volt (TeV) scale (52–57)]. However, it is possible that electroweak symmetry breaking is dynamical; for example, it may result from the formation of a bilinear condensate of fermions *F* (called technifermions) subject to an asymptotically free, vectorial, confining gauge interaction, commonly called TC, that gets strong on the TeV scale (52–57). In such theories there is no fundamental Higgs field. TC theories are challenged by, but may be able to survive, constraints from precision electroweak data. By using our arguments above, in a TC theory, the technifermion and technigluon condensates would have spatial support in the technihadrons and techniglueballs and would contribute to the masses of these states. We stress that, just as was true for the QCD condensates, these technifermion and technigluon condensates would not contribute to *ρ*_{Λ}. Hence, if a technicolor-type mechanism should turn out to be responsible for electroweak symmetry breaking, then there would not be any problem with a supposedly excessive contribution to *ρ*_{Λ} for a Higgs vacuum expectation value. Indeed, stable technihadrons in certain technicolor theories may be viable dark matter candidates.

We next comment briefly on type-(ii) contributions. The formal expression for the energy density *E*/*V* due to zero-point energies of a quantum field corresponding to a particle of mass *m* is [14]where the energy is . However, first, this expression is unsatisfactory, because it is (quadratically) divergent. In modern particle physics one would tend to regard this divergence as indicating that one is using a low-energy effective field theory, and one would impose an ultraviolet cutoff *M*_{UV} on the momentum integration, reflecting the upper range of validity of this low-energy theory. Because neither the left- nor right-hand side of Eq. **14** is Lorentz invariant, this cutoff procedure is more dubious than the analogous procedure for Feynman integrals of the form ∫*d*^{4}*kI*(*k*,*p*) in quantum field theory, where *I*(*k*,*p*_{1},…,*p*_{n}) is a Lorentz-invariant integrand function depending on some set of 4-momenta *p*_{1},…,*p*_{n}. If, nevertheless, one proceeds to use such a cutoff, then, because a mass scale characterizing quantum gravity (QG) is , one would infer that , and hence (*δ*Ω_{Λ})_{QG} ∼ 10^{120}. With the various mass scales characterizing the electroweak symmetry breaking and particle masses in the SM, one similarly would obtain (*δ*Ω_{Λ})_{SM} ∼ 10^{56}. Given the fact that Eq. **14** is not Lorentz-invariant, one may well question the logic of considering it as a contribution to the Lorentz invariant quantity *ρ*_{Λ}. (This criticism of the conventional lore has also been made in refs. 61 and 84.) Indeed, one could plausibly argue that, as an energy density, it should instead be part of *T*_{00} in the energy-momentum tensor *T*_{μν}. Phrased in a different way, if one argues that it should be associated with the Λ*g*_{μν} term, then there must be a negative corresponding zero-point pressure satisfying *p* = -*ρ*, but the source for such a negative pressure is not evident in Eq. **14**. The LF approach to the construction of a quantum field theory, in particular, the SM, provides another perspective to this issue (85—88).

## Concluding Remarks

We have argued from several physical perspectives that, because of color confinement, quark and gluon QCD condensates are localized within the interiors of hadrons. Our analysis is in agreement with the Casher–Susskind model and the explicit demonstration of in-hadron condensates by Roberts and coworkers (14), using the Bethe–Salpeter–Dyson–Schwinger formalism for QCD-bound states. We also discussed this physics using condensed matter analogues, the AdS/CFT correspondence, and the Bethe–Salpeter–Dyson–Schwinger approach for bound states.

In-hadron condensates provide a solution to what has hitherto commonly been regarded as an excessively large contribution to the cosmological constant by QCD condensates. We have argued that these condensates do not, in fact, contribute to Ω_{Λ}; instead, they have spatial support within hadrons and, as such, should really be considered as contributing to the masses of these hadrons and hence to Ω_{b}. We have also suggested a possible solution to what would be an excessive contribution to Ω_{Λ} from a hypothetical Higgs vacuum expectation value; the solution would be applicable if electroweak symmetry breaking occurs via a technicolor-type mechanism.

## Acknowledgments

We thank R. Alkofer, A. Casher, C. Fischer, M. E. Fisher, F. Llanes-Estrada, C. Roberts, L. Susskind, P. Tandy, and G. F. de Téramond for helpful conversations. This research was partially supported by Grant DE-AC02-76SF00515 (to S.J.B.) and by Grant NSF-PHY-06-53342 (to R.S.).

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: sjbth{at}slac.stanford.edu.Author contributions: S.J.B. and R.S. performed research and wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1010113107/-/DCSupplemental.

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- Abstract
- A Condensed Matter Analogy
- A Picture of QCD Condensates
- Chiral Symmetry Breaking in the AdS/CFT Model
- Empirical Determinations of the Gluon Condensate
- Some Other Features of QCD Condensates
- The Case of an Infrared-Free Gauge Theory
- Finite-Temperature QCD
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