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Home , Chiral symmetry breaking, Light front holography, Quantum chromodynamics, Technicolor (physics)

Condensates in chromodynamics and the

Stanley J. Brodskya,b,1 and Robert Shrockb

aSLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309; and bChen Ning Yang Institute for Theoretical , Stony Brook University, Stony Brook, NY 11794

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

Casher and Susskind [Casher A, Susskind L (1974) Phys Rev 9:436– the infinite number of and quark pairs in the LF wavefunc- 460] have noted that in the -front description, spontaneous tion. In fact, the Regge behavior of hadronic structure functions chiral breaking is a property of hadronic wavefunctions requires that LF Fock states of have Fock states with and not of the . Here we show from several physical an infinite number of quark and partons (5–7). Thus, in perspectives that, because of , quark and gluon contrast to formal discussions in , infinite condensates in (QCD) are associated volume is not required for a phase transition in relativistic quan- with the internal dynamics of hadrons. We discuss condensates tum theories. using condensed analogues, the Anti de Sitter/conformal Spontaneous in QCD is often ana- field theory correspondence, and the Bethe–Salpeter–Dyson– lyzed by means of an approximate solution of the Dyson–Schwin- Schwinger approach for bound states. Our analysis is in agreement ger equation for a massless quark ; if the running with the Casher and Susskind model and the explicit demonstra- α ¼ 2∕ð4πÞ s gs exceeds a value of order 1, this equation tion of “in-” condensates by Roberts and coworkers [Maris yields a nonzero dynamical (constituent) quark mass Σ (8–12). Phys Lett B – P, Roberts CD, Tandy PC (1998) 420:267 273], using Because in the path integral, Σ is formally a source for the op- – – – the Bethe Salpeter Dyson Schwinger formalism for QCD-bound erator qq¯ , one associates Σ ≠ 0 with a nonzero quark condensate. states. These results imply that QCD condensates give zero contri- –

However, the Dyson Schwinger equation, by itself, does not PHYSICS bution to the cosmological constant, because all of the gravita- incorporate confinement and the resultant property that tional effects of the in-hadron condensates are already included and have maximum wavelengths (13); further, it does not in the normal contribution from hadron masses. actually determine where this condensate has spatial support or ∣ ∣ imply that it is a space-time constant. vacuum properties In contrast, let us consider a consisting of a light quark q bound to a heavy antiquark, such as a . One can analyze adronic condensates play an important role in quantum the propagation of the light quark q in the background field of Hchromodynamics. Two important examples are hqq¯ i≡ ¯ – 2−1 the heavy b quark. Solving the Dyson Schwinger equation for h∑Nc ¯ ai h μνi ≡ h∑Nc a aμνi a¼1 qaq and GμνG a¼1 GμνG , where q is a light the light quark, one obtains a nonzero dynamical mass and thus quark (i.e., a quark with mass small compared with a nonzero value of the condensate hqq¯ i. But this quantity is not a Λ a ¼ the quantum chromodynamics (QCD) scale QCD), Gμν true ; instead, it is the element ∂ a −∂ a þ b c ¼ 3 ¯ ¯ μAν νAμ gscabcAμAν, a, b, c are color indices, and Nc . of the operator qq in the background field of the b quark; i.e., (For most of the paper we focus on QCD at zero temperature one obtains an in-hadron condensate. ¼ μ ¼ 0 and chemical potential, T .) For QCD with Nf light The concept of in-hadron condensates was in fact established h¯ i¼h¯ þ ¯ i quarks, the qq qLqR qRqL condensate spontaneously in a series of pioneering papers by Roberts and coworkers ð Þ × ð Þ – – – – breaks the global chiral symmetry SU Nf L SU Nf , where (14 16) using the Bethe Salpeter Dyson Schwinger analysis [SUðNÞ denotes the of special unitary N × N matrices] for bound states in QCD in conjunction with the Banks–Casher ð Þ relation −hqq¯ i¼πρð0Þ, where ρðλÞ denotes the density of eigen- down to the diagonal, vectorial subgroup SU NF diag, where ¼ 2 ¼ 3 values iλ of the (antihermitian) euclidean Dirac operator (17). Nf (or Nf because s is a moderately light quark). Thus h¯ i ∼ Λ3 These authors reproduced the usual features of spontaneous in the usual description, one identifies qq QCD and μν 4 chiral symmetry breaking using hadronic matrix elements of the hGμνG i ∼ Λ , where Λ ≃ 300 MeV. These condensates QCD QCD Bethe–Salpeter eigensolution. For example, as shown by Maris are conventionally considered to be properties of the QCD et al. (14), the Gell-Mann–Oakes–Renner relation (18) for a vacuum and hence are constant throughout space-time. A conse- 2 pseudoscalar hadron in the Bethe–Salpeter analysis is f m ¼ quence of the existence of such vacuum condensates is contribu- H H −ρH M M tions to the cosmological constant from these condensates that ζ H , where H is the sum of current-quark masses and 45 are 10 times larger than the observed value. If this disagreement f H is the meson decay constant: were really true, it would be an extraordinary conflict between the Z Λ 4 1 1 1 experiment and the . μ d q μ f P ¼ Z2 T γ5γ S P þ q Γ ðq;PÞS P − q : A very different perspective on hadronic condensates was first H ð2πÞ4 2 H 2 H 2 presented in a seminal paper by Casher and Susskind (1) [1] published in 1974, see also ref. 2. These authors argued that “ spontaneous symmetry breaking must be attributed to the prop- The essential quantity is the hadronic matrix element: erties 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 Author contributions: S.J.B. and R.S. performed research and wrote the paper. physics, which is equivalent to light-front (LF) The authors declare no conflict of interest. and Dirac’s front form (4) rather than the usual instant form. *This Direct Submission article had a prearranged editor. Casher and Susskind also presented a specific model in which 1To whom correspondence should be addressed. E-mail: [email protected]. spontaneous chiral symmetry breaking occurs within the confines This article contains supporting information online at www.pnas.org/lookup/suppl/ of the hadron wavefunction due to a phase change supported by doi:10.1073/pnas.1010113107/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1010113107 PNAS Early Edition ∣ 1of6 Downloaded by guest on September 26, 2021 −hqq¯iH Lagrangian, quarks, and gluons, but instead relative to the actual H ζ iρζ ≡ physical, color-singlet, states. f H In the front form, the analysis is simpler, because the physical Z 4 Λ d q 1 1 1 vacuum is automatically trivial, up to zero modes. There are no ¼ Z4 T γ5S P þ q Γ ðq;PÞS P − q ; [2] ð2πÞ4 2 H 2 H 2 perturbative bubble diagrams in the LF formalism, so the front- form vacuum is Lorentz-invariant from the start. The LF method which takes the place of the usual vacuum expectation value. provides a completely consistent formalism for quantum field ðΛÞ ðΛÞ Here TH is a flavor projection operator, Z2 and Z4 are re- theory. For example, it is straightforward to calculate the cou- normalization constants, SðpÞ is the dressed quark propagator, pling of to physical particles using the LF formalism; Γ ð ; Þ¼ h jψð Þψ¯ð Þj0i and H q P FT H xa xb , where FT is the Fourier in particular, one can prove that the anomalous gravitational transform, is the Bethe–Salpeter bound-state vertex amplitude. magnetic moment vanishes, Fock state by Fock state (19), in H The notation hqq¯iζ in the Bethe–Salpeter analysis thus refers agreement with the equivalence principle (20). Furthermore, to a hadronic matrix element, not a vacuum expectation value. the LF method reproduces quantum corrections to the gravita- The Bethe–Salpeter analysis of Roberts and coworkers (14) re- tional form factors computed in perturbation theory (21). produces the essential features of spontaneous chiral symmetry A Condensed Matter Analogy 2 ∝ ð þ Þ∕ breaking, including mπ mq mq¯ f π as well as a finite value A formulation of using a euclidean path → 0 for f π at mq . integral (vacuum-to-vacuum amplitude), Z, provides a precise One can recast the Bethe–Salpeter formalism into the LF Fock meaning for hOi as state picture by time-ordering the coupled Bethe–Salpeter equa- − − 0 3 δlnZ tion in τ ¼ t þ z∕c or by integrating over dk where k ¼ k − k hOi¼lim ; [3] and using the Wick analysis. This procedure generates a set of J→0 δJ equations which couple the infinite set of Fock states at fixed where J is a source for O. The path integral for QCD, integrated τ – – . Thus the Casher Susskind and Bethe Salpeter descriptions over quark fields and gauge links using the gauge-invariant lattice of spontaneous chiral symmetry breaking and in-hadron conden- discretization exhibits a formal analogy with the partition func- sates are complementary. tion for a statistical mechanical system. In this correspondence, μν In this paper we show from several physical perspectives that, a condensate such as hqq¯ i or hGμνG i is analogous to an ensem- because of color confinement, quark and gluon QCD conden- ble average in statistical mechanics. To develop a physical picture sates can be regarded as being associated with the dynamics of of the QCD condensates, we pursue this analogy. In a supercon- hadron wavefunctions, rather than the vacuum itself. Thus we ductor, the –phonon interaction produces a pairing of h¯ i h μνi analyze the condensates qq and GμνG and address the ques- two with opposite spins and 3-momenta at the Fermi tion of where they have spatial (and temporal) support. We argue, surface, and for T < Tc, an associated nonzero Cooper pair in agreement with the original work of Casher and Susskind (1), h i h…i condensate ee T (22, 23). (Here T means thermal average.) that these condensates have spatial support restricted to the Because this condensate has a phase, the phenomenological interiors of hadrons, as a consequence of the fact that they are Ginzburg–Landau free energy function, due to quark and gluon interactions, and these particles are con- fined within hadrons. Higher-order in-hadron condensates such ¼j∇Φj2 þ ðΦΦÞþ ðΦΦÞ2 [4] 2 μν F c2 c4 ; as hðqq¯ Þ i, hðqq¯ ÞGμνG i, etc. are also present, and our discussion implicitly applies to these too. [The fact that QCD experimentally uses a complex scalar field Φ to represent it. The formal treat- conserves P and T shows that P- and T-noninvariant condensates ment of a phase transition such as that in a superconductor begins h ~ μνi ~ ¼ð1∕2Þϵ αβ such as GμνG , where Gμν μναβG are negligible; with a partition function calculated for a finite d-dimensional explaining this fact is part of the strong CP problem, where C, lattice, and then takes the thermodynamic (infinite volume) limit. P, and T denote conjugation, parity, and time reversal The nonanalytic behavior associated with the superconducting transformations.] Our analysis includes consideration of con- phase transition only occurs in this infinite volume limit; for densed matter analogues, the Anti de Sitter/conformal field the- T < Tc, the (infinite volume) system develops a nonzero value ory (AdS/CFT) correspondence, and the Bethe–Salpeter–Dyson– hΦi of the order parameter, namely T , in the phenomenological Schwinger approach for bound states. Our analysis highlights the – h i – Ginzburg Landau model, or ee T , in the microscopic Bardeen difference between chiral models where are treated as Cooper–Schrieffer theory. In the formal statistical mechanics elementary fields and QCD in which all hadrons are composite context, the minimization of the j∇Φj2 term implies that the order systems. We note that an important consequence of the in-hadron parameter is a constant throughout the infinite spatial volume. nature of QCD condensates is that QCD gives zero contribution However, the infinite-volume limit is an idealization; in reality, to the cosmological constant, because all of the gravitational superconductivity is experimentally observed to occur in finite effects of the in-hadron condensate are already included in the samples of material, such as Sn, Nb, etc., and the condensate normal contribution from hadron masses. clearly has spatial support only in the volume of these samples. We emphasize the subtlety in characterizing the formal quan- This observation is evident from either of two basic properties of tity h0jOj0i in the usual instant form, where O is a product a superconducting substance, namely, (i) zero-resistance flow of of quantum field operators, by recalling that one can render this electric current, and (ii) the Meissner effect that quantity automatically zero by normal-ordering O because one − ∕λ has to divide S-matrix elements by vacuum expectation values. jBðzÞj ∼ jBð0Þje z L [5] It should be noted that perturbative contributions to the vacuum in the instant form are not frame independent as can be seen by for a magnetic field BðzÞ and a distance z inside the superconduct- — ϕ3 λ computing any bubble diagram e.g., in theory. As shown by ing sample, where the London penetration depth L is given 1 λ2 ¼ 2∕ð4π 2Þ ¼ Weinberg (3), these contributions are suppressed by powers of P by L mec ne , where n electron concentration; both for an observer moving at high momentum P. However, such of these properties clearly hold only within the sample. The same contributions are removed by normal-ordering, thus restoring statement applies to other phase transitions such as liquid gas or Lorentz invariance of the instant form vacuum. Such subtleties ferromagnetic; again, in the formal statistical mechanics frame- are especially delicate in a confining theory, because the vacuum work, the phase transition and associated symmetry breaking state in such a theory is not defined relative to the fields in the by a nonzero order parameter at low T occur only in the thermo-

2of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1010113107 Brodsky and Shrock Downloaded by guest on September 26, 2021 dynamic limit, but experimentally, one observes the phase transi- in the Introduction, the vacuum condensate appearing in the tion to occur effectively in a finite volume of matter, and the or- Gell-Mann–Oakes–Renner relation (18) der parameter (e.g., magnetization M) has support only in this ð þ Þ finite volume, rather than the infinite volume considered in 2 ¼ − mu md h¯ i [7] mπ 2 qq the formal treatment. Similarly, the Goldstone modes that result f π from the spontaneous breaking of a continuous symmetry (e.g., waves in a Heisenberg ferromagnet) are experimentally ob- is replaced by the in-hadron condensate, as defined in Eq. 2. served in finite-volume samples. There is, of course, no conflict between the experimental measurements and the abstract theo- Chiral Symmetry Breaking in the AdS/CFT Model rems. The key point is that these samples are large enough for the The AdS/CFT correspondence between in AdS infinite volume limit to be a useful idealization. Finite-size scaling space and CFTs in physical space-time has been used to obtain methods that make this connection precise are standard tools in an analytic, semiclassical model for strongly coupled QCD which studies of phase transitions and critical phenomena (24–26). has scale invariance and dimensional counting at short distances There is another important distinction between condensed and color confinement at large distances (31–34). Color confine- matter physics and relativistic quantum field theories. The ment can be imposed by introducing hard-wall boundary condi- ¼ 1∕Λ ¼ eigenstate in QCD is a summation over Fock states tions at z QCD (z AdS fifth dimension) or by modification of the AdS metric. This AdS/QCD model gives a good represen- ∞ tation of the mass spectrum of light-quark mesons and as j i¼ Ψ ð λ Þj i [6] P ∑ n∕P xi;k⊥i; i n ; well as the hadronic wavefunctions (31–34). One can also study n¼3 the propagation of a scalar field XðzÞ as a model for the dynami- cal running quark mass (31–34). The AdS solution has the form where xi denotes the fraction of the total proton momentum (35, 36) carried by the parton i, k⊥;i denotes the transverse momentum, 3 λ XðzÞ¼a1z þ a2z ; [8] i denotes the helicity, and the summation extends over states with an unlimited number of gluons and sea quarks and anti- 2 1−α where a1 is proportional to the current-quark mass. The coeffi- quarks. In fact, the Regge behavior, F2ðx;Q Þ ∼∑ β x R , of ha- Λ3 h¯ i R R cient a2 scales as QCD and is the analogue of qq ; however, be- dronic structure functions at small x requires that the hadronic cause the quark is a color nonsinglet, the propagation of XðzÞ; PHYSICS wavefunction has Fock states jni with an infinite number of quark and thus the domain of the quark condensate, is limited to the ¼ − 2∕ð2 νÞ and gluon partons. [Here, in standard notation, x q MN , region of color confinement. The AdS/QCD picture of effective ν β where denotes the energy transfer, R denotes the amplitude confined condensates is in agreement with results from chiral bag with which a Regge trajectory contributes to the scattering, models (37–39), which modify the original MIT bag (where MIT α and R denotes the intercept of this trajectory.] This relation ap- denotes the Massachusetts Institute of Technology) by coupling a ≫ Λ2 ¼ − 2 plies in the Regge region, s QCD with t q fixed, i.e., small field to the surface of the bag in a chirally invariant manner. – – x. For example, Mueller (5) has shown that the Balitsky Fadin Because the effect of a2 depends on z, the AdS picture is incon- Kuraev–Lipatov behavior of the structure functions at x → 0 is a sistent with the usual picture of a constant condensate. result of the infinite range of gluonic Fock states. The relation between Fock states of different n is given by an infinite tower Empirical Determinations of the Gluon Condensate hðα ∕πÞ μνi of ladder operators (6). In the analysis by Casher and Susskind The invariant quantity s GμνG , where (1), spontaneous chiral symmetry breaking occurs within the μν ¼ 2 ðjBaj2 − jEaj2Þ [9] confines of the hadron wavefunction due to a phase change sup- GμνG ∑ ported by the infinite number of quark and quark pairs in the LF a wavefunction. Thus, as noted above, unlike the usual discussion in , infinite volume is not required for a can be determined empirically by analyzing vacuum-to-vacuum phase transition in relativistic quantum field theories. current correlators constrained by data for eþe− → charmonium and hadronic τ decays (40–47). [Here we use units where A Picture of QCD Condensates ℏ ¼ c ¼ 1, and our flat-space metric is ημν ¼ diagð1; − 1; − 1; The condensed matter physics discussion above helps to motivate −1Þ.] Some recent values (in GeV4) include 0.006 0.012 our analysis for QCD. The spatial support for QCD condensates (43), 0.009 0.007 (44, 45), and −0.015 0.008 (46, 47). These should be where the particles are whose interactions give rise to values show significant scatter and even differences in sign. These them, just as the spatial support of a magnetization M is inside, are consistent with the picture in which the vacuum gluon ¯ not outside, of a piece of iron. The physical origin of the hqqi condensate vanishes; it is confined within hadrons, rather than condensate in QCD can be argued to be due to the reversal of extending throughout all of space, as would be true of a vacuum helicity () of a massless quark as it moves outward and condensate. reverses its three-momentum at the boundary of a hadron due to confinement (27). This argument suggests that the condensate Some Other Features of QCD Condensates has support only within the spatial extent where the quark is con- In the picture discussed here, QCD condensates would be con- fined; i.e., the physical size of a hadron. Another way to motivate sidered as contributing to the masses of the hadrons where they this observation is to note that in the LF Fock state picture of are located. This observation is clear, because, e.g., a proton hadron wavefunctions (1, 28–30), a valence quark can flip its subjected to a constant electric field will accelerate and, because chirality when it interacts or interchanges with the sea quarks the condensates move with it, they comprise part of its mass. of multiquark Fock states, thus providing a dynamical origin Similarly, when a hadron decays to a nonhadronic final state, such for the quark running mass. In this description, the hqq¯ i and as π0 → γγ, the condensates in this hadron contribute their energy μν hGμνG i condensates are effective quantities which originate to the final-state . Thus, over long times, the dominant from qq¯ and gluon contributions to the higher Fock state LF wa- regions of support for these condensates would be within nu- vefunctions of the hadron and hence are localized within the cleons, because the proton is effectively stable (with lifetime τ ≫ τ ≃ 1 4 × 1010 hadron. There is a natural relation with the sigma term, p univ . y), and the can be stable when σ ¼ð1∕2Þð þ Þh j¯ j i þ − → πN mu md N qq N , where here the nucleon states bound in a nucleus. In a process like e e hadrons, the forma- are normalized as hNðp0ÞjNðpÞi¼ð2πÞ3δ3ðp − p0Þ. As discussed tion of the condensates occurs on the same time scale as hadro-

Brodsky and Shrock PNAS Early Edition ∣ 3of6 Downloaded by guest on September 26, 2021 nization. In accord with the Heisenberg uncertainty principle, QCD and the Cosmological Constant these QCD condensates also affect virtual processes occurring One of the most challenging problems in physics is that of the ≲ 1∕Λ Λ – over times t QCD. cosmological constant ; recent reviews include refs. 61 67. This Moreover, in our picture, condensates hqq¯ i in different quantity enters in the Einstein gravitational field equations as hadrons may be chirally rotated with respect to each other, some- (68–72) what analogous to disoriented chiral condensates in heavy-ion collisions (48–50). This picture of condensates can, in principle, 1 Rμν − gμνR − Λgμν ¼ð8πG ÞTμν; [10] be verified by careful measurements. Note 2 N that the lattice measurements that have inferred nonzero values of hqq¯ i were performed in finite volumes, although these were where Rμν, R, gμν, Tμν, and GN are the Ricci curvature tensor, the usually considered as approximations to the infinite volume limit. scalar curvature, the metric tensor, the stress-energy tensor, and ’ [For an early review of hqq¯ i lattice measurements, see ref. 51; Newton s constant. One defines recent reviews are given in the annual Symposia on Lattice Field Λ Theory.] In SI Text we discuss an application of these ideas to ρ ¼ [11] Λ 8π other asymptotically free gauge theories. GN The Case of an Infrared-Free Gauge Theory and Λ ρ Our discussion is only intended to apply to asymptotically free Ω ¼ ¼ Λ [12] Λ 2 ρ ; gauge theories. However, we offer some remarks on the situation 3H0 c for a particular infrared-free theory here, namely a U(1) gauge theory with gauge coupling e and some set of ψ with where i 3 2 charges q . Here there are several important differences with re- ρ ¼ H0 [13] i c 8π spect to an asymptotically free non-Abelian gauge theory. First, GN whereas the chiral limit of QCD, i.e., quarks with zero current- _ quark masses, is well-defined because of quark confinement, a and H0 ¼ða∕aÞ0 are the Hubble constant in the present era, with U(1) theory with massless charged particles is unstable, owing aðtÞ being the Friedmann–Robertson–Walker scale parameter – ð_∕ Þ2 ¼ 2 ¼ð8π ∕3Þρþ to the well-known fact that these would give rise to a divergent (68 72). The field equations imply a a H GN Λ∕3 − ∕ 2 ¨∕ ¼ −4π ðρ þ 3 ÞþΛ∕3 ρ ¼ Bethe-Heitler pair production cross-section. It is therefore neces- k a and a a GN p , where ∕ ¼ sary to break the chiral symmetry explicitly with bare total mass energy density, p pressure, and k is the curvature 2 1 ¼ Ω þ Ω þ Ω þ Ω Ω ¼ α ¼ ∕ð4πÞ parameter; equivalently, m γ Λ k, where m mass terms mi. If the running coupling 1 e at a given 2 2 2 2 8πG ρ ∕H , Ωγ ¼ 8πG ργ∕H , and Ω ¼ −k∕ðH a Þ. Long energy scale μ were sufficiently large, α1ðμÞ ≳ Oð1Þ, an approxi- N m 0 N 0 k 0 mate solution to the Dyson–Schwinger equation for the propaga- before the current period of precision cosmology, it was known ψ ≪ μ that ΩΛ could not be larger than O(1). In the context of quantum tor of a fermion i with mi would suggest that this fermion Σ – field theory, this empirical fact was very difficult to understand, gains a nonzero dynamical mass i (8 12) and hence, presumably, ρ i there would be an associated condensate hψ¯ ψ i (no sum on i). because estimates of the contributions to Λ from ( ) vacuum i i condensates of quark and gluon fields in QCD and the vacuum However, in analyzing SχSB, it is important to minimize the ef- expectation value of the Higgs field hypothesized in the standard fects of explicit chiral symmetry breaking due to the bare masses model (SM) to be responsible for electroweak symmetry break- mi. The infrared-free nature of this theory means that for any ii α μ ∕μ ing, and from ( ) zero-point energies of quantum fields appear to given value of 1 at a scale , as one decreases mi to reduce α ð Þ be too large by many orders of magnitude. Observations of super- explicit breaking of chiral symmetry, 1 mi also decreases, ap- ∕μ → 0 α ð Þ novae showed the accelerated expansion of the universe and are proaching zero as mi . Because 1 mi should be the rele- consistent with the hypothesis that this accelerated expansion is vant coupling to use in the Dyson–Schwinger equation, it may in due to a cosmological constant, ΩΛ ≃ 0.76 (73–79). The superno- fact be impossible to realize a situation in this theory in which one vae data (73–79), together with measurements of the cosmic mi- has small explicit breaking of chiral symmetry and a large enough crowave background radiation, galaxy clusters, and other inputs, α ð Þ value of 1 mi to induce spontaneous chiral symmetry breaking. e.g., primordial element abundances, have led to a consistent A full analysis would require knowledge of the bound-state spec- determination of the cosmological parameters (80–83). These in- trum of the hypothetical strongly coupled U(1) theory, but this ¼ 73 3 ∕ð − Þ ρ ¼ 0 56 × 10−5 ∕ 3 ¼ clude H0 km s Mpc , c . GeV cm spectrum is not reliably known. ð2 6 × 10−3 Þ4 Ω ≃ 0 24 Ω ≃ 0 042 . eV , total m . with term b . , Ω ≃ 0 20 so that the term is dm . . In the equation of Finite-Temperature QCD state p ¼ wρ for the “dark energy,” w is consistent with being So far, we have discussed QCD and other theories at zero tem- equal to −1, the value if the accelerated expansion is due to a perature (and chemical potential or equivalently here, baryon cosmological constant. [Other suggestions for the source of the density). For QCD in thermal equilibrium at a finite-temperature accelerated expansion include modifications of T,asT increases above the temperature Tdec, and time-dependent wðtÞ, as reviewed in refs. 61–67.] both the hadrons and the associated condensates eventually Here, using our observations concerning QCD condensates, disappear, although experiments at European Organization for we propose a solution to problem (i) of the contributions by these Nuclear Research and Brookhaven National Laboratory-Relati- condensates to ρΛ, which, in the conventional approach, are much ≳ −5 vistic Heavy Ion show that the situation for T Tdec is too large. The QCD condensates form at times of order 10 sin more complicated than a weakly coupled quark–gluon plasma. the early universe, as the temperature T decreases below the – ≃ 200 The picture of the QCD condensates here is especially close confinement deconfinement temperature Tdec MeV. As ≪ to experiment, because, although finite-temperature QCD makes noted above, for T Tdec, in the conventional quantum field use of the formal thermodynamic, infinite volume limit, actual theory view, these condensates are considered to be constants heavy-ion experiments and resultant transitions from confined throughout space. If one accepts this conventional view, then ðδρ Þ ∼ Λ4 to deconfined quarks and gluons take place in the finite-volume these condensates would contribute Λ QCD QCD, so that ðδΩ Þ ≃ 1045 and time interval provided by colliding heavy ions. Indeed, one of Λ QCD . On the contrary, if one accepts the argument the models that has been used to analyze such experiments in- that these condensates [and also higher-order constants such as 2 μν volves the notion of a color glass condensate (58–60). hðqq¯ Þ i and hðqq¯ ÞGμνG i] have spatial support within hadrons,

4of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1010113107 Brodsky and Shrock Downloaded by guest on September 26, 2021 not extending throughout all of space, then one makes consider- Lorentz-invariant integrand function depending on some set of … able progress in solving the above problem, because the effect of 4-momenta p1; ;pn. If, nevertheless, one proceeds to use such these condensates on is already included in the baryon a cutoff, then, because a mass scale characterizing quantum grav- Ω Ω Ω ¼ −1∕2 ¼ 1 2 × 1019 term b in m and, as such, they do not contribute to Λ. ity (QG) is MPlanck G . GeV, one would infer 4 N 2 120 Another excessive type-(i) contribution to ρΛ is conventionally ðδρ Þ ∼ ∕ð16π Þ ðδΩ Þ ∼ 10 that Λ QG MPlanck , and hence Λ QG . With viewed as arising from the vacuum expectation value of the SM the various mass scales characterizing the electroweak symmetry ¼ 2−1∕4 −1∕2 ¼ 246 ðδρ Þ ∼ 4 Higgs field, vEW G GeV, giving Λ v breaking and particle masses in the SM, one similarly would ob- 56F EW EW and hence ðδΩΛÞ ∼ 10 for the electroweak (EW) theory. Si- ðδΩ Þ ∼ 1056 14 EW tain Λ SM . Given the fact that Eq. is not Lorentz- milar numbers are obtained from Higgs vacuum expectation va- invariant, one may well question the logic of considering it as lues in supersymmetric extensions of the SM [recalling that the a contribution to the Lorentz invariant quantity ρΛ. (This criti- breaking scale is expected to be the tera-electron cism of the conventional lore has also been made in refs. 61 – volt (TeV)scale (52 57)]. However, it is possible that electroweak and 84.) Indeed, one could plausibly argue that, as an energy den- symmetry breaking is dynamical; for example, it may result from sity, it should instead be part of T00 in the energy-momentum the formation of a bilinear condensate of fermions F (called tech- tensor Tμν. Phrased in a different way, if one argues that it should nifermions) subject to an asymptotically free, vectorial, confining be associated with the Λgμν term, then there must be a negative gauge interaction, commonly called TC, that gets strong on the ¼ −ρ – corresponding zero-point pressure satisfying p , but the TeVscale (52 57). In such theories there is no fundamental Higgs source for such a negative pressure is not evident in Eq. 14. field. TC theories are challenged by, but may be able to survive, The LF approach to the construction of a quantum field theory, constraints from precision electroweak data. By using our argu- in particular, the SM, provides another perspective to this issue ments above, in a TC theory, the technifermion and technigluon (85—88). condensates would have spatial support in the technihadrons and techniglueballs and would contribute to the masses of these Concluding Remarks states. We stress that, just as was true for the QCD condensates, We have argued from several physical perspectives that, because these technifermion and technigluon condensates would not con- of color confinement, quark and gluon QCD condensates are ρ tribute to Λ. Hence, if a -type mechanism should turn localized within the interiors of hadrons. Our analysis is in agree- out to be responsible for electroweak symmetry breaking, then ment with the Casher–Susskind model and the explicit demon- there would not be any problem with a supposedly excessive con- ρ stration of in-hadron condensates by Roberts and coworkers PHYSICS tribution to Λ for a Higgs vacuum expectation value. Indeed, (14), using the Bethe–Salpeter–Dyson–Schwinger formalism stable technihadrons in certain technicolor theories may be viable for QCD-bound states. We also discussed this physics using con- dark matter candidates. densed matter analogues, the AdS/CFTcorrespondence, and the We next comment briefly on type-(ii) contributions. The for- Bethe–Salpeter–Dyson–Schwinger approach for bound states. mal expression for the energy density E∕V due to zero-point en- In-hadron condensates provide a solution to what has hitherto ergies of a quantum field corresponding to a particle of mass m is commonly been regarded as an excessively large contribution to Z 3 the cosmological constant by QCD condensates. We have argued d k ωðkÞ ∕ ¼ [14] that these condensates do not, in fact, contribute to ΩΛ; instead, E V ð2πÞ3 2 ; they have spatial support within hadrons and, as such, should pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi really be considered as contributing to the masses of these ωð Þ¼ k2 þ 2 where the energy is k m . However, first, this expres- hadrons and hence to Ω . We have also suggested a possible solu- sion is unsatisfactory, because it is (quadratically) divergent. In b tion to what would be an excessive contribution to ΩΛ from a modern one would tend to regard this divergence hypothetical Higgs vacuum expectation value; the solution would as indicating that one is using a low-energy , be applicable if electroweak symmetry breaking occurs via a tech- M and one would impose an ultraviolet cutoff UV on the momen- nicolor-type mechanism. tum integration, reflecting the upper range of validity of this low- energy theory. Because neither the left- nor right-hand side of 14 ACKNOWLEDGMENTS. We thank R. Alkofer, A. Casher, C. Fischer, M. E. Fisher, Eq. is Lorentz invariant, this cutoff procedure is more dubious F. Llanes-Estrada, C. Roberts, L. Susskind, P. Tandy, and G. F. de Téramond than the analogous procedure for Feynman integrals of the form for helpful conversations. This research was partially supported by Grant ∫ 4 ð Þ ð … Þ d kI k;p in quantum field theory, where I k;p1; ;pn is a DE-AC02-76SF00515 (to S.J.B.) and by Grant NSF-PHY-06-53342 (to R.S.).

1. Casher A, Susskind L (1974) Chiral magnetism (or magnetohadrochironics). Phys Rev D 14. Maris P, Roberts CD, Tandy PC (1998) Pion mass and decay constant. Phys Lett B 9:436–460. 420:267–273. 2. Casher A, Noskowitz H, Susskind L (1971) Goldstone–Nambu in dual and parton 15. Maris P, Roberts CD (1997) π and K meson Bethe–Salpeter amplitudes. Phys Rev C: Nucl theories. Nucl Phys B 32:75–92. Phys 56:3369–3383. 3. Weinberg S (1966) Dynamics at infinite momentum. Phys Rev 150:1313–1318. 16. Langfeld K, Markum H, Pullirsch R, Roberts CD, Schmidt SM (2003) Concerning the 4. Dirac PAM (1949) Forms of relativistic dynamics. Rev Mod Phys 21:392–399. quark condensate. Phys Rev C: Nucl Phys 67:065206. 5. Mueller AH (1994) Soft gluons in the infinite momentum and the BFKL 17. Banks T, Casher A (1980) Chiral symmetry breaking in confining theories. Nucl Phys B – . Nucl Phys B 415:373–385. 169:103 125. 18. Gell-Mann M, Oakes RJ, Renner B (1968) Behavior of current divergences 6. Antonuccio F, Brodsky SJ, Dalley S (1997) Light-cone wavefunctions at smallx. Phys Lett underSUð3Þ × SUð3Þ. Phys Rev 175:2195–2199. B 412:104–110. 19. Brodsky SJ, Hwang DS, Ma BQ, Schmidt I (2001) Light-cone representation of the spin 7. Motyka L, Stasto AM (2009) Exact kinematics in the smallx evolution of the color and orbital angular momentum of relativistic composite systems. Nucl Phys B dipole and gluon cascade. Phys Rev D 79:085016. 593:311–335. 8. Lane K (1974) and Goldstone realization of chiral symmetry. Phys 20. Teryaev OV (1999) Spin structure of nucleon and equivalence principle. ArXiv:hep-ph/ Rev D 10:2605–2618. 9904376. 9. Politzer HD (1976) Effective quark masses in the chiral limit. Nucl Phys B 117:397–406. 21. Berends FA, Gastmans R (1976) Quantum electrodynamical corrections to - 10. Kugo T (1978) Dynamical instability of the vacuum in the Lagrangian formalism of the matter vertices. Ann Phys 98:225–236. – – Bethe Salpeter bound states. Phys Lett B 76:625 630. 22. Fetter A, Walecka J (1971) of Many-Particle Systems (McGraw-Hill, 11. Pagels H (1979) Dynamical chiral symmetry breaking in quantum chromodynamics. New York) p 439. Phys Rev D 19:3080–3090. 23. Tinkham M (1996) Introduction to Superconductivity (McGraw-Hill, New York) p 1. 12. Higashijima K (1984) Dynamical chiral symmetry breaking. Phys Rev D 29:1228–1232. 24. Fisher ME, Ferdinand AE (1967) Interfacial, boundary, and size effects at critical points. 13. Brodsky SJ, Shrock R (2008) Maximum wavelength of confined quarks and gluons and Phys Rev Lett 19:169–172. properties of quantum chromodynamics. Phys Lett B 666:95–99 arXiv:0803.2541; 25. Fisher ME, Barber MN (1972) Scaling theory for finite-size effects in the critical region. arXiv:0803.2554. Phys Rev Lett 28:1516–1519.

Brodsky and Shrock PNAS Early Edition ∣ 5of6 Downloaded by guest on September 26, 2021 26. Barber MN (1983) Finite size scaling. Phase Transitions and Critical Phenomena, eds 56. Dimopoulos S, Susskind L (1979) Mass without scalars. Nucl Phys B 155:237–252. C Domb and JL Lebowitz (Academic, New York), Vol. 8, pp 145–266. 57. Eichten E, Lane K (1980) Dynamical breaking of symmetries. Phys Lett 27. Casher A (1979) Chiral symmetry breaking in quark confining theories. Phys Lett B B 90:125–130. 83:395–398. 58. McLerran LD, Venugopalan R (1994) Computing quark and gluon distribution func- 28. Brodsky SJ, Pauli HC, Pinsky SS (1998) Quantum chromodynamics and other field tions for very large nuclei. Phys Rev D 49:2233–2241. theories on the light cone. Phys Rep 301:299–486. 59. McLerran LD, Venugopalan R (1994) Gluon distribution functions for very large nuclei 29. Susskind L, Burkardt M (1994) A Model of mesons based on chi(SB) in the light front at small transverse momentum. Phys Rev D 49:3352–3355. frame. Proceedings of the Fourth International Conference on the Theory of Hadrons 60. McLerran LD, Venugopalan R (1994) Green’s functions in the color field of a large and Light-Front QCD, Zakopane pp 5–14 arXiv:hep-ph/9410313v1. nucleus. Phys Rev D 50:2225–2233. 30. Burkardt M (1998) Dynamical vertex and chiral symmetry breaking on 61. Peebles PJE, Ratra R (2003) The cosmological constant and dark energy. Rev Mod Phys the light front. Phys Rev D 58:096015. 75:559–606. 31. Brodsky SJ, de Téramond GF (2004) Light front hadron dynamics and AdS/CFT 62. Carroll SM (2001) The cosmological constant. Liv Rev Rel 4:1–49 astro-ph/0004075. correspondence. Phys Lett B 582:211–221. 63. Padmanabhan T (2003) Cosmological constant—the weight of the vacuum. Phys Rep 32. de Téramond GF, Brodsky SJ (2005) Hadronic spectrum of a holographic dual of QCD. 380:235–320. Phys Rev Lett 24:201601. 64. Padmanabhan T (2008) Dark energy and gravity. Gen Rel Grav 40:529–564. 33. Brodsky SJ, de Téramond GF (2006) Hadronic spectra and light-front wavefunctions in 65. Sahni V, Starobinsky AA (2006) Reconstructing dark matter. Int J Mod Phys D holographic QCD. Phys Rev Lett 96:201601. 15:2105–2132. 34. Brodsky SJ, de Téramond GF, Shrock R (2008) and hadroniza- 66. Frieman J, Turner MS, Huterer D (2008) Dark energy and the accelerating universe. tion at the amplitude level. Proceedings of the Sixth International Conference on Annu Rev Astron Astrophys 46:385–432. Perspectives on Hadron Physics, American Institute of Physics Conference Proceedings 67. Weinberg S (1989) The cosmological constant problem. Rev Mod Phys 61:1–23. 1056 pp:3–14. 68. Astier P, et al. (2006) The surpernova legacy survey: Measurement of ΩM , ΩΛ, and w 35. Erlich J, Katz E, Son DT, Stephanov MA (2005) QCD and a holographic model of from the first year data set. Astron Astrophys 447:31–48. hadrons. Phys Rev Lett 95:261602. 69. Peebles PJE (1993) Principles of Physical Cosmology (Princeton Univ Press, Princeton) 36. Da Rold L, Pomarol A (2005) Chiral symmetry breaking from five-dimensional spaces. p1. Nucl Phys B 721:79–97. 70. Coles P, Lucchin F (2002) Cosmology: The Origin and Evolution of Cosmic Structure 37. Chodos A, Thorn CB (1975) Chiral invariance in a bag theory. Phys Rev D 12:2733–2743. (Wiley, New York). 38. Brown GE, Rho M (1979) The little bag. Phys Lett B 82:177–180. 71. Dodelson S (2003) Modern Cosmology (Academic, Boston) p 1. 39. Hosaka A, Toki H (1996) Chiral bag models for the nucleon. Phys Rep 277:65–188. 72. Carroll S (2004) and Geometry: An Introduction to General Relativity (Ad- 40. Shifman MA, Vainshtein AI, Zakharov VI (1979) QCD and resonance physics. Sum rules. dison-Wesley, San Francisco) p 1. Nucl Phys B 147:385–447. 73. Weinberg S (2009) Cosmology (Oxford Univ Press, Oxford) p 1. 41. Shifman MA, Vainshtein AI, Zakharov VI (1979) QCD and resonance physics. Applica- 74. Misner CW, Thorne KS, Wheeler JA (1973) Gravity 1. tions. Nucl Phys B 147:448–518. 75. Riess AG, et al. (1998) Observational evidence from supernovae for an accelerating 42. Narison S (1989) QCD Spectral Sum Rules (World Scientific, Singapore) p 1. universe and a cosmological constant. Astron J 116:1009–1038. 43. Barate R, et al. (1998) Studies of quantum chromodynamics with the ALEPH detector. 76. Tonry JL, et al. (2003) Cosmological results from high-z supervovae. Astrophys J Phys Rep 294:1–165. 594:1–24. 44. Geshkenbein BV, Ioffe BL, Zyablyuk KN (2001) The check of QCD based on theτ-decay 77. Riess AG, et al. (2004) Type 1a discoveries at z>1 from the Hubble Space data analysis in the complex q2 plane. Phys Rev D 64:093009. Telescope: Evidence for past deceleration and constraints on dark energy evolution. 45. Ioffe BL, Zyablyuk KN (2003) Gluon condensate in charmonium sum rules with three Astrophys J 607:665–687. loop corrections. Eur Phys J C 27:229–241. 78. Riess AG, et al. (2007) New Hubble Space Telescope discoveries of type 1a supernovae 46. Davier M, Höcker A, Zhang Z (2007) ALEPH tau spectral functions and QCD. Nucl Phys at z ≥ 1: Narrowing constraints on the early behavior of dark energy. Astrophys J B, Proc Suppl 169:22–35. 659:98–121. 47. Davier M, et al. (2008) The determination of αs from τ decays revisited. Eur Phys J C 79. Perlmutter S, et al. (1999) Measurements of Ω and Λ from 42 high-redshift supernovae. 56:305–322. Astrophys J 517:565–586. 48. Bjorken JD (1991) What lies ahead? Proceedings Superconductivity Super Collider 80. Knop RA, et al. (2003) New constraints on ΩM , ΩΛ, andw from an independent set of Workshop, Corpus Christi, In Conclusion: A Collection of Summary Talks in High Energy 11 high-redshift supernovae observed with HST. Astrophys J 598:102–137. Physics (World Scientific, Singapore). 81. Astier P, et al. (2006) The Supernova legacy survey: Measurement of ΩM , ΩΛ, and 49. Bjorken JD (1997) Disoriented chiral condensate: Theory and . Acta w from the first year dataset. Astron Astrophys 447:31–48. Phys Pol B28:2773–2791. 82. Kogut A, et al. (2007) Three-year Wilkinson Microwave Anisotropy Probe (WMAP) ob- 50. Mohanty B, Serreau J (2005) Disoriented chiral condensate: Theory and experiment. servations: Foreground Polarization. Astrophys J 665:355–362. Phys Rep 414:263–358. 83. Tegmark M, et al. (2006) Cosmological constraints from the SDSS luminous red 51. Kogut J (1983) The lattice gauge theory approach to quantum chromodynamics. Rev galaxies. Phys Rev D 74:123507. Mod Phys 55:775–836. 84. Percival WJ, et al. (2007) Measuring the baryon acoustic oscillation scale using the SDSS 52. Weinberg S (1979) Implications of dynamical symmetry breaking: An addendum. Phys and 2dFGRS. Mon Not Roy Astr Soc 381:1053–1069. Rev D 19:1277–1280. 85. Amsler C, et al. (2008) Review of particle properties. Phys Lett B 667:1–1340. 53. Susskind L (1979) Dynamics of spontaneous symmetry breaking in the Weinberg– 86. Jaffe RL (2005) and . Phys Rev D 72:021301. Salam theory. Phys Rev D 20:2619–2625. 87. Srivastava PP, Brodsky SJ (2002) A unitary and renormalizable theory of the standard 54. ’t Hooft G, et al., ed. (1980) Recent Development in Gauge TheoriesNATO Advanced model in free light cone gauge. Phys Rev D 66:045019. Study Institutes Series - Series B: Physics (Plenum, New York), 59, p 438. 88. Brodsky SJ (2004) Light-front QCD. Proceedings of the 58th Scottish University Summer 55. Dimopoulos S, Raby S, Susskind L (1980) Light composite fermions. Nucl Phys B School in Physics: A NATO Advanced Study Institute and EU Hadronic Physics 13th Sum- 173:208–228. mer Institute (Taylor and Francis, Oxford), pp 121–173.

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