Condensates in quantum chromodynamics and the cosmological constant Stanley J. Brodskya,b,1 and Robert Shrockb aSLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309; and bChen Ning Yang Institute for Theoretical Physics, 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 quark and quark pairs in the LF wavefunc- 460] have noted that in the light-front description, spontaneous tion. In fact, the Regge behavior of hadronic structure functions chiral symmetry breaking is a property of hadronic wavefunctions requires that LF Fock states of hadrons have Fock states with and not of the vacuum. Here we show from several physical an infinite number of quark and gluon partons (5–7). Thus, in perspectives that, because of color confinement, quark and gluon contrast to formal discussions in statistical mechanics, infinite condensates in quantum chromodynamics (QCD) are associated volume is not required for a phase transition in relativistic quan- with the internal dynamics of hadrons. We discuss condensates tum field theories. using condensed matter analogues, the Anti de Sitter/conformal Spontaneous chiral symmetry breaking 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 propagator; if the running with the Casher and Susskind model and the explicit demonstra- α ¼ 2∕ð4πÞ coupling s gs exceeds a value of order 1, this equation tion of “in-hadron” 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 quarks tional effects of the in-hadron condensates are already included and gluons 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 gauge theory dark energy 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 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 current quark 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 vacuum expectation value; instead, it is the matrix 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 R, where (14 16) using the Bethe Salpeter Dyson Schwinger analysis [SUðNÞ denotes the group 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 Æiλ 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 standard model. 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) quantization 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 gravitons 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 quantum field theory 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.