Essence of the Vacuum Quark Condensate

SLAC-PUB-14107 Essence of the vacuum quark condensate

Stanley J. Brodsky,1, 2 Craig D. Roberts,3, 4 Robert Shrock,5 and Peter C. Tandy6 1SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309 2Centre for Particle Physics Phenomenology: CP3-Origins, University of Southern Denmark, Odense 5230 M, Denmark 3Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 4Department of Physics, Peking University, Beijing 100871, China 5C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794 6Center for Nuclear Research, Department of Physics, Kent State University, Kent OH 44242, USA We show that the chiral-limit vacuum quark condensate is qualitatively equivalent to the pseudoscalar meson leptonic decay constant in the sense that they are both obtained as the chiral-limit value of well-deﬁned gauge-invariant hadron-to-vacuum transition amplitudes that possess a spectral representation in terms of the current-quark mass. Thus, whereas it might sometimes be convenient to imagine otherwise, neither is essentially a constant mass-scale that ﬁlls all spacetime. This means, in particular, that the quark condensate can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions.

PACS numbers: 11.30.Rd; 14.40.Be; 24.85.+p; 11.15.Tk

Non-zero vacuum expectation values of local operators; in terms of a non-zero vacuum expectation value 0 qq¯ 0 . i.e., condensates, are introduced as parameters in QCD The notion that non-zero vacuum condensates | exist| sum rules, which are used to estimate essentially non- and possess a measurable reality has long been recognized perturbative strong-interaction matrix elements. They as posing a conundrum for the light-front formulation of are also basic to current algebra analyses. It is widely QCD. This formulation follows from Dirac’s front form of held that such quark and gluon condensates have a relativistic dynamics [11], and is widely and efficaciously physical existence, which is independent of the hadrons employed in perturbative and nonperturbative QCD [12, that express QCD’s asymptotically realizable degrees-of- 13]. In the light-front formulation, the ground-state is a freedom; namely, that these condensates are not merely structureless Fock space vacuum, in which case it would mass-dimensioned parameters in a theoretical trunca- seem to follow that DCSB is impossible. In response, it tion scheme, but in fact describe measurable spacetime- was argued by Casher and Susskind [14] that, in the light- independent configurations of QCD’s elementary degrees- front framework, DCSB must be a property of hadron of-freedom in a hadron-less ground state. wavefunctions, not of the vacuum. This thesis has also We share the view that these condensates are funda- been explored in a series of recent articles [15–17]. mental dynamically-generated mass-scales in QCD. How- A non-zero spacetime-independent QCD vacuum con- ever, we shall argue that their measurable impact is en- densate also poses a critical dilemma for gravitational tirely expressed in the properties of QCD’s asymptoti- interactions because it would lead to a cosmological con- cally realizable states; namely hadrons. In taking this stant some 45 orders of magnitude larger than observa- position we have assumed confinement, from which fol- tion. As noted elsewhere [15], this conflict is avoided lows quark-hadron duality and hence that all observable if strong interaction condensates are properties of rigor- consequences of QCD can, in principle, be computed us- ously well-defined wavefunctions of the hadrons, rather ing a hadronic basis. Here, the term “hadron” means any than the hadron-less ground state of QCD. one of the states or resonances in the complete spectrum Given the importance of DCSB and the longstanding of color-singlet bound-states generated by the theory. puzzles described above, we will focus our attention on the vacuum quark condensate. The essential issues be- We focus herein on 0 qq¯ 0 , where 0 is viewed as come particularly clear in the context of the Gell-Mann– some hadron-less ground | state| of QCD.| This is the vac- Oakes–Renner relation [18, 19], which is usually under- uum quark condensate. Its non-zero value is usually held stood as the statement to signal dynamical chiral symmetry breaking (DCSB), 2 2 u d 0 a concept of critical importance in QCD, whose con- fπ mπ = 2(mζ + mζ ) qq¯ ζ , (1) nection with the dressed-quark propagator was antici- − pated [1–5] (see also references therein). As reviewed wherein mπ is the pion’s mass; fπ is its leptonic decay q elsewhere (most recently, e.g., Refs. [6–8]), DCSB is a constant; mζ , with q = u, d, is the current-quark mass 0 remarkably efficient mass-generating mechanism, the ori- at a renormalization scale ζ; and qq¯ ζ is the chiral-limit gin of constituent-quark masses and intimately connected vacuum quark condensate, with a precise definition of the with confinement. It is also the basis for the success- chiral limit given below in Eqs. (8), (9). In arriving at ful application of chiral-effective field theories (see, e.g., Eq.(1) using standard methods, one makes truncations; Refs. [9, 10] for contemporary perspectives). On the face namely, soft-pion techniques [20] have been used to re- of it, this seems far more than can be understood simply late an in-pion matrix element of the current-quark mass-

Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 term to the vacuum quark condensate. (NB. For techni- that fπ is truly an observable; and Z4(ζ, Λ) ensures that cal simplicity, we will only explicitly consider the SU(2)- ρπ(ζ) is independent of Λ and evolves with ζ in just the flavor case of two light quarks, with mu = m = md. way necessary to guarantee that the product mζ ρπ(ζ) is A discussion of SU(3)-flavor and the η-η′ complex is renormalization-point-independent. qualitatively identical and readily accomplished follow- We now can discuss the chiral limit, which is well- ing Ref. [21].) defined in QCD since it is an asymptotically free, con- It is instructive to consider Eq. (1) in another light. fining theory. Recall that Chiral symmetry and the pattern by which it is broken bm in QCD are expressed through the axial-vector Ward- Z2(ζ, Λ) m (Λ) = Z4(ζ, Λ) mζ , (8) Takahashi identity, where mbm(Λ) is the Lagrangian bare-mass parameter. PµΓ5µ(k; P ; ζ)+2imζ Γ5(k; P ; ζ) Then, the chiral limit is defined by −1 −1 = S (k+; ζ)iγ5 + iγ5S (k−; ζ) . (2) bm Z2(ζ, Λ) m (Λ) 0 , Λ ζ , (9) ≡ ∀ ≫ Here Γ (k; P ; ζ) is the axial-vector vertex; Γ (k; P ; ζ) is 5µ 5 which is equivalent to requiringm ˆ = 0 [23], where the pseudoscalar vertex; k± = k P/2; and S(ℓ; ζ) is the mˆ is the renormalization-point-invariant current-quark dressed-quark propagator, which± has the general form1 mass. This means, of course, that we suppress the ef- S(ℓ; ζ)=1/[iγ ℓA(ℓ2; ζ)+ B(ℓ2; ζ)] . (3) fect of electroweak interactions, which explicitly violate SU(2)L SU(2)R chiral symmetry. Making use of the fact that the ground-state pion is Equation× (5) is the exact expression in QCD for the the lowest-mass pole in both vertices if, and only if, chi- pion’s leptonic decay constant.3 It is a property of the ral symmetry is dynamically broken, one can derive the pion and, as consideration of the integral expression re- following identity [22], which is exact in QCD: veals, it can be described as the pseudo-vector projection of the pion’s Bethe-Salpeter wavefunction onto the ori- f m2 =2m ρ (ζ) , (4) π π ζ π gin in configuration space. (It can also be defined as the where (omitting ζ unless necessary for clarity or empha- integral of the pion’s gauge invariant distribution ampli- sis) tude [12].) We note that the product ψ = SΓS is called the Bethe-Salpeter wavefunction because, when a non- ifπPµ = 0 qγ¯ 5γµq π (5) relativistic limit can validly be performed, the quantity | | Λ d4q ψ at fixed time becomes the quantum mechanical wave- = Z2(ζ, Λ) trCD iγ5γµS(q+)Γπ(q; P )S(q−) , Z (2π)4 function for the system under consideration. If chiral symmetry were not dynamically broken, then iρπ = 0 qiγ¯ 5q π − | | in the neighborhood of the chiral limit fπ mˆ [27]. Of Λ ∝ d4q course, chiral symmetry is dynamically broken in QCD = Z4(ζ, Λ) trCD γ5S(q+)Γπ(q; P )S(q−) . (6) Z (2π)4 [28–30] and

Λ d4q 0 Here represents a Poincar´e-invariant regulariza- lim fπ(ˆm)= fπ =0 . (10) (2π)4 mˆ →0 tion ofR the integral, with Λ the ultraviolet regularization Taken together, these last two observations express the mass-scale,2 and Γ (k; P ) is the pion’s Bethe-Salpeter π fact that f 0, which is an intrinsic property of the pion, is amplitude; viz., π a bona ﬁde order parameter for DCSB. A typical estimate

Γπ(k; P )= γ5 [iEπ(k; P )+ γ P Fπ(k; P ) from chiral perturbation theory [9] suggests that the chi- 0 ral limit value, fπ, is 5% below the measured value of +γ k Gπ(k; P ) σµν kµPν Hπ(k; P )] . (7) ∼ − 92.4 MeV; and eﬃcacious DSE studies give a 3% chiral- The quark wavefunction and Lagrangian mass renormal- limit reduction [23]. In connection with the leptonic de- 2 ization constants, Z (ζ, Λ), respectively, depend on the cay, it is interesting to note that Γπ+→µ+ν f mπ. In 2,4 ∝ π gauge parameter in precisely the manner needed to en- contrast, within a constituent-quark model, Γπ+→µ+ν 2 ∝ sure that the right-hand sides of Eqs. (5), (6) are gauge- ψ(0) , where ψ(r) is the pion’s constituent-quark wave- | | invariant. Moreover, Z2(ζ, Λ) ensures that the right- function [31]. Therefore, consistency with DCSB in QCD hand side of Eq.(5) is independent of both ζ and Λ, so requires that a realistic pion constituent-quark wavefunction must satisfy ψ(0) √mπ in the neighborhood of the chiral limit [32|]. | ∝

1 † We are using a Euclidean metric, with {γµ, γν } = 2δµν ; γµ = γµ; 4 2 γ5 = γ4γ1γ2γ3; a · b = Pi=1 aibi; and Pµ timelike ⇒ P < 0. 2 3 In connection with Eq. (15) below, we describe how conﬁnement In the neighborhood of the chiral limit, a value for fπ can be es- and dynamical mass generation regulate the infrared domain in timated via either of two approximation formulae [24–26]. These Eqs. (5), (6), thus also ensuring the absence of infrared diver- formulae both illustrate and emphasize the role of fπ as an order gences. parameter for DCSB. 3

Equation (6) is kindred to Eq. (5); it is the expression On the other hand [35], in quantum field theory which describes the pseudoscalar lim κπ(ˆm; ζ) = constant. (14) projection of the pion’s Bethe-Salpeter wavefunction onto mˆ →∞ the origin in configuration space. Thus it is truly just another type of pion decay constant. Moreover, the physics The result in Eq. (14) contrasts sharply with the be- becomes transparent upon the consideration of its chiral- havior of the trace of the dressed-quark propagator, limit behavior. which is usually considered to provide the context for Complementing the discussion in Ref. [14], a rigorous an extension of the vacuum quark condensate tom ˆ = 0. definition of an “in-hadron condensate” was presented in The latter quantity exhibits a quadratic divergence [33]; Refs. [22, 23]. In our context it is given by Λ d4q 2 viz., 4 Smˆ (q) mˆ Λ form ˆ ΛQCD. As a func- (2π) ∝ ≫ π tion ofR m ˆ , it is thus only rigorously defined at a single qq¯ fπ 0 qγ¯ 5q π = fπρπ(ζ) =: κπ(ˆm; ζ) . (11) − ζ ≡− | | value of its argument; i.e., on a set of measure zero. In Λ d4q Since the dressed-quark propagator has a spectral rep- QCD, therefore, (2π)4 trSmˆ alone provides no informa- resentation when considered as a function ofm ˆ [33], one tion about the current-quark-mass-dependenceR of the dy- can derive from the axial-vector Ward-Takahashi identity namical chiral symmetry breaking phenomenon. That is a collection of Goldberger-Treiman-like relations for the better traced through other means [36]. pion [22, 34], the most important of which herein is It is now clear that both fπ(ˆm = 0) and κπ(ˆm = 0; ζ) are intrinsic properties of the bound-state, in this case 1 2 an isovector pseudoscalar meson constituted from equally Eπ(k;0) = 0 B0(k ) , (12) fπ massive current-quarks. They are also order parameters for DCSB and, as elucidated in Ref. [37], they character- where B0 is the scalar part of the dressed-quark self- ize QCD’s susceptibility to respond to the insertion of energy computed in the chiral-limit. This is Eq. (10) a pseudoscalar probe. Their role as order parameters of Ref. [22]. Equations (10)–(13) therein state that if, is conveyed through the dressed-quark mass function, and only if, chiral symmetry is dynamically broken, then M(p2) = B(p2; ζ)/A(p2; ζ), via the Goldberger-Treiman the solution of the chiral limit one-body problem for relations derived in Ref. [22] and exemplified in Eq. (12). the dressed-quark propagator completely determines the The derivation of Eq. (13) given here makes this con- leading amplitude in the solution of the pion bound-state nection manifest. It also shows that it is the dynam- problem, and it tightly constrains the bound-state’s sub- ical generation of a non-zero quark mass function in leading amplitudes. chiral-limit QCD which expresses DCSB most funda- Using Eq. (12), one finds [22] mentally. The behavior of M(p2) is now well known. Λ Form ˆ = 0 it is power-law suppressed at ultraviolet d4q 2 lim κπ(ˆm; ζ) = Z4(ζ, Λ)trCD S0(q; ζ) momenta; viz., 1/p . On the other hand [28–30], mˆ →0 Z (2π)4 M(0) 0.3 GeV=∼ 1/[0.66 fm]. Gluons also acquire a 0 ∼ = qq¯ ζ . (13) large dynamical mass [38]. Indeed, even in quenched- − 2 QCD a momentum-dependent dynamical mass, mg(k ), Thus the so-called vacuum quark condensate is, in fact, appears in the transverse part of the gluon 2-point func- Q the chiral-limit value of the in-pion condensate; i.e., it de- tion [39, 40], with mg (0) 0.35GeV= 1/[0.55 fm]. The scribes a property of the chiral-limit pion. One can there- inclusion of dynamical quarks≈ leads to an increased value Q 2 2 fore argue that this condensate is no more a property of [41]; viz., mg(0) 0.45 GeV, but with m (k )= mg(k ) ∼ g the “vacuum” than the pion’s chiral-limit leptonic decay for k2 & 10 GeV2. constant. Moreover, Ref. [33] establishes the equivalence Both of these dynamical mass functions owe their long- of all three definitions of the vacuum quark condensate: range enhancement to the non-Abelian nature of QCD a constant in the operator product expansion [1, 2]; via and are intimately connected, as suggested by the similar the Banks-Casher formula [5]; and the trace of the chiral- magnitude of their values at infrared momenta; i.e., limit dressed-quark propagator. Note that Eq. (1) is now readily obtained using Eqs. (4), (11), (13). M(0) mG(0) 0.4 GeV 1/[0.5 fm] =: mir . (15) We reiterate that the in-pion condensate is essentially ∼ ≈ ≈ equivalent to the pion’s leptonic decay constant. As with This mass-scale provides an infrared cutoff within an fπ(ˆm), κπ(ˆm; ζ) has a spectral representation in terms of hadron, such that the role played by constituent field 2 2 the current-quark mass and, for all values of that mass, it modes with p < mir is suppressed. This is the is a well-defined, gauge-invariant and properly renormal- dynamically-induced infrared regulator we referred to in ized function. On one hand, the quantity κπ(ˆm; ζ) has connection with Eqs. (5), (6). Defining a “wavelength”: a non-zero value in the chiral limit if, and only if, chi- λ := 1/ p2, then an equivalent statement is that modes ral symmetry is dynamically broken; and the so-called with λ pincreasing beyond λir = 1/mir play a progres- vacuum quark condensate is merely the result obtained sively smaller part in defining the bound-state’s prop- when evaluating κπ(ˆm; ζ) at this single value of its argu- erties. The phenomenon of dynamical mass generation ment. It is not qualitatively different from fπ(ˆm = 0). is closely related to the maximal wavelength condition 4 discussed in Ref. [16]; and both are consequences of con- u– (a) finement in QCD. Recent DSE studies of ground-state + – pseudoscalar and vector quarkonia, from the u- to the b- u – – – quark region, and the corresponding D- and B-mesons, π– u u indicate that a DCSB-induced infrared regularization of 5 – γ both the self-interaction dressing and binding dynamics d of c- and b-quarks tends to improve the computed values of hadron masses and electroweak decay constants [42]. + – As stated above, color confinement is important in (b) u δm establishing condensates as in-hadron rather than vacuum matrix elements. This may be visualized via the – ¯ B-meson; i.e., a bound-state of a heavy b- and a light- π– – γ5 quark. Dyson-Schwinger equation and lattice-QCD studies demonstrate that the propagator for an apparently d isolated light-quark acquires a momentum-dependent mass function [28–30], with which a non-zero condensate FIG. 1: (Color online) Light-front contributions to ρπ = can be associated [33]. However, in a fully self-consistent −0|qγ¯ 5q|π. Upper panel – A non-valence piece of the me- treatment of the bound state, this phenomenon occurs in son’s light-front wavefunction, whose contribution to ρπ is the background field of the ¯b-quark, whose influence on mediated by the light-front instantaneous quark propagator light-quark propagation is primarily concentrated in the (vertical crossed-line). The “±” denote parton helicity. Lower far infrared and whose presence ensures the manifesta- panel – There are infinitely many such diagrams, which can tions of light-quark dressing are gauge invariant. introduce chiral symmetry breaking in the light-front wave- Equation (13) shows that the so-called vacuum quark function in the absence of a current-quark mass. (The case of f is analogous.) condensate is the chiral-limit value of the in-hadron con- π densate in all reference frames. It is a property of the bound-state in precisely the same manner as f . It is π of Eq.(17) gives the appearance of vanishing identically therefore of interest to consider these two quantities in in the chiral limit, which is defined in Eqs. (8), (9). the light-front framework. Following Ref. [12], one finds + − 2 + Within the light-front formulation of QCD, the ques- in the collinear frame; i.e., P = (P , P = m /P ,0⊥): π tion of DCSB has often led to a consideration of longi- 2 2 1 2 tudinal zero modes. That discussion has almost exclu- − d k⊥ k⊥ + mζ fπP = 2√N c Z2 dx ψ(x, k⊥) sively been conducted in the context of producing: (1) a Z Z 16π3 P + x(1 x) 0 − non-zero value for the vacuum quark condensate; and (2) +instantaneous , (16) a Goldstone boson. (See, e.g., Refs. [43–45].) However, 1 2 d k⊥ mζ with a shift in the paradigm so that DCSB is understood ρπ = √N c Z2 dx ψ(x, k⊥) Z Z 16π3 x(1 x) as being expressed in properties of hadrons rather than of 0 − +instantaneous , (17) the vacuum, it becomes apparent that zero modes cannot provide for nonzero values of fπ and ρπ in Eqs. (16) and where both currents receive contributions from the “in- (17). Indeed, owing to confinement, light-front modes + + + stantaneous” part of the quark propagator ( γ /k ) with k mir are exponentially suppressed within a and the associated gluon emission, which are not∼ written bound-state≪ and hence cannot contribute materially to explicitly. In Eqs. (16) and (17), ψ(x, k⊥) is the valence- these in-hadron properties. only Fock state of the pion’s light-front wavefunction. An alternative is illustrated in Fig. 1. The light-front- Its integral over transverse momentum defines the gauge- instantaneous quark propagator can mediate a contribu- invariant pion distribution amplitude. tion from higher Fock state components to the matrix Recall now that the pion’s leptonic decay constant is elements in Eqs. (16) and (17). Such diagrams connect an order parameter for DCSB. In the context of Eq.(5), dynamically-generated chiral-symmetry breaking compo- this is readily seen: owing to Eq. (12) and linearity of the nents of the meson’s light-front wavefunction to these Bethe-Salpeter equation in the solution Γπ, the magni- matrix elements. There are infinitely many contribu- tude of fπ is determined by that of the scalar piece of tions of this type and they do not depend sensitively the dressed-quark self-energy. This is large in the chiral on the current-quark mass in the neighborhood of the limit when chiral symmetry is dynamically broken, but chiral limit. Thus, DCSB in the light-front formulation, otherwise vanishes form ˆ = 0. Furthermore, from the dis- expressed via in-hadron condensates, is seen to be con- cussion associated with Eqs. (6), (11) and (13), we have nected with sea-quarks derived from higher Fock states. already seen that ρπ is also a DCSB order parameter. This solution is kindred to that discussed in Ref. [14]. However, how are fπ(ˆm = 0) and ρπ(ˆm = 0) to be We note that the role of the instantaneous contributions established as order parameters from Eqs.(16) and (17)? is also emphasized in Ref. [44], with the suggestion that This aspect of these in-hadron properties is difficult to see they conspire to ensure Eqs. (16) and (17) are consistent from these equations. For example, the right-hand-side with Eq. (4). 5

Herein we have presented an alternate view of the a manner that does not involve the introduction of vac- chiral order parameter which is conventionally under- uum condensates, such as via Dyson-Schwinger equations stood as the vacuum quark condensate; namely, that [6–8] or lattice gauge theory [46]. it is qualitatively equivalent to the pion decay constant This study was conceived at a workshop sponsored by and is localized within the hadron. There are also the Argonne/U. Chicago Joint Theory Institute, funded other QCD quantities that are conventionally interpreted by ANL’s LDRD program. We acknowledge valuable dis- as vacuum condensates, uniform in spacetime, such as cussions with P. O. Bowman during this event and subse- µν Gµν G . Condensates are typically introduced as a pri- quent conversations with Guy de Téramond. This work ori -undetermined mass-dimensioned parameters in the was supported in part by: U. S. Department of Energy operator product expansion of a color-singlet current- contract no. DE-AC02-76SF00515; U. S. Department of current correlator. As such, the so-called vacuum con- Energy, Office of Nuclear Physics, contract no. DE- densates have come to be useful theoretical devices. How- AC02-06CH11357; and the U. S. National Science Foun- ever, this should not be permitted to obscure the fact dation, under grants NSF-PHY-06-53342 and NSF-PHY- that their rigorous definition is delicate, owing, e.g., to 0903991. SJB also thanks the Hans Christian Andersen possible dependence on the normal-ordering prescription; Academy and Professor Franceso Saninno for hosting his nor that current-current correlators can be calculated in visit at CP3.

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