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(b) µ + o h eotmeauepaesrcueo etra SU( vectorial a of structure phase exclus zero-temperature for scattering rules the inelastic counting for dimensional deep and as jets, implic such quarkonia, the cond discuss phenomena We for and modified. wavelength also are Thus instantons value. of expectation effects true a to ymtybekn ntebudsaeDsnShigrequat w Dyson-Schwinger condensate state quark bound ¯ The the operator infrared. in t the confinement, breaking of in Because finite physics. stays atomic in coupling effe shift calcula Lamb bound-state normally the by of , modified are equations, scale. Dyson-Schwinger confinement the order N ASnmes 11.q 13.d 12.38.-t 11.30.Rd, 11.15.-q, numbers: PACS k , e f min,atom eas ursadgun r ofie ihnhdos hyh they , within confined are gluons and quarks Because − fmsls . massless of .INTRODUCTION I. ,bgn ihteBteSlee equa- Bethe-Salpeter the with begins ), a tnodLna ceeao etr tnodUniversi Stanford Center, Accelerator Linear Stanford (a) qq c eateto hsc,Uiest fDra,Dra DH1 Durham Durham, of University Physics, of Department (c) ≃ vlae ntebcgon ftefilso h te hadroni other the of fields the of background the in evaluated Z 2 na sifae I)dvret it divergent, (IR) infrared is 1 B ≃ S n 1 α / sterda quantum radial the is em 2 tne .Brodsky J. Stanley n n 2 and m e k , P /α ≃ 1 / 2 em Chromodynamics m eesof levels nthe in ) onot do (1) a,b,c n oetShrock Robert and hr Π where arn.Tu,bcueo ofieet ursadglu- and wavelengths quarks maximum confinement, have the glu- of ons states, because and physical Thus, color-singlet quarks hadrons. within and confined breaking, are ons symmetry chiral neous smttcfedm stemmnu cl decreases scale momentum Λ the through As freedom. asymptotic cin.A hr itne .. ag ulda mo- Euclidean large i.e., α distance; scales short inter- mentum At strong and hadrons actions. of theory [2]. successful markably ref. in given is field strength background field the gauge- of electromagnetic of a analysis the of An in terms expansion in invariant self-energy . electron heavy bound-state the the of field tromagnetic ursadgun ihnhdoshv iiu bound- minimum momenta, have state hadrons within gluons and for quarks wavefunctions mechanical quantum the Equivalently, gtri elcdb h resolvent the by prop- as electron replaced such bound-state is heavy, the very agator muonium, as where taken or case hydrogen be can in the In these particles. of bound-state one two the for tion ie h rm-needn odto teullight- equal AdS/QCD, at dimension [3]: condition fifth of time AdS front frame-independent model the the in hard-wall quarks gives the of confinement in color example, For s unu hooyais(C)hspoie re- a provided has (QCD) chromodynamics Quantum ( µ = ) v ecin.W lodsusimplications discuss also We reactions. ive nvriy tn ro,N 19 and 11794 NY Brook, Stony University, k N toso h aiu ur n gluon and quark maximum the of ations ag hoywt aibenumber variable a with theory gauge ) µ t ncoeaaoyt h calculation the to analogy close in cts e o reqak n losusing gluons and quarks free for ted g n niiain h ea fheavy of decay the annihilation, and eeetv unu chromodynamic quantum effective he s 2 em ihaie rmsotnoschiral spontaneous from arises hich = ( o steepcainvleo the of value expectation the is ion nae eiewti arn.The hadrons. within reside ensates µ ) p / y tnod A94309 CA Stanford, ty, ≃ µ v aiu aeeghof wavelength maximum a ave ζ (4 k λ µ − min = 0 e,teter xiissponta- exhibits theory the MeV, 200 π max h qae C ag coupling, gauge QCD squared the , eoe ml,a osqec of consequence a as small, becomes ) eA q Π b osiuns ncontrast in constituents, c = µ b · ≃ ⊥ 2 γ | and k x Λ − L,UK 3LE, | (1 QCD − min 1 m 1 − A e µ ≃ x + ≃ ) stebcgon elec- background the is Λ iǫ λ < e fQuantum of ies fm 1 QCD , max . . (2) (5) (3) (4) 2

~ 2 2 where b⊥ is the quark-antiquark impact separation and x vanish as q /kmin when this ratio is small, it follows that is the quark light-cone momentum fraction x = k+/P + = (to the extent that one can continue to use the quark and 0 3 0 3 (k +k )/(P +P ). Thus in principle all perturbative and gluon fields and the associated coupling αs to describe nonperturbative QCD analyses should be performed with the physics in this region of momenta), as q2 decreases 2 the infrared imposed by color confinement. in magnitude below ΛQCD, the QCD β function measur- Here we point out and discuss several important conse- ing the evolution of α at the scale µ2 ≃ q2 must also quences of color confinement and the resultant maximum vanish in the infrared. This implies a physical cutoff on 2 wavelength of quarks and gluons that do not seem to have the growth of αs(µ) at small µ ; i.e., infrared fixed-point received attention in the literature. These include impli- behavior of the QCD coupling. In effect, QCD exhibits cations for an infrared fixed point of the QCD β func- a well-defined limiting behavior in the infrared, and the tion and new insights into spontaneous chiral symmetry infrared growth of αs is suppressed, not because the per- breaking which are apparent when one uses bound-state turbative β function exhibits a zero away from the ori- rather than free Dyson-Schwinger equations (DSE’s) in gin, but because confinement provides an infrared cutoff. close analogy to QED computations. An In fact, as summarized in reference [14], effective QCD important consequence is that quark and gluon conden- couplings measured in experiments, such as the effective 2 sates are confined within hadrons, rather than existing charge αs,g1 (Q ) defined from the Bjorken sum rule, dis- throughout space-time. We also discuss mass cal- play a lack of Q2-dependence in the low Q2 domain. culations using Bethe-Salpeter equations (BSE’s), ana- lyze the modifications of instanton physics, and comment on insights that one gains concerning short-distance- dominated processes and dimensional counting for hard III. IMPLICATIONS FOR QCD exclusive processes. PHENOMENOLOGY Parenthetically, we note that if QCD contains Nf ≥ 2 exactly massless quarks and if one turned off electroweak The Bjorken scaling of the deep inelastic - interactions, then the theory would have a resultant set of scattering (DIS) ℓN → ℓ′X cross section is ex- 2 Nf − 1 exactly massless Nambu-Goldstone (e.g., plained in perturbative QCD by arguing that the emis- the , for the case Nf = 2). Because the size of a nu- sion of gluons from the scattered quark is governed by a cleon, rN , is determined by the emission and reabsorption coupling αs which is small because of asymptotic free- of virtual pions and the resultant cloud, and hence is dom. Thus in the leading-twist DIS regime, the DIS −1 2 rN ∼ mπ , this nucleon size would be much larger than structure functions Fi(x, Q ) are mainly functions of x, −1 with only small, logarithmic dependence on Q2. A sim- ΛQCD. Since in the real world, mπ is not << ΛQCD, we do not pursue the analysis of this gedanken world here. ilar argument underlies the application of perturbative QCD and the parton model to deep inelastic annihilation, e+e− → hadrons with center-of-mass energy squared 2 II. IMPLICATIONS OF λmax FOR THE s >> ΛQCD, away from thresholds, leading to the for- INFRARED BEHAVIOR OF QCD mula σ(e+e− → hadrons)/σ(e+e− → µ+µ−) = R with 2 R(s) = Nc j qj , where the sum is over quarks with The fact that quarks and gluons have maximum wave- 2 4mq < s. AsymptoticP freedom in QCD has also been lengths λmax has important consequences for the infrared 3 PC −− used to explain the narrow widths of S1, J = 1 behavior of QCD. For µ2 >> Λ2 , QCD is weakly QCD QQ¯ states of heavy (i.e., mQ >> ΛQCD) quarks with coupled, and the evolution of αs is described by the β masses below threshold for emission of the associated function heavy-flavor . The explanation is, in essence, that 2 hadronic decays proceed by emission of three gluons, and dαs αs b2αs 3 β(t)= = − b1 + + O(αs) , (6) because the corresponding couplings g (µ) are small for dt 2π  4π  s µ ∼ mQ/3 due to asymptotic freedom, the resultant de- where t = ln µ and the one- and two-loop coefficients cays are suppressed [4]. A fourth property of QCD which bℓ, ℓ = 1, 2 are scheme-independent, while higher-order makes use of its property of asymptotic freedom is the coefficients are scheme-dependent. phenomenon of jets [5]. In the standard perturbative calculations of these co- In each of these examples, the theoretical analyses pro- efficients, one performs integrations over Euclidean loop vide successful descriptions of the physical processes in momenta ranging from k = 0 to k = ∞. Although terms of the asymptotic freedom of QCD. The analy- this is a correct procedure for describing the evolution ses involve a factorization between the short-distance, of αs(µ) in the ultraviolet region µ >> ΛQCD where the perturbatively calculable, part of the process and the coupling is weak and effects of confinement are unimpor- long-distance part involving hadronization. However, it tant, it does not incorporate the property of confinement is important to ask whether such calculations are stable < at low scales µ ∼ ΛQCD. Confinement implies that both against multiple gluon emission. For example, if the out- the gluons and quarks have restricted values of momenta going struck quark in DIS or the qq¯ pair in DIA were to k ≥ kmin. Since loop corrections to the gluon radiate a sufficiently large number of gluons, then, since 3 on average each of these would carry only a small momen- a global SU(Nf )L × SU(Nf )R chiral symmetry, broken tum, the associated coupling αs would not be small, and spontaneously to the diagonal, vector isospin subgroup this could significantly change the prediction for the cross SU(Nf )diag, where Nf = 2, by the hqq¯ i condensates with section. A related concern regards the narrow width of q = u, d. (The analogous statement applies to the cor- orthoquarkonium: if the heavy Q and Q¯ annihilate not responding symmetry with Nf = 3, with larger explicit into three gluons, but into a considerably larger number, breaking via ms.) Some studies of spontaneous chiral ℓ, of gluons, then each would carry a much smaller mo- symmetry breaking in QCD are listed in ref. [10]. −1 mentum, k ∼ 2mQ/ℓ, and thus the associated running The inverse quark propagator has the form Sf (p) = 2 2 coupling gs(k) would not be small. Similarly, in a hard A(p )p/ − B(p ). In the one-gluon exchange approxima- −1 scattering process that leads to the production of a qq¯ tion, the DSE for Sf is 2 2 pair with invariant masss ˆ = (pq + pq¯) >> ΛQCD and 2 4 sˆ >> 4mq, the q andq ¯ might radiate a large number of −1 2 d k µ ν Sf (p) − p/ = −iCF g Dµν (p − k) γ Sf (k) γ gluons, each carrying a small relative momentum; again Z (2π)4 the resulting running coupling is consequently not small, (8) and this could dilute the jet-like structure of the event. where CF is the quadratic Casimir invariant and Dµν (k) A similar concern applies to jets involving gluons. is the gluon propagator. Chiral symmetry breaking is a Here we provide a simple physical explanation of why gauge-invariant phenomenon, so one may use any gauge gluon emission is stabilized: because of confinement, the in solving this equation. It is convenient to use the Lan- gluons have minimum momenta of order ΛQCD, so the dau gauge since then there is no wavefunction apparently dangerous scenario involving emission of a renormalization; i.e., A(p2) = 1. Equation (8) has a large number of gluons with momenta of order ΛQCD nonzero solution for the dynamically generated fermion or smaller is kinematically impossible. mass Σ (which can be taken to be Σ(p2)= B(p2) for Eu- An infrared cutoff on the growth of the QCD coupling 2 2 clidean p << Λ ) if αs ≥ αcr, where 3αcrCF /π = 1. also helps to explain the remarkable success of dimen- Since Σ is formally a source in the path integral for sional counting rules in describing data on the differen- the operatorqq ¯ , one associates Σ 6= 0 with a nonzero tial cross sections of exclusive reactions at high-energy quark condensate. Clearly, this only provides a rough and fixed-angle in QCD [6]. Recall that for a reaction estimate of α , in view of the strong-coupling nature of 2 2 2 cr a + b → c1 + ... + ck with s >> ΛQCD, t = Q >> ΛQCD, the physics and the consequent large higher-order per- and fixed s/t (i.e., fixed CM scattering angle), dimen- turbative, and also nonperturbative, contributions. sional counting gives [6] If one now takes quark and gluon confinement into account, then just as for the Lamb shift, the integral dσ − − ∝ s (n 2) , (7) over loop momenta in eq. (8) can extend in the infrared dt only to kmin ∼ ΛQCD. Although the DSE analysis for where the twist n denotes the total number of elemen- the free quark propagator may incorporate some of the tary valence fields entering the hard scattering ampli- physics relevant to spontaneous chiral symmetry break- tude. This implies, for example, that, under these con- ing, it does not incorporate the property of confinement. ditions, dσ(pp → pp)/dt ∝ s−10, dσ(πp → πp) ∝ s−8, This is an important omission since a plausible physical etc. One might worry that since all of the external parti- explanation for spontaneous chiral symmetry breaking cles in these reactions are on-shell, eq. (7) might receive in QCD involves confinement in a crucial manner; this large long-distance corrections. An appealing explana- breaking results from the reversal in helicity (chirality) tion for the absence of such corrections is that they are of a massless quark as it heads outward from the center suppressed by the cutoff in the growth of the running of a hadron and is reflected back at the outer boundary QCD coupling αs due to the kmin of the gluons. This is of the hadron [11]. also consistent with fits to data [7, 8, 9]. The Dyson-Schwinger equation has been used in con- junction with the Bethe-Salpeter equation for approxi- mate calculations of hadron masses and other quantities IV. IMPLICATIONS FOR SPONTANEOUS in QCD [12]. Our observation implies that here again, CHIRAL SYMMETRY BREAKING the integration over virtual loop momenta in the BSE can only extend down to kmin ∼ ΛQCD, not to k = 0. A limit on the maximum of gluon and quark wave- This obviates the need for artificial cutoffs on the growth lengths also has implications for spontaneous chiral sym- of the QCD coupling occurring in the integrand that have metry breaking (SχSB) in QCD. It is clear that because been employed in past studies. Analyses using the DSE of confinement, analyses based on free quark and gluon and BSE have been used to calculate hadron masses in propagators, such as the Dyson-Schwinger equation, need the confining phase of an abstract asymptotically free, to be replaced in principle by analyses which incorporate vectorial SU(N) gauge theories with a variable number, bound-state dynamics, such as the QCD Bethe-Salpeter Nf , of light fermions [13]. It would be worthwhile to in- equation. Recall that since the u and d current-quark corporate the effect of kmin in these studies, as well as in masses are << ΛQCD, the QCD Lagrangian theory has analyses for actual QCD. 4

Thus let us consider the propagator of a light quark they interact and bind to create the pion. For example, ¯ + ¯ + + bound in a light-heavy qQ , such as B = (ub) or at fixed light-front time xa = xb , the pion pole contribu- 0 ¯ 2 ⊥ ⊥ 2 2 Bd = (db). At sufficiently strong coupling αs, the DSE tion appears when −(xa −xb) = (xa −xb ) ≃ Rπ,where (in this case, effectively a bound-state Dyson-Schwinger Rπ is the transverse pion size, of order 1/mπ. The axial equation) yields a nonzero, dynamically generated mass, current that couples to the pion thus involvesqq ¯ creation Σ for the light quark. One can associate this with a bilin- within a domain of the size of the pion. Furthermore, ear quark condensate hqq¯ i, as noted above. However, this any qq¯ condensate that appears in the quark or anti- condensate is the expectation value of the operatorqq ¯ in quark propagator in this process is created in a finite the background (approximately Coulombic) field of the domain of the pion size, since that is where the quark heavy ¯b antiquark, in contrast to a true vacuum expec- and antiquark propagate, subject to the confining color tation value [15]. This is in accord with our argument in interaction. Again, we see that one needs to use a mod- [15] that the quark condensate hqq¯ i and gluon condensate ified form of the DSE for the quark propagator which µν hTr(Gµν G )i have spatial support only in the interior of takes into account the field of the antiquark. And again, hadrons, since that is where the quarks and gluons which the loop momenta have a maximum wavelength corre- give rise to it are confined. In [15] we noted that this con- sponding to the finite size of the domain where color can clusion is the analogue for quantum field theory of the ex- exist. perimental fact that the spontaneous magnetization be- Since the property that quarks and gluons have a maxi- low the Curie temperature in a piece of iron has spatial mum wavelength is a general consequence of confinement, support only within the iron rather than extending to it necessarily appears in specific phenomenological mod- spatial infinity. Our observation in Ref. [15] that QCD els of confined hadrons, such as bag models [18] (see also condensates have spatial support restricted to the interi- [19]) and recent approaches using AdS/CFT methods [9]. ors of hadrons has the important consequence that these It is also evident in lattice gauge simulations of QCD [20]. condensates contribute to the mean mass density Sometimes the results of these simulations are phrased as in the universe, but not to the cosmological constant or the dynamical generation of an effective “gluon mass”; dark energy [16]. As in the case of QED, the bound- here we prefer to describe the physics in terms of a max- state problem of a light quark in a background field can imum gluon wavelength, since this emphasizes the gauge be formulated as a resolvent problem or in terms of an invariance of the phenomenon. effective theory [17]. Of course, in the hypothetical situa- tion in which the current-quark masses mu = md = 0 and one turned off electroweak interactions, so that mπ → 0, V. IMPLICATIONS FOR INSTANTON the size of hadrons, as determined by their meson clouds, EFFECTS IN QCD would become infinitely large, and the condensates would have infinite spatial extent. The maximum wavelength of gluons also has implica- In general, the breaking of a continuous global symme- tions for the role of instantons in QCD. In the semiclassi- try gives rise to Nambu-Goldstone modes. It was noted in cal picture, after a Euclidean rotation, one identifies the Ref. [15] that in a sample of a ferromagnetic material be- gluon field configurations which give the dominant contri- low its Curie temperature, these Nambu-Goldstone spin bution to the path integral as those which minimize the wave modes () are experimentally measured to Euclidean action. This requires that the field strength reside within the sample. Moreover, via the correspon- a tensor Fµν ≡ a TaFµν vanish as |x| → ∞, which implies dence between the partition function defining a statisti- −1 a that Aµ = −(Pi/gs)(∂µU)U , where Aµ ≡ TaAµ and cal mechanical system and the path integral defining a a U(x) ∈ SU(Nc). Since the outer boundaryP of the com- quantum field theory, there is an analogy between the pactified R4 in this |x| → ∞ limit is S3, the gauge fields spin waves in a Heisenberg ferromagnet and the almost thus fall into topologically distinct classes, as described Nambu-Goldstone modes in QCD - the pions. As re- 3 by the class of continuous mappings from S to SU(Nc), quired for the self-consistency of our analysis of conden- i.e., the homotopy group π3(SU(Nc)) = Z. Instantons sates, the wavefunctions for the pions have spatial sup- appear to play an important role in QCD, explaining, for port where the chiral condensates exist, namely within example, the breaking of the global U(1)A symmetry and the pions themselves and, via virtual emission and reab- the resultant fact that the η′ meson is not light [21, 22]; sorption, within other hadrons, in particular . however, one should recognize that, because of confine- We relate these statements to some current algebra re- ment, the gauge fields actually have no support beyond sults. Consider the vacuum-to-vacuum correlator of the length scales of order 1/ΛQCD ∼ 1 fm. The semiclassical µ µ axial-vector current, h0|J5 (xa)J5 (xb)|0i. The Fourier analysis with its |x| → ∞ limit used to derive the result −1 transform of the cut of this propagator can, in prin- Aµ = −(i/gs)(∂µU)U is thus subject to significant cor- + − ∗ ciple, be measured in e e → Z0 axial-vector current rections due to confinement. This is evident in the BPST events. The pion appears as a pole in this propagator at instanton solution for Nc = 2, namely [23] 2 2 + − ∗ 0 q = mπ, corresponding to e e → Z0 → π . Here the µ µ 5 2 axial-vector current J5 =qγ ¯ γ q creates a quark pair at x −1 igsAµ = (∂µU)U , (9) xa which propagates to xb, and as the q andq ¯ propagate, x2 + ρ2  5 where U(x) = (x0 + iτ · x)/|x|. This form only becomes Neither the β function nor the DSE used in this βDS a pure gauge in the limit |x| → ∞. It has long been rec- method includes the effect of instantons. Studies in QCD ognized that in calculating effects of instantons in QCD, have shown that instantons enhance spontaneous chiral which involve integrations over instanton scale size ρ, un- symmetry breaking [29]. Analyses of instanton effects certainties arise due to the fact that there are big con- on fermion propagation were carried out for general Nf , tributions from instantons with large scale sizes, where and these were shown to contribute substantially to SχSB the semiclassical approximation is not accurate [22]. Our [30]. One would thus expect that if one augmented the point is different, although related; namely that the ac- βDS approach to include effects of instantons, the resul- curacy of the semiclassical instanton analysis is also re- tant improved estimate of Nf,cr would be greater than stricted by the property that λmax ∼ 1 fm for the gluon the value obtained from the βDS method without in- field and hence one cannot really take the |x| → ∞ limit stantons. in the manner discussed above. In principle, lattice can provide a fully nonperturbative approach for calculating Nf,cr [31]-[32]. One lattice group has obtained values of Nf,cr con- VI. IMPLICATIONS FOR GENERAL siderably smaller than the respective βDS values (for NON-ABELIAN GAUGE THEORIES N = 2, 3) [31]; however, the most recent study of the N = 3 case finds evidence that the infrared behavior of Our observations are also relevant to the problem of de- the theory is conformal for Nf ≥ 12 but exhibits confine- termining the phase structure (at zero temperature and ment and chiral symmetry breaking for Nf ≤ 8, consis- chemical potential) of a vectorial SU(N) gauge theory tent with the βDS analysis [32]. Since the βDS method with a gauge coupling g and a given content of massless does not include instanton effects, there is thus a ques- fermions, such as Nf fermions transforming according to tion why it appears to produce a rather accurate value the fundamental representation of SU(N). We assume of Nf,cr. Nf < (11/2)N, so that the theory is asymptotically free. Our observation provides a plausible answer to this Since fermions screen the gauge field, one expects that question. Approaching the chiral boundary from within for sufficiently large Nf , the gauge interaction would the phase with confinement and spontaneous chiral sym- be too weak to confine or produce spontaneous chiral metry breaking, we note that the confinement-induced breaking (e.g., [24]). An estimate of the critical value, kmin of the gluons reduces their contribution to the in- Nf,cr, beyond which there would be a phase transition crease of the gauge coupling in the infrared and also to from a phase with confinement and SχSB to one with- the virtual gluon exchange effects on the fermion propa- out such symmetry breaking (and presumably without gator. This reduction of gluonic effects acts in the oppo- confinement) has been obtained combining the pertur- site direction relative to the enhancement of chiral sym- bative β function and the DSE. We denote this as the metry breaking due to instantons, and thus has the po- βDS method. For sufficiently large Nf , the perturbative tential to explain why the βDS estimate for Nf,cr, which β function exhibits a zero away from the origin, at αIR, does not incorporate either confinement or instanton ef- 2 where α = g /(4π). The value of αIR is a decreasing fects, could nevertheless yield a reasonably accurate value function of Nf . The value of Nf,cr is determined by the for Nf,cr. Moreover, as noted above, the role of instan- condition that αIR decreases below the minimal value αcr tons in SχSB is affected by the confinement of gluons and for which the approximate solution to the DS equation the resultant corrections to the semiclassical approach to yields a nonzero solution for the dynamically generated QCD. fermion mass. The βDS analysis, to two-loop accuracy, yields the estimate [25] VII. CONCLUSIONS 2N(50N 2 − 33) N = (N ) = , (10) f,cr f,cr 2ℓβDS 5(5N 2 − 3) Because quarks and gluons are confined within where 2ℓ refers to the two-loop accuracy to which the hadrons, there is maximum limit on their wavelengths. beta function is calculated. This gives Nf,cr ≃ 8 for Propagators, normally calculated for free quark and glu- N = 2 and Nf,cr ≃ 12 for N = 3. The value of Nf,cr is ons using Dyson-Schwinger equations, are modified in the important because if the approximate IR fixed point αIR infrared by bound state effects in close analogy to the cal- is larger than, but near to, αcr, the resultant gauge cou- culation of the Lamb shift in atomic physics. Thus be- pling runs slowly over an extended interval of . cause of confinement, the effective QCD coupling stays This “walking” behavior is useful for models of dynam- finite and flat at low momenta. The quark condensate ical electroweak symmetry breaking [25, 26]. (In such which arises from spontaneous chiral symmetry breaking models there are motivations for choosing N = 2, includ- in the bound-state Dyson-Schwinger equation is the ex- ing a mechanism to explain light masses [27].) pectation value of the operatorqq ¯ evaluated in the back- The effect of the three-loop terms in the β function and ground of the fields of the other hadronic constituents, the next higher-loop terms in the DSE have been studied in contrast to a true . Thus in Ref. [28]. quark and gluon condensates have support only within 6 hadrons. of confinement or instantons. Our observations suggest a We have shown that the limit on the maximum wave- program of future research devoted to incorporating the length of gluons and quarks from confinement leads to effect of the maximum wavelength of quarks and gluons new insights into a number of phenomena in QCD, in- in analytic studies of QCD properties. cluding deep inelastic scattering and annihilation, the narrow widths of heavy orthoquarkonium states, jets, in- stantons, dimensional counting rules for hard exclusive processes, and other phenomena related to the infrared VIII. ACKNOWLEDGMENTS behavior of the theory. We have also given a plausible explanation of how estimates of the value Nf,cr in a gen- We are grateful to Guy de Teramond for helpful dis- eral asymptotically free, vectorial SU(N) theory based cussions and comments. This research was partially on a method using the perturbative β function and the supported by grants DE-AC02-76SF00515 (SJB) and Dyson-Schwinger equation could be reasonably accurate NSF-PHY-06-53342 (RS). Preprint SLAC-PUB-13246, even though this method does not incorporate the effects IPPP/08/37, DCPT/08/74, YITP-SB-08-11.

[1] H. A. Bethe, Phys. Rev. 72 (1947) 339; see also N. M. [14] A. Deur, V. Burkert, J. P. Chen and W. Korsch, Kroll and W. E. Lamb, Phys. Rev. 75 (1949) 388; J. B. arXiv:0803.4119 [hep-ph]. French and V. F. Weisskopf, Phys. Rev. 75 (1949); H. A. [15] S. J. Brodsky and R. Shrock, arXiv:0803.2541. Bethe and E. E. Salpeter, of One- [16] S. J. Brodsky and R. Shrock, arXiv:0803.2554. and Two-Electron (Academic Press, New York, [17] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Rev. Mod. 1957). Phys. 77 (2005) 1423. [2] G. W. Erickson and D. R. Yennie, Ann. Phys. (N.Y.) 35 [18] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. (1965) 271, 447; S. J. Brodsky and G. W. Erickson, Phys. F. Weisskopf, Phys. Rev. D 9 (1974) 3471; T. Degrand, Rev. 148 (1966) 26. R. L. Jafffe, K. Johnson, and J. Kiskis, Phys. Rev. D 12 [3] S. J. Brodsky and G. F. de Teramond, arXiv:0804.3562 (1975) 2060. [hep-ph]. [19] J. M. Cornwall, Phys. Rev. D 26 (1982) 1453; L. von [4] T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34 Smekal, A. Hauck, and R. Allkofer, Phys. Rev. Lett. 79 (1975) 43; Phys. Rev. D 12 (1975) 1404. (1997) 3591; A. C. Aguilar, A. Mihara, and A. A. Natale, [5] A review is G. Sterman, in Proceedings of the Phys. Rev. D 65 (2002) 054011; A. C. Aguilar, A. A. Theoretical Advanced Study Institute (TASI 2004), Natale, and P. S. da Silva, Phys. Rev. Lett. 90 (2003) arXiv:hep-ph/0412013. 152001. [6] S. J. Brodsky and G. Farrar, Phys. Rev. Lett. 31 (1973) [20] J. E. Mandula and M. Ogilvie, Phys. Lett. 185 (1987) 1153; V. A. Matveev, R. M. Muradian, and A. N. Tavkhe- 127; D. Zwanziger, Phys. Lett. B 257 (1991) 168; C. lidze, Lett. Nuovo Cim. 7 (1973) 719. Bernard, C. Parrinello, and A. Soni, Phys. Rev. D 49 [7] S. J. Brodsky, E. Gardi, G. Grunberg, and J. Rathsman, (1994) 1585. Phys. Rev. D 63 (2001) 094017. [21] G. ’t Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D [8] J. H. , Phys. Rev. D 66 (2002) 013013. 14 (1976) 3432. [9] S. J. Brodsky and G. F. de Teramond, arXiv:0802.0514. [22] C. G. Callan, R. Dashen, and D. J. Gross, Phys. Rev. D [10] Early papers include K. Lane, Phys. Rev. D 10 (1974) 16 (1977) 2526. 2605; H. D. Politzer, Nucl. Phys. B 117 (1976) 397; T. [23] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. Kugo, Phys. Lett. B 76 (1978) 625; H. Pagels, Phys. S. Tyupkin, Phys. Lett. B 59 (1975) 85. Rev. D 19 (1979) 3080; K. Higashijima,, Phys. Rev. D [24] T. Banks and A. Zaks, Nucl. Phys. B 196 (1982) 189. 29 (1984) 1228. [25] T. Appelquist, J. Terning, and L. C. R. Wijewardhana, [11] A. Casher, Phys. Lett. B 83 (1979) 395. Phys. Rev. Lett. 77 (1996) 1214; T. Appelquist, A. Rat- [12] K.-I. Aoki, M. Bando, T. Kugo, and M. Mitchard, Prog. naweera, J. Terning, and L. C. R. Wijewardhana, Phys. Theor. Phys. 85 (1991) 355; K.-I. Aoki, T. Kugo, and M. Rev. D 58 (1998) 105017. Mitchard, Phys. Lett. B 266 (1991) 467; V. Miransky, [26] T. Appelquist, D. Karabali, and L. C. R. Wijewardhana, Dynamical Symmetry Breaking in Quantum Field Theo- Phys. Rev. Lett. 57 (1986) 957; T. Appelquist and L. C. ries (World Scientific, Singapore, 1993); P. Jain and H. R. Wijewardhana, Phys. Rev. D 36 (1987) 568; see also Munczek, Phys. Rev. D 48 (1993) 5403; C. J. Burden et B. Holdom, Phys. Lett. B150 (1985) 301. al., Phys. Rev. C 55 (1997) 2649; R. Alkofer and L. von [27] T. Appelquist and R. Shrock, Phys. Lett. B 548 (2002) Smekal, Phys. Rept. 353 (2001) 281; P. Maris and C. D. 204; Phys. Rev. Lett. 90 (2003) 201801; T. Appelquist, Roberts, Int. J. Mod. Phys. E 12 (2003) 297. M. Piai, and R. Shrock, Phys. Rev. D 69 (2004) 015002. [13] M. Harada and Y. Yoshida, Phys. Rev. D 50 (1994) 6902; [28] T. Appelquist, K. Lane, and U. Mahanta, Phys. Rev. T. Appelquist and F. Sannino, Phys. Rev. D 59 (1999) Lett. 61 (1998) 1553. 067702; M. Harada, M. Kurachi, and K. Yamawaki, Phys. [29] D. Caldi, Phys. Rev. Lett. 39 (1997) 121; C. Callan, R. Rev. D 68 (2003) 076001; Prog. Theor. Phys. 115, 765 Dashen, and D. Gross, Phys. Rev. D 17 (1978) 2717. (2006); M. Kurachi and R. Shrock, JHEP 12 (2006) 034; [30] T. Appelquist and S. Selipsky, Phys. Lett. B 400 (1997) Phys. Rev. D 74 (2006) 056003. 364; M. Velkovsky and E. V. Shuryak, Phys. Lett. B 437 7

(1998) 398. [32] T. Appelquist, G. Fleming, and E. Neil, Phys. Rev. Lett. [31] Y. Iwasaki et al., Phys. Rev. Lett. 69 (1992) 21; Phys. 100 (2008) 171607. Rev. D 69 (2004) 014507.