Fractal Propagators in QED and QCD and Implications for the Problem of Conﬁnement

286 Brazilian Journal of Physics, vol.37, no. 1B, March, 2007

S. Gulzari, Y. N. Srivastava, J. Swain, and A. Widom Dept. of Physics, Northeastern University, Boston, MA 02115, USA (Received on 15 September, 2006) We show that QED radiative corrections change the propagator of a charged Dirac particle so that it acquires a fractional anomalous exponent connected with the fine structure constant. The result is a nonlocal object which represents a particle with a roughened trajectory whose fractal dimension can be calculated. This represents a significant shift from the traditional Wigner notions of asymptotic states with sharp well-defined masses. Non- abelian long-range fields are more difficult to handle, but we are able to calculate the effects due to Newtonian gravitational corrections. We suggest a new approach to confinement in QCD based on a particle trajectory acquiring a fractal dimension which goes to zero in the infrared as a consequence of self-interaction, representing a particle which, in the infrared limit, cannot propagate. Keywords: QED; Theory of quantized fields; QCD; Confinement; Nonperturbative methods; Fractals

I. INTRODUCTION In the literature there has not been full agreement about what constitutes the correct function γ(α), nor has there been In [1], a gauge-invariant calculation was presented showing a consensus as to whether or not it can be set to zero by a that the propagator for a charged particle acquires an interest- suitable choice of gauge. ing fractal structure from its self-interaction in quantum elec- Appelquist and Carazzone[12] argued that γ = −(α/π) + trodynamics. In simple terms, the usual Dirac propagator S(k) ... to leading order, in conflict with earlier work based on is replaced by the nonlocal expression summing logarithms[15]. Reference [12] does not actually derive the exponent, rather citing the earlier work, so there is a possibility that a typographical error may be involved. The fourth volume of the Landau and Lifschitz course of S(k) = (κ − k/)D(k), theoretical physics[10] starts off in agreement with γ 6= 0 but   ¡ ¢ ultimately sets γ = 0 by giving a small mass to the photon. ³ κ ´α/π  Γ 1 + (α/π)  D(k) = ¡ ¢ , (1) Such a photon mass explicitly breaks both gauge invariance iΛ £ ¤ 1+(α/π)  and conformal/scale invariance and must be rejected on phys- k2 + κ2 − i0+ ical grounds. Intuitively, what goes wrong is that the fractal where ~κ = mc, Λ is a short distance length scale, the frac- structure of the propagator for a charged particle must be due tional exponent γ is a function of the coupling strength α = to the fact that ever increasing wavelengths one can produce (e2/~c) which we find to be α/π and the usual Gamma func- more and more soft photons without limit. A photon mass tion is implies a maximum photon wavelength and a loss of scale in- Z variance. ∞ ds Γ(z) = e−ssz with ℜe(z) > 0. (2) There have been other approaches in the literature, often 0 s tackling only the case of charged scalar fields, or again ar- guing that the anomalous dimension can somehow be set to This represents a rather radical departure from the usual zero. For example, a path integral[11] approach using the textbook discussions of particle propagators (with the notable Schwinger[8] proper time representation of the propagator exception of [2]) and from the usual Wigner classification of and some work by Bloch and Nordsieck[16] on soft photon elementary particles in terms of a sharp mass and spin (see, for emission gives the same sort of result as ours, but with the example, [3]). This is despite the fact that it is well-known that final answer given only for charged scalar fields. These re- particles coupled to long-range massless fields cannot have sults are argued to be gauge invariant in such a way that the sharp masses[4–6]. A good review of the basic issues can be singularity structure can be returned to a simple pole by a found in [7]. choice of gauge. As argued above, this is unphysical as there Aside from having been neglected in textbooks, the actual is a real meaning to a fractional exponent and the failure of exponent in the above expression has been the subject of some charged particles to have sharp masses. Fried also discusses controversy. There are calculations in the literature based on this problem[17] as do Johnson and Zumino[18], and Stefa- either infinite sums of logarithmic Feynman diagrams[15] or nis and collaborators[19]. Batalin, Fradkin and Schvartsman non-perturbative Schwinger[8, 9, 11] computations which ar- have made a similar gauge dependent calculation for scalar gue that such an electron propagator should be of the form we particles[20]. obtained: ( ) In addition, there are possible experimental consequences ³ κ ´γ κ − k/ of the radical change in the nature of the singularity in the S(k) = Γ(1 + γ) £ ¤ , (3) propagator. Handel has argued[21] that the change from a iΛ 2 2 + (1+γ) k + κ − i0 pole into a branch point has measurable physical implications . for “1/ω” noise in the Schrodinger¨ (non-relativistic) limit. S. Gulzari et al. 287

The result is clearly not analytic in α and requires a non- As we are interested here in the infrared limit, spin-flips perturbative approach. Such non-analyticity in α for an all- are suppressed and we can neglect the corresponding term in orders or non-perturbative calculation could be anticipated on the Hamiltonian. This corresponds to usual Bloch-Nordsieck physical grounds from the arguments of Dyson against the replacement of cγµ by four velocity vµ – one simply thinks convergence of perturbative expansions in QED[13]. of the Dirac spinor having only one nonvanishing component Given the apparent confusion in the literature, we present a representing an electron of given spin state and allows no cou- simple derivation which preserves gauge invariance through- pling to the other spin state (and, of course, no coupling to out. With this in hand, we then discuss the physical inter- positrons). pretation of the result, extend it to include Newtonian grav- With this approximation, Eq. (7) which gives the propaga- ity,discuss the fractal dimension one can associate with paths tion amplitude for an electron can be written in the Lagrangian involved, and finally introduce a new way to think about con- path integral formulation finement in these terms. Z X(τ)=x G(x,y,τ;A) ≈ eiS[X;A]/~ ∏dX(σ), X(0)=y σ II. DERIVATION OF THE DRESSED ELECTRON Z PROPAGATOR τ S[X;A] = L(X˙ (σ),X(σ);A)dσ. (10) 0 For completeness, we rederive the expression for the dressed electron propagator presented in [1]. This expression deserves some comments. The paths X(σ) For an electron in an external electromagnetic field, Fµν = which are integrated over represent virtual histories for the ∂µAν − ∂νAµ, the Dirac propagator electron. The proper time τ for each path in the sum is not given by the classical expression c2τ2 6= −(x − y)2, so in inte- (−id/ + κ)G(x,y;A) = δ(x − y) , µ ¶ grating over paths one is integrating here over all proper times. eAµ dµ = ∂µ − i , (4) For each one of these paths, with its own proper time, the ~c interaction between the electron and an elecromagnetic vector may be solved employing the function ∆(x,y;A); potential Aµ is described by the action Z 4 Z τ ∆(x,y;A) = γ5G(x,z;A)γ5G(z,y;A)d z , Sint(P;A) = Lint(X˙ (σ),X(σ);A)dσ, 0 G(x,y;A) = (id/ + κ)∆(x,y;A) , Z τ ¡ ¢ e ˙ µ d/ 2 + κ2 ∆(x,y;A) = δ(x − y) , Sint(P;A) = Aµ(X(σ))X (σ)dσ, c 0 e Z d/ 2 = −dµd − σµνF . (5) e µ µ µν Sint(P;A) = Aµ(X)dX , (11) 2~c c P The Hamiltonian of the electron can be written where the integral is along the worldline P. Htot = H + Hspin , ½ ¾ For the case of no external field, we want the action due to 1 ³ e ´2 H = p − A + m2c2 , self-interaction (interaction with vacuum fluctuations), so one 2m c µ ¶ wants to apply the rule e~ H = − σµνF (6) spin 4mc µν ˆ eiSint(P;A)/~ → h0|eiSint(P;A)/~ |0i , and with pµ = −i~∂µ, one may define the amplitude for the + electron to go from y to x in a proper time τ as the matrix eiSint(P;A)/~ → eiSsel f (P)/~, element Z Z ~α µ ν −iH τ/~ Ssel f (P) = Dµν(x1 − x2)dx1dx2. (12) G(x,y,τ;A) = hx|e tot |yi. (7) 2 P P

From Eqs.(5), (6) and (7) follows the electron propagator ex- In the above Eq.(12), the subscript “+” denotes time order- pression ing, Aˆ (x) denotes the operator vector potential ﬁeld and the Z µ i~ ∞ photon propagator is given by ∆(x,y;A) = G(x,y,τ;A)dτ, 2m 0 ³ e ´ i ~G(x,y;A) = mc − p/ + A/ (x) ∆(x,y;A) . (8) Dµν(x1 − x2) = h0|Aˆµ(x1)Aˆν(x2)|0i . (13) c ~c + One also has a Lagrangian The action form in Eq.(12) is of a well known form[22]. As 1 ¡ ¢ e (v,x;A) = m vµv − c2 + vµA (x), noted earlier, we have bypassed the usual Bloch-Nordsieck L µ µ µ µ 2 µ ¶ c µ ¶ replacement of cγ by four velocity v by simply evaluating a d ∂L ∂L phase in the soft photon infrared limit that we are considering = ; (9) dτ ∂vµ ∂xµ here. 288 Brazilian Journal of Physics, vol.37, no. 1B, March, 2007

The propagator may be written Thus, Eq.(17) is renormalized to

µ ¶ Z X(τ)=x ∂µ∂ν ˜ iS˜ [X;A]/~ Dµν(x − y) = ηµν − (1 − ξ) D(x − y), G(x − y,τ) = e ∏dX(σ), ∂2 X(0)=y σ Z Z τ 4π d4k ˜ ˙ D(x − y) = eik·(x−y) , S[X] = Lm(X(σ))dσ + iW[X] k2 − i0+ (2π)4 0 ½ ¾ 1 i 1 L (X˙ ) = m(X˙ µX˙ − c2) D(x − y) = . (14) m 2 µ π (x − y)2 + i0+ W[X;τ] = ℑmSsel f [X], Z τ Z τ µ where the parameter ξ fixes a gauge. ~α X˙ (σ1)X˙µ(σ2)dσ1dσ2 W[X;τ] = 2 . (21) Because the worldline of the electron never begins nor ends 2π 0 0 (X(σ1) − X(σ2)) the partial derivative terms in Eq.(14) do not contribute to µ µ µ 2 the self action in Eq.(12). This requirement that the world- For a straight-line path X (σ) = V σ with V Vµ = −c , one line have no endpoints is required by charge conservation (or, finds Z τ Z τ equivalently, by gauge invariance). ~α dσ1dσ2 Independent of any choice of the gauge parameter ξ then, W(τ) = 2 . (22) 2π 0 0 (σ − σ ) we have 1 2 Z Z This expression as it stands is infinite and requires regulariza- ~α µ tion. Using differential regularization[23] we have Ssel f (P) = D(x1 − x2)dx1µdx2 . (15) 2 P P d2W(τ) ~α = . (23) In the absence of any external field (other than that due to dτ2 πτ2 vacuum fluctuations) we have now derived expressions for the renormalized vacuum electron propagator The solution to the differential equation Eq.(23) with a logarithmic cut-off Λ is Z 4 µ ¶ ³ ´ ik·(x−y) d k ~α cτ G˜(x − y) = S(k)e , W(τ) = − ln . (24) (2π)4 π 2Λ ˜ ˆ G(x − y) = h0|G(x,y;A)|0i+ , From Eqs.(21) and (24), one finds G˜(x − y) = (i∂/ + κ)∆˜ (x − y) , Z ˜ −W(τ)/~ ˜ i~ ∞ G(x − y,τ) ≈ e Gm(x − y,τ) (25) ∆˜ (x − y) = G˜ (x − y,τ)dτ. (16) 2m 0 wherein G˜ m(x − y,τ) is the proper time Green’s function for a particle of mass m with the corresponding free Lagrangian The functional integral expression for G˜ (x − y,τ) is given by Lm(X˙ ). To exponentially lowest order in α, one then has

Z X(τ)=x Z n o 4 ˜ 2 2 d k ˜ (x − y,τ) = eiS[X;A]/~ dX(σ), ˜ −i~(k +κ )τ/2m ik·(x−y) G ∏ Gm(x − y,τ) = e e 4 , X(0)=y σ (2π) Z τ ³ cτ ´α/π S˜ [X] = L0(X˙ (σ))dσ + Ssel f [X], (17) G˜ (x − y,τ) = G˜ m(x − y,τ). (26) 0 2Λ wherein the free electron Lagrangian is Eqs.(16) and (26) then imply Z 4 ik·(x−y) d k 1 µ 2 ∆˜ (x − y) = D(k)e , L0(X˙ ) = m0(X˙ X˙µ − c ), (18) (2π)4 2 Z 4 ik·(x−y) d k and the self action is given by Eqs.(14) and (15) as G˜(x − y) = S(k)e . (27) (2π)4 Z τ Z τ µ i~α X˙ (σ1)X˙µ(σ2)dσ1dσ2 and we obtain Eq. (1). Ssel f [X] = 2 + . (19) 2π 0 0 (X(σ1) − X(σ2)) + i0 We chose differential regularization as simple and conve- nient, and preserving gauge invariance, but other regulariza- The divergent piece of the self action tions can also be used as long as they are also gauge invariant. We plan to return to this issue in a later publication in more Z τ ∆m µ 2 detail. In a sense, the form of the answer is intuitively clear ℜeSsel f [X] = (X˙ (σ)X˙µ(σ) − c )dσ, 2 0 since, on dimensional grounds, the expression to be regular- |∆m| = ∞. (20) ized can only be logarithmic. The sign, as we shall see below, has a physically sensible interpretation, and is in agreement The formally infinite self-mass can be absorbed into a rede- with reference [2] and with reference [12] if one assumes that finition of the finite physical mass 0 < m = (m0 + ∆m) < ∞. there was a typographical error going from reference [15]. S. Gulzari et al. 289

III. A PHYSICAL INTERPRETATION one simply goes to α(q2 = 0) which is the Thompson value and there are no additional complications. In the ultraviolet It is interesting to consider what the physical interpreta- limit, one expects a continuously changing exponent and thus tion of the non-integer exponent in the radiatively corrected a multifractal as opposed to fractal structure. In addition, the Dirac propagator means. First of all, the fact that the exponent lack of asymptotic freedom leads one to expect trouble at very is non-integer means that the renormalized Dirac operator is short distance scales unless, as noted above, other fields enter non-local[24]. This was, of course, to be expected since the significantly. electromagnetic field has infinite range. Non-locality has been previously introduced ad-hoc[25, 26] as a regularization tool. Here it appears naturally, suggesting IV. NEWTONIAN QUANTUM GRAVITY that some form of regularization of otherwise formally divergent expressions may be implicit in at least some quantum For quantum gravity one can do the analysis in much the field theories with massless fields, but appearing only when same way as for quantum electrodynamics. The ultraviolet one goes beyond perturbation theory. We will also see later divergences need not worry us since we are dealing with a that there is evidence that the corrections to the exponent can strictly infrared problem. There should be graviton-graviton be changed when other fields are added, and need not always interactions, but if we neglect these as small compared to be of the same sign as in pure QED. The appearance of this graviton-electron interactions we can just repeat what was sort of non-locality also makes the analytic but rarely used done for electrostatics but now with the Newtonian limit of regularization proposed by Speer[27] more physically moti- gravity. vated. This approximation would not be as reasonable in the QCD So far we have only been able to treat the long-wavelength case for a quark propagator since gluon-gluon couplings are approximation since we expect a quenched approximation in comparable to gluon-quark couplings, but there is evidence which we can ignore electron loops to be a reasonable one. from other calculations of the appearance of anomalous di- The physical picture would be one where, as one backs away mensions for infrared propagators. (For a review see [32]. from the worldine of an electron, one continues to see photons Very recent calculations can also be found in references [33].) radiated and absorbed, but now of longer and longer wave- Perhaps the simplest way to think of this is to look at Eq. length. This would suggest a fractal[28] structure, which is (15) and regard the (singular) double integral as the limit of made precise by the above derivation. Such notions of scaling two paths which must be taken as approaching each other arbi- and fractality are not new in QED and in quantum field theory trarily closely. This is an alternative to the differential regular- in general, but are often considered in the high energy, ultravi- ization we used in Eq.(23) and physically corresponds to two olet limit[29, 30]. This continues without limit at longer and electron worldlines interacting via the exchange of an arbitary longer distance scales since there is no minimum energy pho- number of photons between them and at all points of their ton (photons are massless) and this self-similarity reflects the paths. In the limit of the paths coinciding, this becomes self- scale invariance of Maxwell’s equations. interaction, with the electron continuously exchanging pho- If the photon is given a mass, however small, this structure tons with itself along its worldline. will break down asymptotically, since now there is a minimum For gravity (at least in the Newtonian limit) we want to re- place the repulsive electrostatic self-interaction, say +(e2/r), energy required to create a virtual photon, and at distances 2 greater than the corresponding Compton wavelength one will with the attractive gravitational self-interaction, say −Gm /r, get the non-interacting Dirac propagator. This was done by suggesting an asymptotic form of the Dirac propagator expo- Lifshitz et al.[10], and this argument makes clear how break- nent ing gauge invariance, i.e. including a photon mass, removes 1 γ ≈ (e2 − Gm2) + .... (28) the anomalous scaling behavior here derived for gauge invari- π~c ant QED. √ The fact that a particle is non-localizable, at least in part If m = |e/ G|, which is the ADM[34, 35] mass of charged due to its electromagnetic field which extends over all space, shell of charge e, regularized by its own gravity, then one is interesting. The feeling of this calculation is such that recovers an effectively free propagator. Since one generally at least part of what one thinks of as quantum-mechanical makes measurements using the electromagnetic interaction, about an otherwise point-particle (its lack of localizability) and the suggestion that quantum mechanics might be linked may arise from the non-perturbative quantum mechanics of to self-interaction[31], it is interesting to consider what this its self-interaction[31]. might imply for the role of gravity in the quantum measure- At shorter distance scales one might again expect an anom- ment problem[36]. In particular, since one has a connection alous dimension, but now the calculations are more compli- between mass and charge which is non-perturbative in New- cated since one must imagine more and more electron (and ton’s G, and implies a mass near the Planck mass ∼ 10−5 gm, other charged particle) loops contributing, with an ever “froth- which may be thought to be in the neighborhood of a putative ier” structure at smaller and smaller distance scales. In fact, classical-quantum boundary. one would not even expect a constant exponent which is inde- Intuitively, electromagnetic interactions roughen the path pendent of momentum since one expects a running α with val- of an electron, making it “spread out more”, which is natural ues which change with momentum scale. In the infrared limit since the electromagnetic force between two identical charges 290 Brazilian Journal of Physics, vol.37, no. 1B, March, 2007

(or an electron and itself) is repulsive. Gravitational inter- nonlinear interactions, and (by hypothesis), are not even ob- actions, on the other hand, make it “spread out less”, since servable asymptotically. the gravitational force between two identical objects (here an Perturbative treatments make QCD look very much like electron and itself) is attractive. QED, and involve propagators which look pretty much like those of QED with the main difference being in the nonabelian nature of the gauge fields and their associated self-couplings. V. FRACTAL PATHS The question which we now raise is whether or not one can imagine a situation whereby anomalous exponents in the in- The notion of a path somehow “spreading out” can be made frared will give rise to such strong qualitative changes in how precise with the notion of fractal dimensions. propagators behave that the originally postulated degrees of The fractal nature of particle paths has been discussed in the freedom will simply not propagate (at least at low energies) – literature (see, for example [37, 38]). Abbott and Wise[39], that is, that some form of confinement appears. working within nonrelativistic quantum mechanics, found 2 What could confinement mean in terms of an anomalous as the dimension of a quantum mechanical path, as opposed dimension? If in Eq. 29 we had a γ which drove d to zero, one to 1 for a classical path. Cannata and Ferrari[40] extended this would have a zero-dimensional path, which is no path at all! work for spin-1/2 particles and find different results not only This would correspond to a confined particle. in the classical and quantum mechanical limits, but also in the Let us see how far we can argue that such a phenomenon non-relativistic and relativistic limits. would occur. One could start from Eq.(15) with an appropri- Intuitively one can understand the dimension 2 result for ate form of D(x1 − x2) for QCD valid for any separation. Un- the nonrelativistic quantum mechanical case by thinking of fortunately, we do not have an exact expression for D(x1 −x2) the Schrodinger¨ equation as a diffusion equation in imaginary in the infrared limit and in fact every expectation is that (even time[41]. For diffusion one has the distance r a particle covers aside from colour indices) it is very different from the QED in time t satisfying a relationship of the form t ∝ rd wherein one – presumably having a confining term, roughly linear in d is the fractal dimension of the “path”. For example, t ∝ r2 separation. Of course this would be assuming confinement. in the diffusion limit, and t ∝ r in the ballistic (simple path) A simpler approach is to imagine that d = 2(1 + γ) where limit[42]. γ ∝ αs and γ < 0. If the strong coupling constant αs is large Here we have a closely analogous situation but with a enough then one could get d → 0. Let us follow this line of 4-dimensional Hamiltonian H and with fractal diffusion in argument more closely. The first thing to check is that the proper time, and as was showed in [1], one has energy of self-interaction of a quark with itself is indeed of the same sign as that due to gravity (and opposite to that due to electromagnetism). While the interaction energy between 2α d = 2(1 + γ) ≈ 2 + + ..., (29) a quark and an antiquark in the singlet state separated by a π 4 αs(r) distance r is given perturbatively by 3 r , the correspond- 2 αs(r) All previous discussions of fractal propagators in quantum ing quark-quark expression in the triplet is − 3 r . In other field theory and quantum mechanics have, to the best of our words, it is of the correct sign to lead to confinement. knowledge, ignored the effects of self-interaction via long- Now let us revisit Eq.(15) and regard the (singular) double range fields. integral as the limit of two paths which must be taken as ap- The fact that self-interaction can qualitatively change the proaching each other arbitrarily closely as we did in section nature of a propagating particle even when coupled to a weak IV. We are now looking at two paths of a red, say, quark, (small coupling constant) Abelian gauge field naturally leads exchanging an arbitrary number of gluons. If we now argue one to wonder whether or not coupling to a strong (large cou- 2 αs(r) as we did for gravity, we have d = 2(1 − 3 π ). It is well- pling constant) non-Abelian gauge field might have significant known[43, 44] (and can be shown on very general grounds physical effects. We turn now to a discussion of possible im- from dispersion relations) that αs must increase without bound plications of this sort of phenomenon for an understanding of as r goes to infinity (or squared momentum transfer q2 goes to confinement in QCD. zero). Happily here one only needs αs to reach a value making d = 0. At the corresponding value of q2 the dimension of a quark path goes to zero and the quark is confined. Lower VI. QCD values q2 (and negative dimensions) make no sense since the integrals in Eq. (15) now have no paths any more as soon as In this section we consider how propagators with anom- d hits zero. Note that this approach to thinking about confine- alous dimensions could shed light on the confinement mech- ment does not require any singular (i.e. infinite) values of q2 anism in QCD. Precise and rigorous calculations are beyond (or length) but should happen at some well-defined value of the scope of this paper, but we do indicate approaches to such q2 (or length) presumably related to the confinement scale. calculations and what one would expect qualitatively. A rigorous calculation along these lines would be difficult, Quantum chromodynamics is an incredibly difficult theory but a few points can be made in defense of this overall picture. in which to say anything precise. The fundamental degrees First of all, why would one imagine in the regularized view of of freedom carry strong (colour) charges, interact via highly Eq. (15) that one was always looking at a red quark interacting S. Gulzari et al. 291 with another red quark? Surely each emitted gluon changes dimension zero. At higher energy scales they partially escape, the colour of the particle and complicates matters. This is in- with a dimension which rises as the effective coupling strength deed true, but if one has defined the quark as being red (in drops until they become, to a good approximation, free. It some gauge) then, even with various colours of gluons be- is interesting to note that the view of confinement suggested ing continuously emitted, one expects that it would remain on here applies to any coloured object with no need for it to have average red. Any tendency to be green or blue would be a vi- neighbours present to “bind” to it and ensure that a colour sin- olation of SU(3) symmetry since it would indicate a preferred glet state propagates – here interaction of a single coloured direction in colour space other than “red” (The effective value object with its own colour field is enough to confine it. of the colour charge would be expected to be scale-dependent, but this is exactly what a running coupling constant is meant to describe.) This sort of argument could be the basis for a more VII. CONCLUSIONS rigourous mean field theory approach. Second αs is usually 2 studied as a function of q , but its form is not even known at We have reviewed the simple and intuitive path integral de- long distance scales and thus a rather difficult (!) function to scription of how the propagator for a charged Dirac particle work out in coordinate space to use in Eq.(15). The point we is modified by soft self-energy radiative corrections as shown want to make here is that it is very generally known that α in- in reference [1]. The result is a self-similar (fractal) object creases in the infrared limit, and this alone is enough to argue with the non-locality one would expect for a particle carrying for confinement. Even without an explicitly known confining an infinite range field. Arguments are made for a similar, but form for D(x−y), all one needs to know is that α continues to qualitatively different, effect due to attractive self-interactions rise. At some point it will rise to a value high enough to drive such as gravity and a calculation made for Newtonian gravity. d to zero. Beyond that, the quark is non-propagating (our ver- The results are linked to the fractal dimensions of the paths sion of “confined”) and there is no need to evalute paths in that particles take in quantum field theory, and the effects of Eq.(15) since the quark has none! The expected results from repulsive (QED) and attractive (gravity) self-interactions are asymptotic freedom are, of course, reproduced, since as αs discussed. Finally an attempt is made to estimate what effects goes to zero the propagator goes over to its free form (aside would be expected in QCD, with a link made to confinement from whatever corrections remain due to charge and mass as at a finite energy scale in terms of a fractal dimension which discussed earlier – particles coupled to long-range fields never goes to zero. really get free!). A similar argument would be expected to hold for any coloured objects, including gluons and hypothetical objects Acknowledgements in other representations of SU(3) colour. We would like to thank the organizers of the IRQCD 2006 The physical picture is an interesting one and rather differ- conference for a wonderful and stimulating conference where ent from the assumption that quarks are bound due to strong one of us (J.S.) presented an earlier version of this work. We attractive forces to other quarks which increase linearly with would also like to thank Reinhard Alkofer for pointing out separation in some approximation. Here, below some energy references [2] and [32] to us, and Christian Fischer for point- scale (that at which the dimension d goes to zero), coloured ing out [33]. This work was supported in part by the National objects interact so much with their own glue that they get Science Foundation under grant NSF-0457001. “stuck” in the sense that their would-be paths are reduced to

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