Chaos in QCD? Gap equations and their fractal properties
CChaos in QCD? Gap equations and their fractal properties.
Thomas Kl¨ahn ∗ Department of Physics & Astronomy, California State University Long Beach, Long Beach, CA 90840, U.S.A.
Lee C. Loveridge † Los Angeles Pierce College, Woodland Hills, CA 91371, U.S.A.
Mateusz Cierniak
Institute of Theoretical Physics, University of Wroc(cid:32)law, 50-204 Wroc(cid:32)law, Poland (Dated: January 5, 2021)We discuss how iterative solutions of QCD inspired gap-equations at finite chemical potentialshow domains of chaotic behavior as well as non-chaotic domains which represent one or the otherof the only two - usually distinct - positive mass gap solutions with broken or restored chiralsymmetry, respectively. In the iterative approach gap solutions exist which exhibit restored chiralsymmetry beyond a certain dynamical cut-off energy. A chirally broken, non-chaotic domain withno emergent mass poles and hence with no quasi-particle excitations exists below this energy cut-off. The transition domain between these two energy separated domains is chaotic. As a result, thedispersion relation is that of quarks with restored chiral symmetry, cut at a dynamical energy scale,determined by fractal structures. We argue that the chaotic origin of the infrared cut-off could hintat a chaotic nature of confinement and the deconfinement phase transition.
Keywords: Confinement, Dynamical Chiral Symmetry Breaking, Quantum Chaos, Quantum Chromodynam-ics, QCD phase transitions
I. INTRODUCTION
In the early 1980’s Benoit Mandelbrot pioneered themethodical study and computational visualization of theiteration of quadratic functions and began to cartographthe emerging fractal landscape [1] which subsequently hasbeen named in his honor as the Mandelbrot set. Withthe advance of personal computers during the mid 80’sfractals gained broad attention, scientifically as well asin popular science.In 1986, Leo Kadanoff, in an article with the title”Fractals: Where’s the physics?”[2], expressed concernedcuriosity about an understanding of fractal properties inphysics which goes beyond the identification of fractaldimensions for certain problems. Kadanoff stated thatwithout a better understanding of how physical mecha-nisms result in geometrical form it is difficult to trace types of questions with interesting answers.
We wish toadd that even in lack of such a deep understanding itis of course possible to find this kind of questions, asmentioned by Mandelbrot: ”I was asking questions whichnobody else had asked before, because nobody else had ac-tually looked at certain structures.” [3].An example for this explorative approach is Hofs-tadter’s butterfly which is publicly less known. In 1976,ten years before Kadanoff asked his curious questionand four years before Mandelbrot’s famous work on thequadratic map, Douglas Hofstadter observed what hecalled recursive structure in the computed spectrum of ∗ [email protected] † [email protected] electrons in electromagnetic fields [4] which has beennamed after the visual appearance as Hofstadter’s butter-fly . A first experimental confirmation of this theoreticalprediction has been reported nearly twenty years later in1997 [5].There is no strict definition of what a fractal is but stillmost people would know one when they see it. Com-mon descriptors of fractals refer to their non-analycity,self-similarity, non-linearity, iterative origin, chaotic be-havior, non-integer (Hausdorff and other) dimension, toname a few. This paper has been motivated by the factthat QCD’s gap equations are by definition highly non-linear and self-consistent. Self-consistency equates thequantity of interest, our gaps, to a functional which de-pends on these gaps themselves. QCD’s gap equationsare organized in a hierarchy of inter-dependencies of aninfinite number of n-point Green-functions and it is atthe heart of contemporary approaches in this field toidentify methods which reduce this infinite number ina manageable way while preserving key features of QCDlike dynamical mass generation and confinement. Whileone can argue how to obtain physically meaningful gapequations, viz. which set of approximations, truncations,etc. is the most reasonable, the self-consistent nature ofthese equations is not debated. Already at the seeminglysimple level of 2-point Green functions for a single quarkflavor appropriate truncation schemes allow to computethe mass spectrum of confined and deconfined quarks.The same methods allow for computation of meson andbaryon spectra. Nothing of this is new and, although nei-ther trivial nor brought to a final solution, it is in a struc-tural sense reasonably well understood and dealt with inDyson and Schwinger’s functional approach which provedto be a powerful tool to investigate the theories of QCD a r X i v : . [ nu c l - t h ] J a n and QED. We refer to recent reviews for examples andmore detailed information [6–13].Practitioners in the field of Dyson-Schwinger equationsfrequently deal with problems that can arise from theirself-consistent nature. As an example, one technique tosolve gap equations is by means of iteration starting froman initial guess. There is no guarantee for the conver-gence of such an iteration in general nor that the ob-tained solution is physical. In order to cover ’all possible’solutions in this approach one would scan over differentinitial guesses. Typically, one can ’tame’ diverging it-erations by damping the impact of the iteration itself.Instead of g = F [ g ] one can write g = αg + (1 − α ) F [ g ]where g is the gap, F a functional of the gap, and α a damping parameter close to but less than one, thusavoiding strong responses of g to the iteration. One canwonder - we claim one should - whether it is justified toapply such an algorithm. It looks innocent in that sensethat technically any solution of the original gap equationis a solution of the damped iteration equation. Never-theless, at identical initial value both may give differentanswers and thus one can claim that the damping param-eter might bear unwanted physical significance as it hasbeen introduced ad hoc. What happens if the gap equa-tion is allowed to iterate itself freely? We found only one,recently published, paper which asks exactly this ques-tion and comes to a clear conclusion: if the system isstrongly coupled, chaos emerges and one can observe aninfinite spectrum of ’unexpected’ gap solutions with in-creasing coupling strength[14]. In the paper we present,we give a brief explanation why these unexpected solu-tions actually should be expected. Further, we employa model with momentum dependent gap solutions. Inan iterative and inherently fractal context this led us ona surprising journey which answered not all but plentyof the questions we asked and at the end of which weare left to wonder whether looking at QCD as a fractaltheory might be a key to understand confinement as anemergent fractal phenomenon.Section II briefly motivates how iterative mapping gen-erates new solutions of an equation while preserving thesolutions of the non-iterated ’seed’ equation, Section IIIreviews the quark matter model by Munczek and Ne-mirovski (MN) in an extension for dense quark matter.We chose it for our exploration as it exhibits confinementand dynamical chiral symmetry breaking while being suf-ficiently simple to make it well suited for iterative map-ping and analytic treatment. The following Section IVillustrates and cartographs chaotic features which emergeupon iteration of the gap equation, Section V is a cau-tious attempt to interpret physical meaning into the in-terplay of chaotic and non-chaotic structures we observe.Our study focuses at the structure of the mass pole. Ap-pearance and disappearance of the mass pole are highlydriven by chaotic behavior. Further, the mass gap it-self is allowed to switch between different, usually dis-tinct solutions. To our surprise, the physical propertiesof the iterative solutions provide a reasonable picture of how de-confinement could present itself in a model whichpossesses a gap equation with a single solution only. Fi-nally, we estimate how a finite width gluon interactioncould affect the observed behavior of the quark disper-sion relation under iteration in Sec.VI before we concludein Sec.VII. II. SELF-CONSISTENCY AND THEEMERGENCE OF NEW ROOTS AMONGST THEOLD
We investigate possible consequences of chaos thatappears in iterative solutions of non-linear and self-consistent equations in the complex domain. For clarityof what we consider physics and what is math we startwith the latter and briefly review Mandelbrot’s fractalwhich is obtained by the iteration z z =0; n →∞ ←−−−−−−− f ( z ) withthe explicit choice f ( z ) = z + c to obtain the Mandel-brot set. We chose to use the symbol z ; n ←−−− to have adistinguished notation for the iterative mapping process- specifying the number of iterations n and the initialvalue z - over the equal sign = which appears in theanalytic equation z = z + c . It is worthwile to lookat the differences between these two. First, the poly-nomial equation has exactly two solutions z , for anygiven c which are defined by the roots of the polynomial P ( z ) = f ( z ) − z = z + c − z . It is further easily seen thatone can determine c for a desired root z . For example, P ( z = 0) = 0 if c = 0.In the iterative approach, each iteration generates anew polynomial, f ( z, c ) = (cid:16) z z, ←−− z + c (cid:17) = f ( z ) = z + c,f ( z, c ) = (cid:16) z z, ←−− z + c (cid:17) = f ( f ( z )) = (cid:0) z + c (cid:1) + c,...f m ( z, c ) = (cid:16) z z,m ←−− z + c (cid:17) = f ( f ( ..f ( z )))= f m − ( z, c ) + c, etc. ad infinitum. There is one trivial but fascinatingproperty of this infinite set of equations which essentiallyinspired the presented work. The left hand side of eachof the previous equations has been set to f i ( z, c ) = z inorder to obtain the next iteration f i +1 ( z, c ) = z . It isthus safe to state that the roots of P ( z, c ) = f ( z, c ) − z are guaranteed to be roots of P ( z, c ) = f ( z, c ) − z = f ( f ( z, c )) + c . As P ( z, c ) is a 4th order polynomial,there are two more roots which, of course, did not appearfor P ( z, c ), a 2nd order polynomial. The important les-son to be learned is that for a self consistent non-linearequation z = f ( z ) the iteration z z ; n ←−− f ( z ) generatesa new self consistent equation. While the solutions ofthe non-iterated equation remain a subset of solutions ofthe iterated equation, the iterated equations can developadditional solutions.
This is a peculiar, almost awkward situation, if onewishes to assign physical meaning to the original solu-tions of the equation f ( z ) = z . What makes these rootssuperior with respect to any of the iterative clones if all,original and clones, share these very same original solu-tions? Evidently, there is an infinite number of (iterated)functions which share the original roots. Is the originalfunction with only these roots a superior or inferior func-tion? Is it worth to ponder the meaning of the additionalroots of iterated clones? Can we safely omit them? Dowe miss important information when we ignore the du-ality of the gap equation as root defining equation andmapping rule? We decided to explore and ponder possi-ble meaning. III. THE MUNCZEK-NEMIROVSKY MODEL
One approach to move towards an understanding ofQCD is based on evaluating QCD’s partition functionby testing its response to external sources. This is theDyson-Schwinger formalism which results in sets of cou-pled n-point Green functions. Out of these we are inter-ested in the quark propagator which is obtained from thegap eguation S ( p ; µ ) − = i(cid:126)γ · (cid:126)p + iγ ( p + iµ ) + m + Σ( p ; µ ) , (1)with the self-energyΣ( p ; µ ) = (cid:90) d q (2 π ) g ( µ ) D ρσ ( p − q ; µ ) × λ a γ ρ S ( q ; µ )Γ aσ ( q, p ; µ ) . (2)Here, m is the quark bare mass, D ρσ ( p − q ; µ ) is thedressed gluon propagator and Γ aσ ( q, p ; µ ) is the dressedquark-gluon vertex. This is the first of an infinite towerof gap equations which, without further approximations,couple back to this one. Further, there are similar equa-tions for the dressed gluon-propagator and the quark-gluon vertex. Note that the gap equation is a self-consistent non-linear (in most cases integral) equation, S − = F [ S ].Within the Munczek-Nemirovsky model [15] thedressed quark-gluon vertex is approximated by the freequark-gluon vertex, Γ aσ ( q, p ; µ ) = λ a γ σ . Gap equationsapplying this approximation are referred to as rainbowgap equations. For the dressed gluon propagator themodel is specified by the choice g ( µ ) D ρσ ( k ; µ ) = (cid:18) δ ρσ − k ρ k σ k (cid:19) π η δ ( k ) . (3)Due to the δ -function which in configuration space cor-responds to a constant, this is a very simplified approx-imation of the gluon-propagator, specified by the cou-pling strength we set to η = 1 .
09 GeV in accordancewith [15]. For non-zero relative momentum k the inter-action strength in this model vanishes, thus making it super-asymptotically free. Furthermore, the infrared en-hanced δ -function is sufficient to provide for dynamicalchiral symmetry breaking and confinement, both featuresof QCD which we wish to address. Finally, the δ -functioneffectively turns the integral gap equation into an alge-braic equation which can be solved analytically.In order to obtain these solutions for the in-mediumdressed-quark propagator one employs the general solu-tion S ( p ; µ ) − = i(cid:126)γ · (cid:126)pA ( (cid:126)p , p )+ iγ ( p + iµ ) C ( (cid:126)p , p ) + B ( (cid:126)p , p ) . (4)Substitution into the dressed-quark gap-equation and ap-propriate tracing over the Dirac γ -matrices results inthree coupled gap equations of which two (for A and C )are identical: A ( p, µ ) = 1 + η A ( p, µ )˜ p A ( p, µ ) + B ( p, µ ) (5) B ( p, µ ) = m + η B ( p, µ )˜ p A ( p, µ ) + B ( p, µ ) . (6)We introduced ˜ p = (cid:126)p + ( p + iµ ) . In the chiral limit( m = 0) one finds two distinct sets of solutions, one ofthem chirally symmetric and referred to as the Nambuphase, A ( p, µ ) = 12 (cid:32) ± (cid:115) η ˜ p (cid:33) (7) B ( p.µ ) = 0 (8)whereas for the other solution, the Wigner phase, chiralsymmetry is broken for R (˜ p ) < η / A ( p, µ ) = 2 (9) B ( p, µ ) = (cid:112) η − p . (10)For R (˜ p ) > η / p E → ˜ p M , with ˜ p E = (cid:126)p + ( p + iµ ) and ˜ p M = (cid:126)p − ( p + iµ ) .Due to our interest in particle mass poles our investiga-tion of the model is performed in Minkowski metric. Forthe next section however, the specific metric is not rele-vant; we will only work with the fact that ˜ p is complex-valued and thus can be decomposed into real and imag-inary part, viz. ˜ p = z R + iz I . We chose to label realand imaginary part with squared quantities as a reminderthat they come in units of energy square. IV. ITERATIVE CHAOS
Gap equations (5) and (6) lead to 4 th order polyno-mial equations with up to four distinct complex valuedsolutions at given ˜ p for each gap. Generally, this is thewhole solution space one would consider and the onlytask left is to identify the one physical solution. How-ever, the self-consistent nature of (5) and (6) is evidentand, according to our reasoning in the previous sectionthere is a possibility for iterated functions with the samefour and additional solutions.Before we discuss our analysis, a few comments shouldbe made. Defined by contact interaction in momentumspace we chose a very simple model for the effective gluonpropogator. For dressed-gluon propagators with finitewidth in momentum space the corresponding gap equa-tions turn into integral equations. Thus, momenta coupleand the simplicity of the MN model which we take ad-vantage of for this exploration is lost. We address thisissue in more detail in the last section of this paper.As sketched in the Introduction, for models with so-phisticated non-trivial interaction-kernel, iteration is apractical path to find gap solutions. We outlined before,that this leaves us with the possibility that iteration gen-erates new functions which possess roots that correspondto solutions of the original gap equation and potentiallyan infinite number of additional roots.We start our iteration from the non-interacting solu-tion ( B = m , A = 1) and treat ˜ p = z R + iz I asone would consider the constant c for the Mandelbrotset z ←− z + c . For the moment, this reduces the num-ber of independent variables from three ( (cid:126)p , p , µ ) to two.The result of such an iteration is shown in Fig.1 for thereal part of the scalar gap B at two different bare-quarkmasses of 10 MeV and 100 MeV, respectively.Unlike the Mandelbrot fractal, this fractal does not di-verge; chaos exhibits in domains in which the gaps forinfinitesimally changes of energy and momentum takevastly different but finite values in a seemingly randompattern. This fractal region is contained within an al-most perfectly shaped ellipsoid which we fit accordinglywith (cid:32) z R + z R, R R (cid:33) + (cid:18) z I R I (cid:19) = 1 . (11)( z R, , R R , R I ) differs slightly for m = 10 MeV(0 . , . , .
77) MeV and for m = 100 MeV(0 . , . , .
80) MeV . The inner almond shape withless obvious chaotic behavior is well approximated by thesame function with (1 . , . , . for m = 10MeV, and (1 . , . , . for m = 100 MeV.As for the Mandelbrot set one would be ill advised tounderstand these figures as a valid representation of thefractal; the appearance of the fractal changes with eachnew iteration. We identify regions with identical period-icity, ranging from a stable, period one solution in theregion outside of the covering ellipse over a period tworegion within the almond shape up to higher and higherperiodicity in between these two regions. This is illus-trated in Fig.3 for m = 100 MeV for a periodicity of upto ten.Keeping in mind, that the analytic gap equations pos-sess four distinct solutions it seems interesting, that there FIG. 1. Real part of the scalar gap B after 300 iterationsstarting from A = 1 (top and bottom) and B = m = (10MeV (top), 100 MeV (bottom)). is an extended stable domain (periodicity one) which fa-vors one, and only one solution. We follow the gap solu-tion along a vertical path at fixed z R and vary z I . Forreasons which become more clear at later stage we chose z R = 0 . . Along this line, one notices that onepasses from an outer stable region into an inner stableregion by traversing a small chaotic domain. This is il-lustrated in Fig.3. As within this chaotic domain thevalue of the gap function can change with each itera-tion, we plot all obtained values of R ( B ) over 300 iter-ations in gray scale according to how frequently a par-ticular solution has been obtained. Evidently, there is atransition between two distinct analytic solutions of thenon-iterated forth order polynomial gap equations. Thisresult seems remarkable if one recalls how one would usu-ally deal with different gap solutions for a given model:each solution is understood as a distinct phase, then oneexamines the stability of each individual solution andpicks the energetically favored solution as the physicalone. Upon iteration we are lead to a different conclu-sion. Although each of the analytic solution indeed isa solution of the gap equations, only one of them canbe stable upon iteration at a given energy and momen-tum. However, the stable iterative solution over a finite FIG. 2. Periodicity of the iterative mass gap solution at m = 100 MeV. The outer, indigo region of the plot is abso-lutely stable under iteration, the inner almond shape has peri-odicity two, the area in between exhibits chaos with increasingperiodicity. For this plot, areas with periodicity larger thanten are plotted in black.FIG. 3. Real part of the mass gap B at z R = 0 . .The color coding indicates how frequently a solution has beenfound over 300 iterations after the first 100 iterations whichare sufficient to shape the fractal as seen. For reference, allanalytic solutions to the polynomial gap equations are plot-ted in color. Iteration switches from massive solutions (blue)at small I ( z ) to bare-mass solutions (green) at larger values.Except for the chaotic transition domain, the iterative ap-proach picks positive mass-gap solutions, only. Note, that thechaotic domain has solutions of periodicity two and higher; itis truly unstable and hence we add a gray scale to measure thefrequency of a particular solution over the final 300 iterations. range of energies can switch between distinct analyticsolutions. It is further remarkable, that the iterativelystable solution is massive at low and massless at high en-ergy. Amongst all the possibilities chaos seems to offer,this seems a very reasonable one. The notion of analyticsolutions describing different phases however, is not sup-ported from an iterative perspective; there is one, and only one, iterative solution to the gap equation.Before we go into further interpretation of what thisresult implies, we wish to address a question relatedto the previous paragraph. Initially we remarked thatour iteration starts from the non-interacting solution A = 1 , B = m . As we try to proceed as careful as pos-sible, let us investigate the iterative stability of the fouranalytic gap solutions as plotted in the upper panels ofFig.4 where we show again the real part of the mass gap.The lower panel of Fig.4 shows the result after 300 it-erations of these algebraic solutions as initial value. Itis safe to say, that none of them is stable under itera-tion. Further, there is a visibly favored solution at largevalues of z R which does not depend on the initial gapthat seeded the iteration. From a global perspective, thefractal keeps the general shape but shows differences foreach different seed solution. This is to be expected andwould happen in similar fashion to the Mandelbrot set ifthe initial value is arbitrarily changed.Comparing the iterations to the algebraic solutions ofthe gap equations in the upper panel of Fig.4 one cangraphically identify which of them is stable under itera-tion and in which domain. As seen, this is the case onlyfor the positive mass-gap solutions 1 and 3 from Fig.4, asillustrated in Fig.5. In other words, although the chaoticdomain will vary, the described features of Fig. 3 withrespect to the analytic gap solutions do not critically de-pend on the chosen initial gap. V. MASS POLES
Up to this point we refrained from searching for mean-ing in our study. In spite of the fact that iterative massgap solutions result in a large domain of chaotic behavior,which may or may not hide future surprises, we cannothelp but wonder whether the switching between mas-sive and mass-less gap solutions in the stable domainsoffers meaning. Before we go further, we want to recallthat MN is considered to be a confining model. Thisis seen by the fact that the inverse propagator has noroots in the chirally broken phase and therefore inte-gration over four-momentum does not pick up weight togenerate a finite particle number. Hence, although con-fined quarks generate mass via chiral symmetry break-ing, the absence of a mass pole results in the absenceof a dispersion, viz. there is no explicit relation be-tween specific momenta and energy. For the vacuum MNmodel, this is easily understood by the realization thatin Minkowski metric p + M ( p ) has no real root if atany p , M > − p = p − (cid:126)p . This running away of themass in the chirally broken phase is exactly what hap-pens in the MN model. However, as we have shown inthe previous section iteration erases the distinction be-tween chirally broken and restored phases and suggeststhat instead there might be a discontinuous gap solu-tion which is confinement-like affected by dynamical chi-ral symmetry at small momenta, and at large momenta FIG. 4. Upper panel: Solutions of the polynomial gap equations for m = 100 MeV . Each is plotted on a scale that mostaccentuates its structure. Solution 1 and 3 (from the left) are stable in some, mutually exclusive domains under iteration asillustrated in Fig.3. Lower panel: After 300 iterations using the corresponding solution of the polynomial gap equations fromthe upper panel as initial seed for the iteration. In the outer, non-chaotic domain all four cases produce nearly identical resultswith positive mass gap only.FIG. 5. Difference between gap solution 1 and 3 (from the left)in the top panel of Fig.4 and iterative solutions seeded with thenon-interacting solution ( A = 1, B = m ) after 500 iterations. White domains show no difference between iterative solutionsseeded with an analytical model solution or seeded with the non-interacting solution. Solution 2 and 4 show no agreementanywhere in the stable domain of periodicity one (not shown). chirally unconfined-like and chirally restored. The tran-sition between these domains is characterized by chaoticand unstable solutions (see Fig. 3).At finite chemical potential the real poles of the prop-agator A ( (cid:126)p − ( p + iµ ) ) + B are represented by (cid:126)p − p + µ + (cid:60) ( M ) = 0, with M = B/A . We notethat the shift of the pole due to chemical potential shouldnot be confused with the physical mass pole of the parti- cle. This becomes evident if one considers an ideal non-interacting gas with M being constant and real valued.For the purpose of this study we refer to the physicalmass pole, defined by (cid:126)p − p + (cid:60) ( M ) = 0. From thedefinition ˜ p = z R + iz I we identify the pole position inour contour plots as z R = µ and z I = − p µ for anideal particle with constant and real M . This representsa vertical line in our plots which does not depend on mo- FIG. 6. Natural logarithm of (cid:0) p + M − p (cid:1) for the iterative solution for µ = (100 , , m = 100 MeV. The vertical line shaped by minimal negative values indicate a physical mass pole, viz. a quasi-particle.In the chaotic domain, this pole structure is absent, viz. the vertical line (or any distinct pole) pattern is absent. This impliesan infrared energy gap below which quarks show no quasi-particle properties. As the chemical potential increases, the quasi-particle pole line moves to the right and simultaneously decreases the gap, viz. the gap region without a pole traces the outershape of the fractal. Once the chemical potential is sufficiently large, the gap closes entirely. Note, that the absence of a masspole does not imply that there is no mass gap solution, as illustrated in Fig.3. mentum and measures energy with increasing distancefrom the real axis. It shifts to higher z R with increasingchemical potential.In Fig.6 we trace the physical mass pole in Minkowskimetric by plotting the logarithm of the quantity (cid:0) p + M − p (cid:1) which gives zero and hence a large neg-ative logarithm at the physical mass pole. As the verticalaxis does not depend on the mass ( z I = 2 p µ ) a verticalpole line indicates constant dressed quark masses. Weobserve the absence of such a well ordered pole struc-ture within the chaotic domain. Since the vertical axisis a measure of the particle energy at fixed chemical po-tential one can conclude that the transition to the mas-sive solution (Fig. 3) suppresses quasi-particle behaviorin the infrared domain of the model. Again, we can tracethe physical pole indicated by the vertical line and find z R = µ − M since the pole is found at p = p + M .Following our elliptic fit of the outer boundary of the frac-tal domain this allows to determine the critical chemicalpotential where the infrared energy gap entirely disap-pears µ C,IR = (cid:113) m + R R − z R, . (12)We find µ C,IR ≈ M eV for m = 100 M eV and µ C,IR ≈ M eV for m = 10 M eV . At these chemical potentialsand beyond, quarks can be considered as completely chi-rally restored.In order to estimate when mass-pole states can be oc-cupied, we determine at which chemical potential the en-ergy p and the Fermi energy or chemical potential µ turnequal, that is when z I = µ on the elliptic boundary ofthe fractal at the position of the physical mass-pole with M = m . We choose this scenario as this is the criti-cal potential starting from where the particle energy islarger than the chemical potential and thus large enough to populate quasi-particle states. This is the case when (cid:32) µ − m + z R, R R (cid:33) + (cid:18) µ R I (cid:19) = 1 , (13)and holds for the light quark with m = 10 M eV at µ m ≈ M eV , for the heavy quark with m = 100 M eV at µ m ≈ M eV .Although this is not a rigorous statement, one canroughly relate the critical chemical potential for the tran-sition from a chirally broken into the restored phase tothe in-vacuum dressed-quark mass. In our case, the situa-tion is a bit different. We estimate a hypothetical chirallybroken quark vacuum mass based on the previous esti-mate of the critical potential for the complete disappear-ance of the infrared gap by setting them approximatelyequal. Relating µ m as onset of a deconfined, chirally re-stored quark phase with an estimate of the constituentquark mass seems to give rather reasonable results incomparison to other model calculations. This is interest-ing, considering that in the MN model the vacuum massat zero 4-momentum is defined by the coupling strength η , which is of the order of 1 GeV.It is noteworthy that our simple approach reproducesquantities related to the effective constituent masses atreasonable values. We state explicitly, that in this modelconstituent masses are nowhere realized for a physicalparticle, viz. an entity with a mass pole of that magni-tude. We can compare the light quark critical chemicalpotential µ m ≈ M eV with the deconfinement criticalpotential obtained within the MN model in Euclideanmetric with a value of 300 MeV [16] or with subsequentwork based on a widened version of the effective gluonpropagator [17] which predicts deconfinement at a chemi-cal potential of 380
M eV . There is a satisfying agreementof these values with ours. We point out though, thatboth of these models are defined within a different met-ric, slightly different bare quark masses and, most impor-tantly, are based on entirely different assumptions. Whilethe two previous papers employed distinct gap solutionsand compare the pressure of the corresponding mass-lessWigner and massive Nambu phase, our approach resultsin only one gap solution which exhibits a transition fromthe Nambu to the Wigner phase through a chaotic do-main as depicted in Fig. 3. Our quarks are either bare-mass quarks with poles or entities with a chaotic massfunction or a dressed quark mass different from the baremass with no associated pole. In the latter case there isa chaotic transition from dressed quark masses to barequark masses with increasing energy.
VI. FINITE INTERACTION WIDTH
We begin the final section of this paper with a plot ofthe particle pole in energy momentum space which weobtain by transforming ( z R , z I ) to ( p = | (cid:126)p | , p ) coor-dinates under explicit choices of the chemical potentialas noted in Fig.7. Although this switch in representa-tion does not provide additional information we find itinstructive to provide an actual dispersion relation ob-tained from the iterative approach. In this example ata chemical potential of 700 MeV no chaotic behavior isvisible and the dispersion is exactly that of a free quarkat bare-mass 100 MeV. With decreasing chemical poten-tial chaos emerges at energies higher than that of theexpected (now absent) free particle dispersion. The ac-tual dispersion branch is cut clean at some critical value(as we discussed in the previous section), thus illustratingour interpretation of the fractal boundary as the causefor a dynamical infrared cutoff below which quarks aremass-pole free.We address a last question which relates to the factthat the MN model bases on the very particular choiceof the effective gluon propagator as a δ -function in 4-momentum space. The reason that we could easily per-form the presented study bases on this Ansatz and thesubsequent decoupling of momenta which allows to iter-ate point-wise for any given 4-momentum without cou-pling to other momenta. This might raise the suspicion,that momentum coupling could destroy the fractal struc-ture we observed. In order to keep the simplicity of thegap equations but still get an idea about the stability ofthe emergent fractal we averaged at each point in ourplane after each iteration step and thus mimicked somekind of momentum coupling. The averaging is based onGaussian weights around a given point according to g D µν ( k ) = 3 π η δ µν exp( k /w ) (cid:82) exp( k /w ) d k , (14)where w is the width of the Gaussian. To further simplify,we assumed that the widening only happens in the direc-tion of momentum and energy, i.e. there is no wideningperpendicular to the momentum.As seen in Fig.7 the separation into chaotic pole-free,and non-chaotic mass-pole domains remains, even when we change the momentum dependence of the gluon froma delta function to a gaussian with a half width as muchas 0.02 times the gluon mass. We find numerical evidencethat this feature remains even with a width as much as20% of the gluon mass, ≈
200 MeV. This correspondsto a spatial width of about 1 fm; about the size of aproton. Based on this - certainly simplified - treatmentof momentum coupling we conclude, that the statementswe make in this paper may indeed survive a more com-plete treatment, involving self-interactions with globallycoupled momenta - which has been our main concernprompting this final analysis.
VII. CONCLUSIONS
As we have shown, a strictly iterative solution of theMN gap equations results in fractal gap structures whichcan be characterized by the existence of three qualita-tively very different, yet co-existing domains of a sin-gle and unique gap solution: a bare-mass quark quasi-particle domain with physical mass-poles extending in-finitely into the ultraviolet, a dressed-mass quark domainwithout mass-poles and hence no quasi-particle interpre-tation in the infrared, and a chaotic domain of transi-tion between the first two phases. Remarkably, the twonon-chaotic domains correspond to distinct analytic so-lutions which would usually represent individual phaseswith either dynamically broken or restored chiral sym-metry. The fractal approach offers an alternative to thisseparation which is rooted in the iterative nature of thegap equation.It is further noteworthy, that the iterative massgap solution is always positive in the smooth, viz.non-chaotic domain of the fractal. The appearance ofa chaotic boundary between two qualitatively differentdomains results in interesting properties: I) The iterative approach provides an ultraviolet cut-off for the massive and mass pole free Nambu solution, asthis solution appears only within the elliptic region of the( z R , z I )) plane. Thus the approach avoids the appear-ance of an infinitely increasing dressed-quark mass withincreasing momentum and energy. In the MN model,this running mass results in the absence of mass poles forthe massive gap solution and thus relates to confinement. II)
It provides an infrared cut-off for the bare-quarkmass Nambu solution and thus ensures that quasi-particle states are not populated at small chemicalpotential although the quark can virtually exist as aquasi-particle with well defined dispersion.
III)
Both cutoffs more or less coincide (as seenin Fig.5) although there is a transition region whichis chaotic in nature. The resulting effective Nambu-UV/Wigner-IR cutoff depends dynamically on energy,momentum, bare mass, and chemical potential. As a side
FIG. 7. Plotted is the logarithm of the mass-pole condition log( | (cid:126)p − p + µ + (cid:60) ( M ) | ) which shows a dispersion relation withdistinct, chaos-induced infrared cut-off. With increasing chemical potential ( m = 0 . η ; µ = (0 . , . , . η from left to right)the infrared cut-off decreases and eventually disappears. With increasing widening ( σ = (0 . , . , . η from top to bottom)chaotic domains blur but the observed IR cut-off remains. note we add that plotting gap solutions in the ( z R , z I )plane removes much of the dynamical arbitrariness andleaves the ratio of bare mass m and coupling constant η as the only ’true’ degree of freedom, viz. a change of thechemical potential µ would rescale the plot but cause noqualitative change, whereas plots like Fig.1 show indeed’the’ gap solution at arbitrary chemical potential. IV)
At sufficiently large chemical potential bare-quark mass-pole states will form at energies which can be populated and thus physical quarks can exist asquasi-particle excitations.A mechanism with these properties can be interpretedas a deconfinement mechanism. The appearance of one,and only one, iterative solution of the MN gap equationsbears a certain elegance. First, by the very fact that thereis only one gap solution with expected properties as theexistence of only a positive mass gap, asymptotically re-stored chiral symmetry and the absence or appearance of0physical mass poles. Next, it builds on distinct solutionswhich one would obtain in the non-iterative approach butgives them a new meaning by slicing them into a singlenew solution with the aforementioned properties.A simple treatment of a widened, δ -like gluon-interaction indicates that momentum coupling blurschaotic domains but does not necessarily change the qual-itative results we describe if the widening is moderate. Asthis study is an exploration and qualitative in nature welook forward to further analyses of this perspective on understanding confinement and the deconfinement tran-sition as highly non-linear and to a certain extent possi-bly chaotic phenomena. VIII. ACKNOWLEDGEMENTS
We are grateful to Prashanth Jaikumar, Pok Man Lo,and Craig D. Roberts for helpful comments and discus-sions which helped us a great deal to find order in chaos. [1] B. B. Mandelbrot, Fractal aspects of the iteration of z → λz (1 − z ) for complex λ and z , Annals of the New YorkAcademy of Sciences , 249.[2] L. Kadanoff, Fractals: Where’s the Physics?, Physics To-day , 6 (1986).[3] W. of Stories, B.Mandelbrot about: Drawing; the abilityto think in pictures and its continued influence ((accessedJuly, 2020)).[4] D. R. Hofstadter, Energy levels and wave functions ofbloch electrons in rational and irrational magnetic fields,Phys. Rev. B , 2239 (1976).[5] U. Kuhl and H.-J. St¨ockmann, Microwave realization ofthe hofstadter butterfly, Phys. Rev. Lett. , 3232 (1998).[6] C. D. Roberts, Three Lectures on Hadron Physics, J.Phys. Conf. Ser. , 022003 (2016), arXiv:1509.02925[nucl-th].[7] T. Horn and C. D. Roberts, The pion: an enigma withinthe Standard Model, J. Phys. G , 073001 (2016),arXiv:1602.04016 [nucl-th].[8] G. Eichmann, H. Sanchis-Alepuz, R. Williams,R. Alkofer, and C. S. Fischer, Baryons as relativis-tic three-quark bound states, Prog. Part. Nucl. Phys. , 1 (2016), arXiv:1606.09602 [hep-ph].[9] V. D. Burkert and C. D. Roberts, Colloquium : Roperresonance: Toward a solution to the fifty year puzzle,Rev. Mod. Phys. , 011003 (2019), arXiv:1710.02549 [nucl-ex].[10] C. S. Fischer, QCD at finite temperature and chemicalpotential from Dyson–Schwinger equations, Prog. Part.Nucl. Phys. , 1 (2019), arXiv:1810.12938 [hep-ph].[11] C. D. Roberts and S. M. Schmidt, Reflections upon theEmergence of Hadronic Mass (2020) arXiv:2006.08782[hep-ph].[12] S.-x. Qin and C. D. Roberts, Impressions of theContinuum Bound State Problem in QCD, (2020),arXiv:2008.07629 [hep-ph].[13] M. Barabanov et al. , Diquark Correlations in HadronPhysics: Origin, Impact and Evidence, Prog. Part. Nucl.Phys. , 103835 (2021), arXiv:2008.07630 [hep-ph].[14] A. Mart´ınez and A. Raya, Solving the Gap Equation ofthe NJL Model through Iteration: Unexpected Chaos,Symmetry , 492 (2019), arXiv:1904.02732 [hep-ph].[15] H. J. Munczek and A. M. Nemirovsky, Ground-state qq mass spectrum in quantum chromodynamics, Phys. Rev.D , 181 (1983).[16] T. Klahn, C. D. Roberts, L. Chang, H. Chen, and Y.-X.Liu, Cold quarks in medium: an equation of state, Phys.Rev. C , 035801 (2010), arXiv:0911.0654 [nucl-th].[17] H. Chen, W. Yuan, L. Chang, Y.-X. Liu, T. Klahn, andC. D. Roberts, Chemical potential and the gap equation,Phys. Rev. D78