Evolving center-vortex loops

We consider coarse-graining applied to nonselfintersecting planar center-vortex loops as they emerge in the confining phase of an SU(2) Yang-Mills theory. Well-established properties of planar curve-shrinking predict that a suitably defined, geometric effective action exhibits (mean-field) critical behavior when the conformal limit of circular points is reached. This suggests the existence of an asymptotic mass gap. We demonstrate that the initially sharp mean center-of-mass position in a given ensemble of curves develops a variance under the flow as is the case for a position eigenstate in free-particle quantum mechanics under unitary time evolution. A possible application of these concepts is an approach to high-$T_c$ superconductivity based (a) on the nonlocal nature of the electron (1-fold selfintersecting center-vortex loop) and (b) on planar curve-shrinking flow representing the decrease in thermal noise in a cooling cuprate.


Introduction
Four dimensional SU(2) Yang-Mills theory occurs in three phases: a deconfining, a preconfining, and a confining one. While the former two phases possess propagating gauge fields a complete decoupling thereof takes place at a Hagedorn transition towards the confining phase [1,2,3]. Namely, by the decay of a preconfining ground state, consisting of collapsing magnetic (w.r.t. the gauge fields in the defining SU (2) Yang-Mills Lagrangian) flux lines of finite core-size d, see also [4], into nonselfintersecting or selfintersecting center-vortex loops [5] the mass m D of the dual gauge field diverges. This, in turn, implies d → 0. As a consequence, center-vortex loops (CVLs) with nonvanishing selfintersection number N become stable solitons in isolation. These solitons are classified according to their topology and center charge. That is, for d → 0 the region of negative pressure P is confined to the vanishing vortex core, and the soliton becomes a particle-like (P = 0) excitation whose stability is in addition assured topologically by its selfintersection(s).
The purpose of our paper is to investigate the sector with N = 0 in some detail. Topologically, there is no reason for the stability of this sector's excitations, and we will argue that on average and as a consequence of a noisy environment a planar CVL with N = 0 shrinks to nothingness within a finite 'time'. Here the role of 'time' is played by a variable measuring the decrease of externally provided resolving power applied to the system. By 'planar CVL' we mean an embedding of the N = 0 soliton into a 2D flat and spatial plane. For an isolated SU(2) Yang-Mills theory the role of the environment is played by the sectors with N > 0. If the SU(2) theory under consideration is part of a world subject to additional gauge symmetries then a portion of such an environment arises from a mixing with these theories. In any case, a planar CVL at finite length L is acquiring mass by frequent interactions with the environment after it was generated by a process subject to an inherent, finite resolution Q 0 . At the same time, the CVL starts shrinking towards a circular point. The latter becomes unresolvable starting at some finite resolution Q * . That is, all properties that are related to the existence of extended lines of center flux, observable for Q 0 ≥ Q > Q * , do not occur for Q ≤ Q * , and the CVL vanishes from the spectrum of confining SU(2) Quantum Yang-Mills theory. Since CVLs with N > 0 have a finite mass (positive-integer multiples of the Yang-Mills scale Λ) we 'observe' a gap in the mass spectrum of the theory when probing the system with resolution Q ≤ Q * .
Notice that by embedding an N = 0 CVL of an isolated SU(2) Yang-Mills theory into a flat 2D surface at m D < ∞, d > 0, a hypothetical observer measuring a positive (negative) curvature along a given sector of the vortex line experiences more (less) negative pressure in the intermediate vicinity of this curve sector leading to this sector's motion towards (away from) the observer, see Fig. 1. The speed of this motion is a monotonic function of curvature. On average, this shrinks the CVL. Alternatively, one may globally consider the limit m D → ∞, d → 0, that is, the confining phase of an SU(2) Yang-Mills theory, but now take into account the P i < 0 P as = 0 e P < 0 P i e < P n Figure 1: Highly space-resolved snapshot of an N = 0 CVL curve-sector. The pressure P i in the region pointed to by the normal vector n is more negative than the pressure P e thus leading to a motion of the sector along n.
effects of an environment which locally relaxes this limit (by collisions) and thus also induces curve-shrinking. One possibility to describe this situation is by the following flow equation in the (dimensionless) parameter τ where s is arc length, x is a point on the planar CVL, and σ is a string tension effectively expressing the distortions induced by the environment. After a rescaling, x ≡ √ σx, ξ = √ σs, Eq. (1) assumes the following form where u is a (dimensionless) curve parameter (dξ = |∂ ux | du), n the (inwardpointing) Euclidean unit normal, k the scalar curvature, defined as |v| ≡ √ v · v, and v · w denotes the Euclidean scalar product of the vectors v and w lying in the plane. In the following we resort to a slight abuse of notation by using the same symbolx for the functional dependence on u or ξ.
It is worth mentioning that Eq. (2) expresses a special case of the local condition that the rate of decrease of the (dimensionless) curve length L(τ ) = L(τ ) 0 dξ = 2π 0 du |∂ ux (u, τ )| is maximal w.r.t. a variation of the direction of the velocity ∂ τx of a given point on the curve at fixed |∂ τx | [6]: The 1D heat-equation (2) is well understood mathematically [7,8] and represents the 1D analogue of the Ricci equation describing curvature-homogenization of 3manifolds [9,10,11] thus proving the geometrization conjecture [12].
The present work interpretes the shrinking of closed and 2D-flat embedded (planar) curves as a Wilsonian renormalization-group evolution governed by an effective action which is defined purely in geometric terms. In the presence of an environment represented by the parameter σ, this action possesses a natural decomposition into a conformal and a nonconformal factor. One of our goals is to show that the transition to the conformal limit of vanishing mean curve length really is a critical phenomenon characterized by a mean-field exponent if a suitable parameterization of the effective action is used. To see this, various initial conditions are chosen to generate an ensemble whose partition function is invariant under curve-shrinking. A (second-order) phase transition is characterized by the critical behavior of the coefficient associated with the nonconformal factor in the effective action. That is, in the presence of an environment the (nearly massless) N = 0 sector in confining SU(2) Yang-Mills dynamics, generated during the Hagedorn transition, practically disappears after a finite 'time' leading to an asymptotic mass gap. We also believe that N = 0 CVLs play the role of Majorana neutrinos in physics: Their disappearance after a finite time and the absence of antiparticles would be manifestations of lepton-number violation [13] forbidden in the present Standard Model of Particle Physics and may provide for an explanation to the solar neutrino problem alternative to the oscillation scenario, see also [14].
In Sec. 2 we provide information on proven properties of curve-shrinking evolution. The philosophy underlying the statistics of geometric fluctuations in the N = 0 sector is elucidated in Sec. 3. In Sec. 4 we present our results for the renormalizationgroup flow of the effective action, give an interpretation, and compute the evolution of a local quantity. Finally, in Sec. 5 we summarize our work and give an outlook to the N = 1 case which we suspect is relevant for high-T c superconductivity.

Prerequisites on mathematical results
In this section we provide knowledge on the properties of the shrinking of embedded (nonselfintersecting) curves in a plane [7,8]. It is important to stress that only for curve-shrinking in a plane are the following results valid. To restrict the motion of a CVL to a plane is a major assumption, and additional physical arguments must be provided for its validity.
The properties of the τ -evolution of smooth, embedded, and closed curvesx(u, τ ) subject to Eq. (2) were investigated in [7] for the purely convex case and in [8] for the general case. The main result of [8] is that an embedded curve with finitely many points of inflection remains embedded and smooth when evolving under Eq. (2) and that such a curve flows to a circular point for τ ր T where 0 < T < ∞. That is, asymptotically the curve converges (w.r.t. the C ∞ -norm) to a shrinking circle: lim τ →T L(τ ) = 0 and lim τ →T A(τ ) = 0, A being the (dimensionless) area enclosed by the curve, such that the isoperimetric ratio L 2 (τ ) A(τ ) approaches the value 4π from above. For later use, we present the following two identities, see Setting A(τ = 0) ≡ A 0 , the solution to Eq. (6) is By virtue of Eq. (7) the critical value T is related to A 0 as

Geometric partition function
We now wish to interprete curve-shrinking as a Wilsonian renormalization-group flow taking place in the N = 0 planar CVL sector. A partition function, defined as a statistical average (according to a suitably defined weight) over N = 0 CVLs, is to be left invariant under a decrease of the resolution determined by the flow parameter τ . Notice that, physically, τ is interpreted as a strictly monotonic decreasing (dimensionless) function of a ratio Q Q 0 where Q (Q 0 ) are mass scales associated with an actual (initial) resolution applied to the system.
To device a geometric ansatz for the effective action S = S[x(τ )], which is a functional of the curvex representable in terms of integrals over local densities in ξ (reparametrization invariance), the following reflection on symmetries is in order.
(i) scaling symmetryx → λx , λ ∈ R + : For both, λ → ∞ and λ → 0, implying λL → ∞ and λL → 0 at fixed L, the action S should be invariant under further finite rescalings (decoupling of the fixed length scale σ −1/2 ), (ii) Euclidean point symmetry of the plane (rotations, translations and reflections about a given axis): Sufficient but not necessary for this is a representation of S in terms of integrals over scalar densities w.r.t. these symmetries. That is, the action density should be expressible as a series involving products of Euclidean scalar products of ∂ n ∂ξ nx , n ∈ N + , or constancy. However, an exceptional scalar integral over a nonscalar density can be deviced. Consider the area A, calculated as The densityx · n in Eq. (9) is not a scalar under translations.
We now resort to a factorization ansatz as where in addition to Euclidean point symmetry In principle, infinitely many operators can be defined to contribute to F c . Since the evolution generates circles for τ ր T higher derivatives of k w.r.t. ξ rapidly converge to zero [7]. We expect this to be true also for Euclidean scalar products involving higher derivatives ∂ n ∂ξ nx . To yield conformally invariant expressions such integrals need to be multiplied by powers of √ A and/or L or the inverse of integrals involving lower derivatives. At this stage, we are not capable of constraining the expansion in derivatives by additional physical or mathematical arguments. To be pragmatic, we simply set F c equal to the isoperimetric ratio: We conceive the nonconformal factor F nc in S as a formal expansion in inverse powers of L. Since we regard the renormalization-group evolution of the effective action as induced by the flow of an ensemble of curves, where the evolution of each member is dictated by Eq. (2), we allow for an explicit τ dependence of the coefficient c of the lowest nontrivial power 1 L . In principle, this sums up the contribution to F nc of certain higher-power operators which do not exhibit an explicit τ dependence. Hence we make the following ansatz The initial value c(τ = 0) is determined from a physical boundary condition such as the mean lengthL at τ = 0 which determines the mean massm of a CVL as m = σL.
Since F c (τ ր T ) = 4π independence of the 'partition function' under the flow in τ implies that That is, F nc approaches constancy for τ ր T which brings us back to the conformal limit by a finite renormalization of the conformal part F c of the action. In this parameterization of S, c(τ ) can thus be regarded as an order parameter for conformal symmetry with mean-field critical exponent. We are interested in a situation where all curves in E M shrink to a point at the same value τ = T . Because of Eqs. (7) and (8) we thus demand that at τ = 0 all curves in E M initially have the same area A 0 . The effective action S in Eq. (10) (when associated with the ensemble E M we will denote it as S M ) is determined by the function c M (τ ), compare with Eq. (12), whose flow follows from the requirement of τ -independence of Z M : This is an implicit, first-order ordinary differential equation for c(τ ) which needs to be supplemented with an initial condition c 0,M = c M (τ = 0). A natural initial condition is to demand that the quantitȳ coincides with the geometric meanL M (τ = 0) defined as FromL M (τ = 0) =L M (τ = 0) a value for c 0,M follows. We also have considered a modified factor F nc (τ ) = 1 + c(τ ) A(τ ) in Eq. (10). In this case the choice of initial con-ditionL M (τ = 0) =L M (τ = 0) leads to F nc (τ ) ≡ 0. While the geometric effective action thus is profoundly different for such a modification of F nc (τ ) physical results such as the evolution of the variance of center-of-mass position agree remarkably well, see Sec. 4.2. That is, the geometric effective action itself is not a physical object. Rather, going from one ansatz for S M to another describes a particular way of redistributing the weight in the ensemble which seems to have no significant impact on the physics. This is in contrast to quantum field theory and conventional statistical mechanics where the action in principle is related to the physical properties of a given member of the ensemble.
For the curves depicted in Fig. 3 we make the convention that A 0 ≡ 2π × 100. It then follows that T = 100 by Eq. (8). The dependence c 2 M (τ ) is plotted in Fig. 4. According to Fig. 4 it seems that the larger the ensemble the closer c 2 M (τ ) to the evolution of a single circle of initial radius R = A 0 π . That is, for growing M the function c 2 M (τ ) approaches the form where the slope k M depends on the strength of deviation from circles of the representatives in the ensemble E M at τ = 0, that is, on the variance ∆L M at a given value A 0 . Physically speaking, the value τ = 0 is associated with a certain initial , are associated with the conditions at which the to-be-coarse-grained system is prepared. Notice that this interpretation is valid for the action S M = L(τ ) 2 only.

Statistical uncertainty of center of mass position
We are now in a position to compute the flow of a more local 'observable', namely, the mean 'center-of-mass' (COM) position in a given ensemble and the statistical variance of the COM position. The COM positionx COM of a given curvex(ξ, τ ) is defined as:x We will below present only results on the statistical variance of the COM position. Let as assume that at τ = 0 the ensembles E M are modified such that a translation is applied to each representative letting its COM position coincide with the origin. Recall that such a modification E M → E ′ M does not alter the (effective) action (Euclidean point symmetry). That is, at τ = 0 the statistical variance in the position of the COM is prepared to be nil, physically corresponding to an infinite resolution applied to the system by the measuring device. The mean COM positionx COM over ensemble E ′ M is defined aŝ The scalar statistical deviation ∆ M,COM ofx COM over the ensemble E ′ M is defined as where and similarly for the coordinate y. In Fig. 5 plots of ∆ M,COM (τ ) are shown when ∆ M,COM (τ ) is evaluated over the ensembles E ′ 3 , · · · , E ′ 8 with the action and subject to the initial conditionL M (τ = 0) =L M (τ = 0). In Fig. 6 the according plots of ∆ M,COM (τ ) are depicted as obtained with the action and subject to the initial conditionL M (τ = 0) =L M (τ = 0). In this case, one has c M (τ ) = −A(τ ) leading to equal weights for each curve in E ′ M .

Quantum mechanical versus statistical uncertainty
. Notice the rapid generation of an uncertainty in the COM position under the flow and its saturation when approaching the conformal limit τ ր T . There also is a saturation of this limiting value with a growing ensemble size.
. Notice the qualitative agreement with the results displayed in Fig. 5.
where H = p 2 2m is the freeparticle Hamiltonian, p is the spatial momentum, and a(τ ) ≡ a 0 1 + τ ma 2 0 2 . In agreement with Heisenberg's uncertainty relation one has during the process that ∆x∆p = 2 1 + τ ma 2 0 2 ≥ 2 . Time evolution in quantum mechanics and the process of coarse-graining in a statistical system describing planar CVLs share the property that in both systems the evolution generates out of a small initial position uncertainty (corresponding to a large initial resolution ∆p) a larger position uncertainty in the course of the evolution. Possibly, future development will show that interference effects in quantum mechanics can be traced back to the nonlocal nature of the degrees of freedom (CVLs) entering a statistical partition function.

Summary of present work
In this exploratory article an attempt has been undertaken to interprete the effects of an environment on 2D planar center-vortex loops, as they emerge in the confining phase of an SU(2) Yang-Mills theory, in terms of a Wilsonian renormalization-group flow carried by purely geometric entities. Our (mainly numerical) analysis uses established mathematics on the shrinking of embedded curves in the plane. In the case of nonintersecting CVLs (N = 0) the role of the environment is played by the entirety of all sectors with N > 0 and possibly an explicit environment. In a particular parametrization of the effective action we observe critical behavior as the limit of circular points is approached during the evolution. That is, planar N = 0 CVLs on average disappear from the spectrum for resolving powers smaller than a critical, finite value. Using this formalism to compute the evolution of the mean values of local observables, such as the center-of-mass position, a behavior is generated that qualitatively resembles the associated unitary evolution in quantum mechanics. We also have found evidence that this situation is practically not altered when changing the ansatz for the effective action.

Outlook on strongly correlated electrons in a plane
Let us conclude this article with a somewhat speculative outlook on planar N = 1 CVLs. Setting the Yang-Mills scale (mass of the intersection in the CVL) of the associated SU(2) theory equal to the electron mass, this soliton is interpreted as an electron or positron, see [1,15,16] for a more detailed discussion on the viability of such an assignment and Fig. 7 for a display. Important for our purpose are the facts that the two-fold directional degeneracy of the center flux represents the two-fold degeneracy of the spin projection and that a large class of curve deformations (shifts It is a remarkable fact that a high level of mathematical understanding exists for the behavior of curve-shrinking in a 2D plane [7,8] even in the case of one selfintersection [17]. Incidentally, 2D quantum systems exist in nature which exhibit unconventional behavior. Specifically, the phenomenon of high-T c superconductivity appears to be strongly related to the two-dimensionality of electron dynamics as it is enabled by rare-earth doping of cuprate materials [18]. Apparently, the Coulomb repulsion between the electrons moving in the would-be valence band within the cuprate planes of high-T c superconductors (Mott insulators) is effectively screened by the interplane environment also providing for the very existence of these electrons by doping. The question then is how the long-range order of electronic spins, which at given (optimal) doping and at a sufficiently low temperature leads to superconductivity, emerges within the cuprate planes. As it seems, quantum Monte Carlo simulations of a transformed Hubbard model (t − J model) subject to Gutzwiller projection yield quantitative explanations of a number of experimental results related to the existence of the pseudogap phase (Nernst effect, nonlinear diamagnetic susceptibility), see [19] and references therein.
We would here like to offer a sketch of an alternative approach to high-T c superconductivity being well aware of our ignorance on the details of present-day research in this field. The key idea is already encoded in Fig. 7. Namely, according to confining SU(2) Yang-Mills theory the electron is a nonlocal object with the physics of its charge localization being only loosely related to the physics of its magnetic moment (spin): The magnetic moment, carried by the core of the flux line, microscopically manifested by (oppositely) moving (opposite) electric charges, receives contributions from vortex sectors that are spatially far separated (on the scale of the diameter of the intersection point) from the location of the isolated electric charge. This suggests that in certain physical circumstances, where the ordering effect of interacting vortex lines becomes important, the postulate of a spinning point particle fails to x y Figure 8: An array of strongly correlated electrons in the plane possibly representing the superconducting state in a cuprate. The equally directed center flux in adjacent vortex sections provides for an attractive force (Ampere's law) at intermediate distances. For a given electron, out of six neighboring vortex sectors there are four sectors with attraction and two sectors with repulsion. At short distance there is repulsion since an overlap of CVL sections, leading to new intersection points each of mass ∼ m e , is topologically forbidden. Thus there is a typical equilibrium configuration contributing to long-range order in the 2D system. If the externally provided resolution (temperature) falls below a critical value then statistical fluctuations of the position of an intersection point relative to another one (the location of the electronic charge) will vanish. That is, electrons no longer can disperse energy provided by the heat bath (phonons, spin fluctuations) and thus provide for a 2D material free of electric resistivity. describe reality.
Concerning the strong correlations in 2D electron dynamics responsible for high-T c superconductivity we imagine a situation as depicted in Fig. 8. Each electron's spin in the plane interacts with the spin of its neighbors as follows. Equally directed electric fluxes (dually interpreted center fluxes of SU(2) e ) attract one another, and there is attraction for four out of six vortex sectors defined by the neighboring electrons (Ampere's law) while two vortex sectors experience repulsion 2 . Notice that for a given electron two of the adjacent electrons exhibit equal spin projection while four of the adjacent electrons have opposite spins. This is in agreement with the observation that high-T c superconductivity is an effect not related to s-wave Cooper-pair condensation.
An overlap of vortex sectors, hypothetically leading to the creation of extra intersection points, is topologically forbidden. That is, the fluctuations in the energy density of the system are far to weak to create an intersection point of mass m e = 511 keV. Therefore, at very small spatial separation repulsion must occur between adjacent vortex sectors. At a sufficiently low temperature and an optimal screening of Coulomb repulsion by the interplane environment (doping) this would lead to a typical equilibrium configuration as depicted in Fig. 8 where the intersection points (electronic charge) do not move relative to one another. A local demolition of this highly ordered state would cost a finite amount of energy manifested in terms of the (gigantic) gaps measured experimentally in the cuprate systems. Applying an electric field vector with a component parallel to the plane would set into resistivityfree motion the thus locked electrons. For a macroscopic analogue imagine a stiff table cloth being pulled over the table's surface in a friction-free fashion. The occurrence of the pseudogap phase would possibly be explained by local defects in the fabric of Fig. 8 due to insufficient Coulomb screening and/or too large of a thermal noise (macroscopic vorticity, liquid of pointlike defects in 2D).