Resolving the scales of the Yang-Mills theory by means of an extra dimension
aa r X i v : . [ h e p - l a t ] S e p Resolving the scales of the Yang-Millstheory by means of an extra dimension
Artan Boric¸ i a,ba
University of TiranaBlvd. King Zog ITiranaAlbania b Academy of Sciences of AlbaniaSq. Fan NoliTiranaAlbania
E-mail: [email protected] A BSTRACT : The Yang-Mills theory is part of the Standard Model of particle physics. Thelack of the mathematical understanding of the theory stands out in theoretical physics. Inorder to address this problem we observe that a recently proposed general model beyondthe Standard Model resolves the energy scales of a lattice regularized Yang-Mills theory bymeans of an extra dimension. The extra dimension ensures that all intermediate length scalesof the physical system are available by definition. In this paper we study the role of the extradimension also in the case of the free boson field. We find that if the extra dimension sizeis large, the model describes the classical motion of the system. In the opposite limit werecover its standard quantum mechanical motion without loss of information. Therefore, theHilbert space of states in the presence of the extra dimension describes physical phenomenaat all energy scales. This observation allows us to raise the description of length scales by anextra dimension at the level of a principle for the theories beyond the Standard Model, theonly modeling constraint being the correspondence principle. As it was shown recently, thefermion ground state energy of a gauge invariant Hamilton operator of Dirac fermions givesa particular lattice regularization of the Yang-Mills theory. Integration of gauge fields givesa pure fermion theory of color singlet fermion-antifermion pairs at each lattice site evolvingalong the extra dimension. Color confinement follows directly from this property. It allowsfor a saddle point solution in the limit of a large number of colors. In this paper we findthat the glueball spectrum of the Yang-Mills regularized theory is of the Hagedron type andbounded below by a positive value. We show also that the color charge is screened and thequark-antiquark potential is constant. ontents – 1 –
Introduction
The Yang-Mills theory is part of the Standard Model of particle physics. It is the basisof current understanding of the strong nuclear force. Its basic properties are asymptoticfreedom and confinement of color. While asymptotic freedom is mathematically proven closeto the continuum limit [1, 2], linear confinement is rigorously shown to hold in the strongcoupling limit of the Wilson regularized theory [3, 4]. Monte Carlo simulation of the latterhas established the linear confinement as well as the computation of physical quantities withinterest like masses of light hadrons and decay constants. While there are a lot of unansweredproblems, lattice simulations remain the direct non-perturbative tool of investigation. Aslong as there is no mathematical understanding the only way forward are larger computers.The ADS/CFT correspondence [5] offers the possibility to understand gravity from fieldtheory. Therefore, understanding the Yang-Mills theory is important for both strong forceand gravity.In order to address the solution of the theory we stay with the lattice regularization. Atheory with a cutoff in place offers a mathematically precise definition of its continuum limit.In addition, we observe that the length scales of the theory may be resolved in the frameworkof a general model beyond the Standard Model by means of an extra dimension [6]. The extradimension ensures that all intermediate length scales of the physical system are available bydefinition. In appendix A we give another example, the one of the free boson field. If theextra dimension has a large size, the model describes the classical motion of the system. Inthe opposite limit one recovers the standard quantum mechanical motion of the same system.However, the Hilbert space of states is unitarily inequivalent to the Hilbert space obtained inthe standard quantization. Therefore, the Hilbert space of states in the presence of the extradimension describes physical phenomena at all energy scales.These observations allow us to raise the extra dimension at the level of a principle dealingwith the length scales of a physical system for the theories beyond the Standard Model. Theonly modeling constraint is the correspondence principle, i.e. the Standard Model shouldbe derived by a theory which uses the extra dimension. In case of the Yang-Mills theory,the model adopted in reference [6] is a gauge invariant Hamilton operator of Dirac fermionsin d dimensions. There, it was shown that the fermion ground state energy of the model isa particular lattice regularization of the Yang-Mills theory. Moreover, it is was shown thatthe theory is written as a pure fermion theory of color singlet fermion-antifermion pairs ateach lattice site evolving along the extra dimension. This property allows for a saddle pointsolution in the limit of a large number of colors. While there is no modeling recipe for theother sectors of the Standard Model, fermions and gauge invariance appear to by necessaryingredients. We comment more on this matter at the end of the appendix A. – 2 –n the next section we give a short review the solution. In section 3 we discuss the prop-erties of the saddle-point effective theory. Equations of motion are derived, the propagator iscomputed in detail and the spectrum of the saddle point theory is studied. The Yang-Millstheory glueball spectrum and the quark-antiquark potential are computed in section 4. Inthe last section we summarize and discuss the results. In appendix A we discuss the extradimension in the case of a free boson. In appendix B we compute the ground state energy ofthe theory using the action formulation.
In this section we set up notations and review the solution of the theory discussed in reference[6]. We begin the description with the Hamilton operator.
The Hamilton operator on a lattice with d space-time dimensions and periodic boundaryconditions is given by the staggered version of the lattice Dirac operator [7] (more details aregiven in appendix B) ˆ H = m X x,c ˆΨ( x ) ∗ c ˆ γ ( x ) ˆΨ( x ) c + κ X x,c,c ′ µ ˆ γ ( x ) η µ ( x ) h ˆΨ( x ) ∗ c U µ ( x ) cc ′ ˆΨ( x + ˆ µ ) c ′ + ˆΨ( x + ˆ µ ) ∗ c U µ ( x ) ∗ cc ′ ˆΨ( x ) c ′ i , (2.1)where ˆΨ( x ) ∗ c , ˆΨ( x ) c are fermion creation and annihilation operators of site x and color com-ponent c = 1 , . . . , N , where ˆ γ ( x ) = ± depending on the parity of the lattice site x and η ( x ) = 1 , η µ ( x ) = ( − x + ··· + x µ − , µ = 2 , . . . , d . m is the fermion mass and U µ ( x ) cc ′ areSU(N) matrix elements at each directed link ( x, x + ˆ µ ) on the lattice. The coupling constantof the theory κ > is the strength of the hopping term. In the next subsection we deal withthe emerging Yang-Mills theory from this formulation. The Yang-Mills theory is derived from the Hilbert space trace Z F ( U ) = tr e − N τ ˆ H , (2.2)where N τ is the size of extra dimension. The effective action of the pure gauge theory is theground state energy of the fermionic theory S eff ( U ) = c o N τ − c N τ κ X µν Tr U µ U ν U ∗ µ U ∗ ν + O ( κ ) + h.c. , (2.3) Matrices of the complex Gaussian ensemble will do as well. See appendix A of [6]. – 3 –here the trace is taken in the tensor product space of the lattice sites and the SU(N) group, c o is real, c = 1 / and U µ are the hopping matrices (see appendix B for the derivation) ( U µ ) xy ; cc ′ = U µ ( x ) cc ′ δ x +ˆ µ,y . (2.4)The first non-trivial term is the Wilson plaquette action if the length of the extra dimensionis fixed by the relation N τ = 4 κ δ , (2.5)where δ is a positive integer. The rest of the terms are larger Wilson loops, which make theaction long ranged. However, the contribution of the Wilson loop of length n decreases as apower of κ n − − δ , n = 2 , , . . . . Since the series converges for κ ≤ / (2 d − , the theory islocal in the weak coupling limit. In this limit larger Wilson loops will not change the essenceof the Yang-Mills theory. Asymptotic freedom and color confinement are properties of thetheory.The value of δ should be chosen such that only the Wilson term survives continuumlimit. In section 4 we show that Wilson loops are analytic functions of κ . In particular themean value of the first term (see eq. (4.10)) X µν h Tr U µ U ν U ∗ µ U ∗ ν i = w o − w κ + O ( κ ) , (2.6)where w o and w are positive, gives a finite Wilson term if we set δ = 2 . This way, thecoupling constant κ coincides, in the weak coupling limit, to the one of the Wilsonian theoryand the length of extra dimension is fixed to the value N τ = 4 κ (2.7)Note however that since the theory is solved by integrating first the gauge field the value of δ does not effect the structure of Green’s functions of the theory. It does however effect thelevel spacing of the spectrum. Nonetheless, the value δ = 1 set in reference [6] is in errorsince with this value we may not relate the solution of the theory to that of the Yang-Millstheory.The Dirac theory defined in equation (2.1) is a theory beyond the Standard Model whereforces are absent. The emerging Yang-Mills theory is a general example that the fermionground state induces a holonomy of gauge fields. The idea may be tested experimentallyusing the technique of trapped ions. If we put, for example, four fermions in a ring topologyand gauge fields are U (1) phase factors, we may define the Hamilton operator H θ = c ∗ e iθ c + c ∗ e iθ c + c ∗ e iθ c + c ∗ e iθ c + h.c. , (2.8)– 4 –here c ∗ j , c j , j = 1 , , , are fermion operators that satisfy anicommutation relations { c j , c ∗ l } = δ jl and θ j , j = 1 , , , are the phases of the U (1) field. In this case we expect the emergenceof a magnetic field perpendicular to the plane containing the ring. The exact experimentalprocedure is outside the scope of this paper. Next, we introduce the action of the theory. The action of the theory is I = X x,τ,τ ′ ,c ¯ ψ ( x, τ ) c h mδ τ,τ ′ + ˆ γ ( x ) ˆ ∂ t ( τ, τ ′ ) i ψ ( x, τ ′ ) c + κ X x,τ,µ,c,c ′ η µ ( x ) (cid:2) ¯ ψ ( x, τ ) c U µ ( x ) cc ′ ψ ( x + ˆ µ, τ ) c ′ − ¯ ψ ( x + ˆ µ, τ ) c U µ ( x ) ∗ cc ′ ψ ( x, τ ) c (cid:3) , (2.9)where τ labels lattice sites along the extra dimension, ψ ( x, τ ) c , ¯ ψ ( x, τ ) c are fermion fieldswith color index c . They satisfy antiperiodic boundary conditions in τ , whereas ˆ ∂ τ is thelattice derivative, in our case, the symmetric differences matrix ˆ ∂ τ ( τ, τ ′ ) = 12 ( δ τ +1 ,τ ′ − δ τ − ,τ ′ ) . (2.10)Integration of gauge fields in the small κ regime gives the pure fermion action in terms ofcolor singlet fermion-antifermion pairs S = X x,t,t ′ ,a ¯ ψ a ( x, t ) h mδ t,t ′ + ˆ γ ( x ) ˆ ∂ t ( t, t ′ ) i ψ ( x, t ′ ) a + N X x,µ,t F " − κ N X t ′ ,a,b ¯ ψ ( x, t ) b ψ ( x, t ′ ) b ¯ ψ ( x + ˆ µ, t ′ ) a ψ ( x + ˆ µ, t ) a . (2.11)Therefore, color confinement is trivial in this formulation. We use the first order expansionof F ( . ) and have S = X x,τ,τ ′ ,c ¯ ψ ( x, τ ) c h mδ τ,τ ′ + ˆ γ ( x ) ˆ ∂ t ( τ, τ ′ ) i ψ ( x, τ ′ ) c + κ N X x,µ,τ,τ ′ ,c,c ′ ¯ ψ ( x, τ ) c ψ ( x, τ ′ ) c ¯ ψ ( x + ˆ µ, τ ′ ) c ′ ψ ( x + ˆ µ, τ ) c ′ . (2.12)Bosonization of fermions with the field Σ( x, τ, τ ′ ) gives the action S Σ = N X x,τ n ln h m + ˆ γ ( x ) ˆ ∂ τ + Σ( x ) io ( τ, τ ) − N κ X x,y,τ,τ ′ Σ( x, τ, τ ′ )( A − )( x, y )Σ( y, τ ′ , τ ) , (2.13) Gauge invariance along the extra dimension gives another model which deserves a separate study. As it is explained in [6], the leading term dominates the solution. – 5 –here A is the matrix A ( x, y ) = P µ ( δ x +ˆ µ,y + δ x − ˆ µ,y ) . In the following we recall Green’sfunction equalities of the theory. The Green’s functions of the theory are given by the expression h ψ ( y, τ ′ ) a ¯ ψ ( x, τ ) b i I = G ( x, τ, τ ′ ) δ xy δ ab , (2.14)where G ( x, τ, τ ′ ) is the solution of saddle point equations G − ( x, τ ′ , τ ) = mδ τ,τ ′ + ˆ γ ( x ) ˆ ∂ t ( τ, τ ′ ) + κ X µ [ G ( x + ˆ µ, τ ′ , τ ) + G ( x − ˆ µ, τ ′ , τ )] . (2.15)The Σ -field solution is Σ( x, τ ′ , τ ) = κ X µ [ G ( x + ˆ µ, τ ′ , τ ) + G ( x − ˆ µ, τ ′ , τ )] . (2.16)Note that the two-point fermion correlators are color singlets, i.e. they are invariant undercolor group transformations. They show that fermions do not propagate in space-time irre-spective of their mass. This way, the net effect of the gauge field in a gauge invariant Diractheory is to turn fermions static and colors into degenerate flavors of fermion-antifermionpairs. Two-point functions G ( x, τ, τ ′ ) are space-time local fields as well as correlators alongthe extra dimension. The four-point function of the theory is defined by the equation h ψ ( x, τ ) a ¯ ψ ( x, τ ′ ) a ψ ( y, τ ′ ) b ¯ ψ ( y, τ ) b i I = G ( x, τ ′ , τ ) G ( y, τ, τ ′ )(1 − δ xy δ ab ) . (2.17)There are other expressions for various combinations of space, time and color. Like thetwo-point function the four-point function is a color singlet quantity. It vanishes identicallyfor equal spatial lattice sites and equal color. Therefore, the theory does not allow for twofermion-antifermion pairs of identical color at the same place. Next we recall the effectivefield solution of the theory. In order to solve the Green’s function equalities one starts by Fourier transforming G ( x, τ, τ ′ ) in time and get the local field ˆ G ( x, ω ) at each lattice site x for each frequency ω . Then, thesolution is split in two pieces ˆ G ( x, ω ) = e − iθ ( x,ω ) h ˜ G o ( ω ) + ˜ G ( x, ω ) i , θ ( x, ω ) = arg [ m + ˆ γ ( x ) i sin ω ] (2.18) In [6] it was derived for for x = y . This slight generalization uses two Wick contractions. – 6 –here ˜ G o ( ω ) is the site-independent piece of the solution, whereas ˜ G ( x, ω ) is a fluctuation.The substitution in the saddle point equations (2.15) gives G o ( ω ) + ˜ G ( x, ω ) = µ ( ω ) + 2 dκ ˜ G o ( ω ) + κ X µ [ ˜ G ( x + ˆ µ, ω ) + ˜ G ( x − ˆ µ, ω )] (2.19)with µ ( ω ) = | m + i sin ω | . (2.20)The field ˜ G ( x, ω ) is taken to be small compared to ˜ G o ( ω ) and one finds [6] G o ( ω ) = µ ( ω ) + 2 dκ ˜ G o ( ω ) ⇒ ˜ G o ( ω ) = − µ ( ω ) + p µ ( ω ) + 8 dκ dκ (2.21)as well as the linear equations ˜ G ( x, ω ) + κ ˜ G o ( ω ) X µ [ ˜ G ( x + ˆ µ, ω ) + ˜ G ( x − ˆ µ, ω )] = 0 . (2.22)These equations are valid as long as the neglected terms in the quadratic equations (2.19) aresmall. The effective action of the field ˜ G ( x, ω ) is quadratic[6] S ˜ G = N κ X x,ω M ( ω ) ˜ G ( x, ω ) + N κ X x,µ,ω h ˜ G ( x + ˆ µ, ω ) − ˜ G ( x, ω ) i , (2.23)where M ( ω ) = 1 κ ˜ G o ( ω ) − d = µ ( ω ) + µ ( ω ) p µ ( ω ) + 8 dκ κ (2.24)are the masses of the scalar fields. In the next section we study the properties of the effectivefield theory. We start with the derivation of the equations of motion of the theory.
The linear equations (2.22) furnish the equation of motion for the field ˜ G ( x, ω ) . Its Fourierspace counterpart reads " κ ˜ G o ( ω ) X µ cos q µ ˜ G F ( q, ω ) = 0 , (3.1)– 7 –here ˜ G F ( q, ω ) is the Fourier transformed field of ˜ G ( x, ω )˜ G ( x, ω ) = 1 V X p ˜ G F ( p, ω ) e ipx . (3.2)This is a free scalar field on the lattice. Its momentum space expression is given by ˜ G F ( p, ω ) = ˜˜ G F ( p, ω ) δ f ( p,ω ) , , (3.3)where ˜˜ G F ( p, ω ) is a regular function of momenta and frequency and f ( p, ω ) = 0 is thedispersion relation of the theory. Its explicit form is κ ˜ G o ( ω ) X µ cos q µ = 0 . (3.4)However, the dispersion relation deriving from the effective action (2.23) is − κ ˜ G o ( ω ) X µ cos p µ = 0 . (3.5)Therefore, we have two dispersion relations. Note that (3.4) may be derived from (3.5) ifwe take the origin of the momenta at the ( π, π, . . . , π ) corner of the Brillouin zone. Sincethere are two solutions for each site parity (see (2.18)), we have a total of four solutions. Theorigin of this degeneracy is the staggered fermion formulation used in this model. Staggeredfermions describe four degenerate flavors of fermions. In the following we compute thepropagator of the theory.
We begin by writing the action (2.23) in terms of the new field ˜ ϕ ( x, ω ) = ˜ G o ( ω ) − ˜ G ( x, ω ) . (3.6)It reads S ˜ ϕ = N X x,y,ω " δ x,y − κ ˜ G o ( ω ) X µ ( δ x +ˆ µ,y + δ x − ˆ µ,y ) ˜ ϕ ( x, ω ) ˜ ϕ ( y, ω ) . (3.7)The propagator of this field is h ˜ ϕ ( x, ω ) ˜ ϕ (0 , ω ) i S ˜ ϕ = 1 N V X p e ipx − κ ˜ G o ( ω ) P µ cos p µ = (cid:2) M ( ω ) + 2 d (cid:3) N V X p e ipx M ( ω ) + 2 P µ (1 − cos p µ ) (3.8) Had we used Wilson fermions we would have had one solution. – 8 –ith x µ = 1 , , . . . , N µ , µ = 1 , , . . . , d . In the infinite volume limit it is the integral ˜ C ( x, ω ) = 1 N (cid:2) M ( ω ) + 2 d (cid:3) Z d d p (2 π ) d e ipx M ( ω ) + 2 P µ (1 − cos p µ ) . (3.9)We use two approaches to compute the propagator as a function of the lattice sites. Residue theorem
The standard method to compute the propagator is the residue theorem. Introducing themomentum dependent mass ν ( ~p, ω ) = M ( ω ) + 2 d − X µ =1 (1 − cos p µ ) (3.10)the propagator is written in the form ˜ C ( ~x, x d , ω ) = 1 N (cid:2) M ( ω ) + 2 d (cid:3) Z d d − p (2 π ) d − e i~p ~x Z dp d π e ip d x d ν ( ~p, ω ) + 2(1 − cos p d ) . (3.11)Using the Feynman contour for the integral over p d for x d > we get ˜ C ( ~x, x d , ω ) = 1 N (cid:2) M ( ω ) + 2 d (cid:3) Z d d − p (2 π ) d − e i~p ~x − ˜ ν ( ~p,ω ) x d p ν ( ~p, ω ) + 4 (3.12)with e ˜ ν ( ~p,ω ) = ν ( ~p, ω ) + p ν ( ~p, ω ) + 42 . (3.13)For ~x = 0 , x d = R and since ν ( ~p, ω ) ≥ M ( ω ) we get the upper bound ˜ C (0 , . . . , , R, ω ) ≤ N M ( ω ) + 2 d p M ( ω ) + 4 e − ˜ M ( ω ) R (3.14)with e ˜ M ( ω ) = M ( ω ) + p M ( ω ) + 42 . (3.15)For small M ( ω ) and large R the integral is dominated by small momenta. In this case itsvalue approaches the upper bound with ˜ M ( ω ) ≈ M ( ω ) and M ( ω ) ≈ and therefore wehave ˜ C (0 , . . . , , R, ω ) ≈ dN e − M ( ω ) R . (3.16)In the following, we give another expression valid in the large mass limit, which is also thecase of the vanishing κ limit. – 9 – odified Bessel functions of the first kind We write the denominator of (3.9) as an exponential integral ˜ C ( x, ω ) = 1 N (cid:2) M ( ω ) + 2 d (cid:3) Z ∞ du e − M ( ω ) u d Y µ =1 Z π − π dp µ π e ip µ x µ − u (1 − cos p µ ) . (3.17)The integral over momenta in the right hand side of (3.17) yields ˜ C ( x, ω ) = 1 N (cid:2) M ( ω ) + 2 d (cid:3) Z ∞ du e − [ M ( ω ) +2 d ] u d Y µ =1 I x µ [2 u ] , (3.18)where the modified Bessel functions of the first kind I x µ (2 u ) = u x µ +2 · x µ + 0)! + u x µ +2 · x µ + 1)! + u x µ +2 · x µ + 2)! · · · (3.19)have been used. In the large mass M ( ω ) limit the leading term dominates and we find ˜ C ( x, ω ) = 1 N h M ( ω ) +2 d i x + x + ··· + x d x ! x ! . . . x d ! ( x + x + · · · + x d )! . (3.20)As expected, the lattice propagator is not rotationally invariant. However, if we set x = x = · · · = x d = R we can get an expression in the equal coordinates case. Using theapproximation x ! ≈ √ πx x x e − x we have ˜ C ( R, . . . , R, ω ) = √ dN e − Rd ln (cid:20) M ( ω )2 d (cid:21) (2 πR ) d − . (3.21)In terms of r = R √ d we find ˜ C (cid:18) r √ d , . . . , r √ d , ω (cid:19) = d d +14 N e − r √ d ln (cid:20) M ( ω )2 d (cid:21) (2 πr ) d − . (3.22)Another useful case is when x = x = · · · = x d − = 0 and x d = R are substituted in (3.20) ˜ C (0 , . . . , , R, ω ) = 1 N (cid:20) M ( ω ) + 2 d (cid:21) R . (3.23)Using (2.24) in the vanishing κ limit we have M ( ω ) + 2 d ≃ µ ( ω ) κ and therefore ˜ C (0 , . . . , , R, ω ) ≃ N (cid:20) κ µ ( ω ) (cid:21) R . (3.24)We get the same result from the expression (3.21) specialized in the vanishing coupling limitand d = 1 . In the next section we compute the mass spectrum of the saddle point theory.– 10 – .3 Hagedorn spectrum Zero momentum energies are computed by averaging the propagator (3.9) over space coor-dinates ˜˜ C ( x d , ω ) = 1 N N . . . N d − X x ,x ,...,x d − ˜ C ( x, ω ) . (3.25)In the infinite volume limit, the right hand side is the one-dimensional integral ˜˜ C ( x d , ω ) = 1 N Z π − π dp d π e ip d x d − d − κ ˜ G o ( ω ) − κ ˜ G o ( ω ) cos p d . (3.26)Using the residue theorem and the Feynman contour as in the previous subsection one has ˜˜ C ( x d , ω ) = 1 N M ( ω ) + 2 d p M ( ω ) + 4 e − ˜ M ( ω ) x d . (3.27)The spectrum is massive with spectral gap ˜ M (0) = 2 ln " M (0) + p M (0) + 42 . (3.28)For vanishing coupling the spectrum has the form (see (2.24)) ˜ M ( ω ) = ln (cid:18) m + sin ωκ (cid:19) . (3.29)Defining the mean exponential mass e M = m + sin ωκ = m + κ (3.30)and since N τ = 4 /κ we find that the number of states grows exponentially with MN τ ∼ e M , (3.31)i.e. the mass spectrum is of the Hagedorn type. In the next section we discuss the propertiesof the Yang-Mills theory. In this section we study the main properties of the Yang-Mills theory, the glueball spectrumand the quark-antiquark potential. Our calculations are based on the equivalent insertions inthe path integral [6] U µ ( x ) ab ←→ X τ ψ ( x, τ ) a ¯ ψ ( x + ˆ µ, τ ) b (4.1)– 11 –s well as the use of Green’s functions of the theory. For example, taking the expectationvalue of the right hand side and using the definition of Green’s functions (2.14) one has X τ (cid:10) ψ ( x, τ ) a ¯ ψ ( x + ˆ µ, τ ) b (cid:11) = δ ˆ µ, δ ab X τ G ( x, τ, τ ) . (4.2)Since δ ˆ µ, = 0 one finds h U µ ( x ) ab i = 0 , which is the Elitzur theorem. Before studyingglueballs we go into more examples involving gauge-invariant operators, such as plaquette,Polyakov loop and the Polyakov loop correlator. This subsection illustrates in some detail calculations involving the plaquette h P µν ( x ) i = 1 N X a,b,c,d h U µ ( x ) ab U ν ( x + ˆ µ ) bc U µ ( x + ˆ ν ) ∗ cd U ν ( x ) ∗ da i . (4.3)Using the fermion-antifermion insertion (4.1) in the path integral we define the plaquette interms of the fermion theory h P µν ( x ) i = c P N N τ X τ τ τ τ ,abcd (cid:10) ψ ( x, τ ) a ¯ ψ ( x + ˆ µ, τ ) b ψ ( x + ˆ µ, τ ) b ¯ ψ ( x + ˆ µ + ˆ ν, τ ) c ψ ( x + ˆ µ + ˆ ν, τ ) c ¯ ψ ( x + ˆ ν, τ ) d ψ ( x + ˆ ν, τ ) d ¯ ψ ( x, τ ) a (cid:11) , (4.4)where c P is the plaquette normalization constant. Note that the normalization of Wilsonloops computed in this way is arbitrary. We set c P to a value which gives h P µν ( x ) i = 1 inthe continuum limit. Using (2.14) and taking the Fourier transform in the extra dimensionone has h P µν ( x ) i = c P N τ X ω D ˆ G ( x, ω ) ˆ G ( x + ˆ µ, ω ) ˆ G ( x + ˆ µ + ˆ ν, ω ) ˆ G ( x + ˆ ν, ω ) E , (4.5)where the last expectation is taken with respect to the effective saddle point theory (3.7).Using the solution Ansatz (2.18) and the definition of the scalar field ˜ ϕ (3.6) one writes h P µν ( x ) i = c P N τ X ω ˜ G o ( ω ) h [1 + ˜ ϕ ( x, ω )] [1 + ˜ ϕ ( x + ˆ µ, ω )] [1 + ˜ ϕ ( x + ˆ µ + ˆ ν, ω )] [1 + ˜ ϕ ( x + ˆ ν, ω )] i = c P N τ X ω ˜ G o ( ω ) h C (ˆ µ, ω ) + 2 ˜ C (ˆ µ + ˆ ν, ω ) i , (4.6)– 12 –here the definition (3.9) of infinite volume boson propagators has been used. In the lead-ing order of the large N expansion and vanishing coupling constant propagators may beneglected and we get h P µν ( x ) i = c P N τ X ω (cid:20) µ ( ω ) − dκ µ ( ω ) + O ( κ ) (cid:21) . (4.7)Computing the sums by definite integrals N τ X ω µ ( ω ) k → Z π − π dω π ω ) k , k = 2 , , (4.8)the expression takes the form h P µν ( x ) i = c P (cid:20) √ − dκ √ O ( κ ) (cid:21) . (4.9)Setting c P = √ the mean plaquette is h P µν ( x ) i = 1 − dκ O ( κ ) . (4.10)In the next subsection we discuss the Polyakov loop. The Polyakov loop is computed in reference [6]. Here we extend it in the next to leadingorder of the large N expansion. In terms of Green’s functions of the theory we define it to be P ( ~x ) = X ω * N d Y x d =1 ˆ G ( ~x, x d , ω ) + , (4.11)where ~x = ( x , x , . . . , x d − ) and N d is the length of the Euclidean time in lattice spacingunits. Substituting the solution Ansatz (2.18) and using the definition of the scalar field ˜ ϕ (3.6) we have P ( ~x )= X ω ˜ G o ( ω ) N d h [1 + ˜ ϕ ( ~x, , ω )] [1 + ˜ ϕ ( ~x, , ω )] · · · [1 + ˜ ϕ ( ~x, N d , ω )] i = X ω ˜ G o ( ω ) N d h N d −
1) ˜ C (0 , , ω ) + · · · + 1 ˜ C (0 , N d − , ω ) + O (cid:16) ˜ C (cid:17)i . (4.12)– 13 –ince boson propagators fall off exponentially we stay with the next to leading term ( N d −
1) ˜ C (0 , , ω ) in the right hand side. Substituting ˜ C (0 , , ω ) from (3.24) and for large N d weget P ( ~x ) = X ω ˜ G o ( ω ) N d (cid:20) N d N κ µ ( ω ) (cid:21) . (4.13)In the large N d limit the sum is dominated by a few low frequencies. We approximate P ( ~x ) ≈ G o (0) N d (cid:18) N d κ N (cid:19) , (4.14)where the factor two comes from the doubler. The free energy of the static quark is definedfrom the large N d exponential fall off of the Polyakov loop aF o = − ln ˜ G o (0) − N d ln 2 − N d ln (cid:18) N d κ N (cid:19) , (4.15)where we have reintroduced the lattice spacing. Substituting ˜ G o (0) and sending N d to infinitywe get aF o = 2 dκ + O ( κ ) . (4.16)We take the continuum limit of the theory by keeping the free energy fixed as the couplingconstant goes to zero. The renormalization group beta function [6] β ( κ ) = − a dκda = a ( ∂F o /∂a )( ∂F o /∂κ ) = − κ O ( κ ) (4.17)shows that the theory is asymptotically free. In the following we study the glueball spectrumof the Yang-Mills theory. Glueball states are computed from the decay of the plaquette correlation functions. We studythe correlations of the scalar glueball operator S ( x d ) = X ~x,kl P kl ( ~x, x d ) , (4.18)where the sum is over all positions of the space-like plaquette. The connected correlator isdefined by the expression h S ( x d ) S (0) i c = X ~x,~y,kl,mn h P kl ( ~x, x d ) P mn ( ~y, i c = X ~x,kl h P kl ( ~x, x d ) P kl ( ~x, i c . (4.19)– 14 –n terms of Green’s functions it is written in the form h S ( x d ) S (0) i c = X ω,~x,kl D ˆ G ( ~x, x d , ω ) ˆ G ( ~x + ˆ k, x d , ω ) ˆ G ( ~x + ˆ k + ˆ l, x d , ω ) ˆ G ( ~x + ˆ l, x d , ω )ˆ G ( ~x, , ω ) ˆ G ( ~x + ˆ k, , ω ) ˆ G ( ~x + ˆ k + ˆ l, , ω ) ˆ G ( ~x + ˆ l, , ω ) E c . (4.20)Using the solution Ansatz (2.18) and evaluating Wick’s contractions of the scalar field ˜ ϕ (3.6)we find the leading term correlator h S ( x d ) S (0) i c ∝ X ω ˜ G o ( ω ) x d C (cid:16) ~ , x d , ω (cid:17) . (4.21)Using (3.24) the glueball spectrum is given by studying the exponential decay of each termof the right hand side aM g ( ω ) = 8 dκ + O ( κ ) + 1 x d ln N µ ( ω ) κ , (4.22)where we have reinstated the lattice spacing. In the large x d limit and vanishing coupling itis aM g ( ω ) = ln (cid:18) m + sin ωκ (cid:19) . (4.23)The lightest glueball mass aM g (0) = ln m κ (4.24)is a twice degenerate state at ω = 0 and ω = π . The mass gap of the Yang-Mills theory isinfinitely large in the continuum limit. The same argument as in the end of subsection 3.3shows that the mass spectrum of the theory is of the Hagedorn type. In the following wecompute the quark-antiquark potential. We compute the energy of a quark-antiquark pair from the decay of Polyakov loop correlatorslocated at ~x = (0 , . . . , and ~y = (0 , . . . , R ) C P ( R, N d ) = X ω * N d Y x d =1 ˆ G (0 , . . . , , R, x d , ω ) N d Y y d =1 ˆ G (0 , . . . , , , y d , ω ) + c (4.25)where the subscript indicates the connected correlator. Using the solution Ansatz (2.18) andthe definition of the scalar field ˜ ϕ (3.6) we have C P ( ~x ) = X ω ˜ G o ( ω ) N d h [1 + ˜ ϕ (0 , . . . , , R, , ω )] · · · [1 + ˜ ϕ (0 , . . . , , R, N d , ω )][1 + ˜ ϕ (0 , . . . , , , , ω )] · · · [1 + ˜ ϕ (0 , . . . , , , N d , ω )] i c . (4.26)– 15 –he leading term of the right hand side is C P ( R, N d ) ≈ X ω ˜ G o ( ω ) N d N d ˜ C (0 , . . . , , R, , ω ) . (4.27)For large N d the sum is dominated by the vanishing frequency term and its doubler at ω = π .Using (3.24) we have C P ( R, N d ) ∝ ˜ G o (0) N d N d N κ R . (4.28)Then, the quark-antiquark potential follows from the large N d limit of the exponential decayof the correlator aV ( R ) = − N d ln C P ( R, N d ) ∝ dκ + O ( κ ) + 1 N d ln NN d + RN d ln 1 κ . (4.29)Sending N d to infinity the continuum limit potential is twice the free energy of the staticquark V ( R ) = 2 F o . (4.30)In the following we summarize and discuss the results obtanied in this paper. We have shown that the Yang-Mills theory, as regularized in this paper, has a positive massgap in the continuum limit. We have shown also that the number of glueball states growsexponentially with the glueball mass. The quarks are screened and the quark-antiquark po-tential is constant. These conclusions are the consequence of the solution of the theory in thelimit of large number of colors, as reviewed in section 2.Color confinement follows from the color singlet property of pure fermion action of thetheory. There is no linear confinement in the continuum limit. The string tension is zero inthe infinite volume limit. Monte Carlo data with the Wilson regularization show that linearconfinement is a property of the theory. In two dimensions, the exact solution of Gross andWitten with the Wilson action gives also a non-zero tring tension [8]. The discrepancy maybe explained by the presence of larger Wilson loops in the action: although vanishingly smallin the continuum limit, they destroy the area law in the same way as dynamical quarks screenthe linear potential of the Wilson action in the infinite volume QCD.We studied also the continuum limit of the free energy of the static quark. The result doesnot change if one considers the quark-antiquark potential. These quantities give direct accessto the coupling constant of the theory. We find that the renormalization group beta functionvanishes linearly with κ , unlike the κ behavior of the asymptotic perturbation theory of theWilson theory. However, the exact solution with the Wilson action at d = 2 shares the linear– 16 –ehavior of the beta function in the vanishing coupling limit. At d = 4 , there are no MonteCarlo data with the Wilson action at vanishingly small couplings.The present regularization of the Yang-Mills theory in d dimensions is the ground state ofa Dirac theory beyond the Standard Model. The net effect of the gauge field is a color singletfermion-antifermion theory in d + 1 dimensions. These pairs, which are elementary fermionsfixed at each lattice, form a color singlet composite field condensate. The fluctuations ofthe latter propagate in space-time and may be observed. The color confinement observed inNature may be described in terms of such a fermion-antifermion condensate. The spectrumof the fluctuation field is the glueball spectrum of the Yang-Mills theory. The Hagerdorn typeof the spectrum, which is otherwise obscure in the Wilson regularization, is an evidence thatthe model with an extra dimension resolves the multiple scales of the Yang-Mills theory. Acknowledgement
I thank my wife Mirela for frequent discussions related to this research and especially for thequestions related to the physical meaning of the extra dimension. I thank Philippe de For-crand for sending his comments on the first draft of the paper. Special thanks go to MichaelCreutz for the correspondence related to various drafts of the paper and useful suggestionsregarding multiple scales and the role of the extra dimension.
A Multiple scales of a free boson
The size of the extra dimension used in the regularization of the Yang-Mills theory is fixedby the coupling constant value. According to the renormalization group beta function, eachvalue of the coupling constant corresponds to a given length scale. The evolution of the sys-tem along the extra dimension is the evolution to reach that scale. In doing so, the systemsteps along all intermediate scales and accesses all energy scales. Therefore, the extra di-mension represents the length scales of the physical system. In this section we study anotherexample, the free boson field. The field operator ˆ φ ( x ) is a complex valued operator definedon lattice sites x of a one dimensional lattice with periodic boundary conditions. Its Fourierspace representation is ˆ φ ( x ) = 1 √ N N X k =1 a k e ip k x , (A.1)where ladder operators satisfy commutation relations [ a k , a ∗ k ′ ] = δ kk ′ , k, k ′ = 1 , , . . . , N , (A.2)– 17 – is the number of lattice sites and p k are the lattice momenta p k = πkN , k = 1 , , . . . , N .The Hamilton operator reads H = 12 N X x =1 h φ ( x ) ∗ ˆ φ ( x ) − ˆ φ ( x ) ∗ ˆ φ ( x + 1) − ˆ φ ( x + 1) ∗ ˆ φ ( x ) i + n ˆ φ ( x ) ↔ ˆ φ ( x ) ∗ o . (A.3)In terms of ladder operators it is written in the form H = E o + 2 N X k =1 (1 − cos p k ) a ∗ k a k . E o = N X k =1 (1 − cos p k ) . (A.4)Operators a k annihilate the ground state with energy E o . The system is coupled to the extradimension of size N τ . The partition function of the theory is Z ( N τ ) = Tr H e − N τ H , (A.5)where the trace is taken in the Hilbert space of states H . While in the interacting theories,the size of the extra dimension is related to the interaction strength, in a free theory it is anoverall coupling. In the large N τ limit the right hand side is dominated by the ground state,whereas in the small N τ limit all energy states contribute. In the following we show theconsequences of this property in the Green’s functions of the theory.We define a complex valued field φ ( x, τ ) on a two-dimensional lattice with periodicboundary conditions. The action of the theory is S = 12 X xt [ φ ( x, τ ) ∗ φ ( x, τ + 1) − φ ( x, τ + 1) ∗ φ ( x, τ )]+ X xt [2 φ ( x, τ ) ∗ φ ( x, τ ) − φ ( x, τ ) ∗ φ ( x + 1 , τ ) − φ ( x + 1 , τ ) ∗ φ ( x, τ )] . (A.6)We are interested in the infinite volume limit propagator h φ ( x, τ ) φ (0 , ∗ i = Z π − π dp π Z π − π dω π e iωτ + ipx i sin ω + 2(1 − cos p ) . (A.7)The first integral may be computed using the residue theorem for τ > . We find h φ ( x, τ ) φ (0 , ∗ i = Z π − π dp π e ipx − τ (1 − cos p ) p − cos p ) . (A.8)As τ → ∞ the second integral allows the saddle point evaluation. In this case, low momentadominate and the integral becomes Gaussian h φ ( x, τ ) φ (0 , ∗ i ≈ Z ∞−∞ dp π e ipx − τp = 1 √ πτ e − x τ = 1 √ πτ − x √ πτ + O (cid:18) x √ τ (cid:19) . (A.9)– 18 –his result shows that, in the large τ limit, the standard deviation of the field h φ (0 , τ ) φ (0 , ∗ i ≈ √ πτ (A.10)is vanishingly small. In this limit, the expression on the right hand side of (A.9) showsalso that the standard deviation of the derivative dφdx exists and vanishes in the large τ limit.Therefore, in the ground state, i.e. in the large N τ limit, the pair of functions (cid:8) φ , dφdx (cid:9) may beused to define a classical theory, where x has the meaning of physical time. This property islost in the small N τ limit. For example, taking N τ = 1 and since ω l = πlN τ , l = 1 , , . . . , N τ ,we have ω = 2 π . The propagator in this case is h φ ( x, φ (0 , ∗ i N τ =1 = Z π − π dp π e ipx − cos p ) = 1 , (A.11)where the result is obtained using the residue theorem for x > . Although the derivative dφdx is well defined and has vanishing standard deviation, the field standard deviation is finite. Synthesis
In the large N τ limit the free boson behaves classically. In the opposite limit we get theexpected behavior of the free field theory in one dimension, which describes a free quantummechanical particle. In the formulation studied here, the motion of a free particle is classicalif it is in the ground state. If it accesses all energy levels, which are otherwise obscure andnot modeled when the extra dimension is missing, its motion is quantum mechanical. There-fore, classical and quantum mechanical behaviors of the same physical system are related bya unitary evolution along the analytically continued extra dimension. However, the Hilbertspace of a free particle is spanned by plane waves, whereas the Hilbert space of the bosonfield is the product of harmonic oscillator Hilbert spaces. These are unitarily inequivalentspaces which share the same Green’s function in the quantum regime of the latter space.Nonetheless, the free boson Hilbert space describes physical phenomena at all energy scales.At low energies the system behaves classically, whereas at high energies quantum mechan-ically. Therefore, for the system studied here, classical mechanics is a well defined limit ofquantum mechanics without loss of information.In principle, all other sectors of the Standard Model may be derived using the physicalmodel in 3+1+1 dimensions. As a starting point one may replace the gauge field by a generalmatrix with Gaussian distribution entries, as it is done in appendix A of reference [6] for thederivation of the regularized Yang-Mills action. The axioms of quantum field theory, such asthe Osterwalder and Schrader axioms [9], may, in principle, be extended without difficultiesto the physical model in 3+1+1 dimensions.– 19 – From action to Yang-Mills theory
In this section we expand definitions given in section 2 as well as give a new derivation ofthe Yang-Mills theory starting from the fermion action. We start with the Hamilton operatorof the theory. Let ˆΨ c ( x ) , ˆΨ c ( x ) ∗ , c = 1 , , . . . , N be N fermion annihilation and creationoperators at each site x = ( x , x , . . . , x d ) on a regular Euclidean lattice in d dimensionsacting on the Hilbert space H . They obey the anticommutation relations n ˆΨ c ( x ) ∗ , ˆΨ c ′ ( x ′ ) o = δ cc ′ δ xx ′ . (B.1)The lattice is finite and we assume it to be a torus with V = N N · · · N d number ofsites, where N , N , . . . N d are the number of sites along each dimension. The Hamilto-nian operator (2.1) used in this theory is the Kogut-Susskind Hamiltonian [7]. In princi-ple, any lattice formulation of Dirac fermions will do the job. We have chosen the stag-gered formulation because it is simple enough for our purpose. Here, ˆ γ is the lattice siteparity operator taking ± values on even/odd lattice sites, i.e. ˆ γ ( x ) = ( − x + ··· + x d and η ( x ) = 1 , η µ ( x ) = ( − x + ··· + x µ − , µ = 2 , . . . , d . Both, ˆ γ and η µ are diagonal matrices onthe lattice. The Hamilton operator may be written in temrs of the Hermitian fermion matrix h = m ˆ γ + κ ˆ γ X µ η µ ( U µ − U ∗ µ ) , (B.2)where the hopping matrices U µ (see 2.4 for their definition) satisfy the following commuta-tion relations ˆ γ η µ − η µ ˆ γ = 0 , ˆ γ U µ + U µ ˆ γ = 0 , η µ U µ − U µ η µ = 0 . (B.3)This way, we write ˆ H = X x,y,c,c ′ ˆΨ( x ) ∗ c h ( x, y ) cc ′ ˆΨ( y ) c ′ , (B.4)where h ( x, y ) cc ′ are the matrix elements of the V N × V N fermionic matrix h .A new Hamilton operator may be derived by the action (2.9). We do this in the following.Introducing Grassmann valued fermion fields Θ( x, τ ) = ψ ( x, τ ) and ¯Θ( x, τ ) = ¯ ψ ( x, τ )ˆ γ ( x ) with antiperiodic boundary conditions along the extra direction the action is written in theform ˜ I = 12 X x,τ,c (cid:2) ¯Θ( x, τ ) c Θ( x, τ + 1) c − ¯Θ( x, τ + 1) c Θ( x, τ ) c (cid:3) + X x,y,t,c,c ′ ¯Θ( x, τ ) c h ( x, y ) cc ′ Θ( y, τ ) c ′ . (B.5)The partition function of the theory is the Berezin integral ˜ Z F ( U ) = Z Y x,τ,c d Θ( x, τ ) c d ¯Θ( x, τ ) c e ˜ I . (B.6)– 20 –ne finds ˜ Z F ( U ) = det h
12 12 − h − h . . . . . . . . . − h − − h , (B.7)where each entry is a block V N × V N matrix of the
V N N τ × V N N τ matrix. Relabelingthe last column as the first one and defining × block matrices A = (cid:18) − h (cid:19) , B = (cid:18) h (cid:19) , C = (cid:18) − (cid:19) , (B.8)one has: ˜ Z F ( U ) = ( − V N det
AC B − A B . . . . . . − A BBC − A , (B.9)where we have assumed N τ even. Finally, using the cyclic reduction of the resulting matrixdeterminant one obtains (modulo a constant factor) ˜ Z F ( U ) = det h (cid:0) B − A (cid:1) − Nτ i , (B.10)where B − A = (cid:18) − h − h h (cid:19) . (B.11)This matrix is brought to block diagonal form and we get ˜ Z F ( U ) = det (cid:16) e − N τ ˜ h (cid:17) , (B.12)with ˜ h = (cid:20) ln (cid:0) h + 2 √ h + h (cid:1) − ln (cid:0) h + 2 √ h + h (cid:1)(cid:21) . (B.13)Therefore, the corresponding Hamilton operator is defined by ˜ H = X x,y,a,b ˜Ψ( x ) ∗ a ˜ h ( x, y ) ab ˜Ψ( y ) b , (B.14)– 21 –here ˜Ψ( x ) a is a doublet of two copies of ˆΨ( x ) a operators. The degeneracy comes form thesymmetric differences in the approximation of lattice derivative, eq. (2.10).Finally, we show that both Hamilton operators defined in (2.1) and (B.14) give Yang-Mills theories with the same Wilson loop expansion structure. The effective action of thetheory may be written in terms of gauge fields only ˜ S eff ( U ) = −
12 ln ˜ Z F ( U ) , (B.15)where we have discounted by a factor of two the overcounting that comes from the degener-acy in the action formulation. The energy of the ground state of the Hamiltonian is computedas the large N τ limit of the effective action. The largest fermion mass on a lattice is propor-tional to m/a and we fix it to be exactly /a . Then, form (B.2) we have h = 1 + κ h o , h o = ˆ γ X µ η µ ( U µ − U ∗ µ ) . (B.16)For large N τ only one of the blocks of the ˜ h matrix (B.13) survives in the exponential functionof (B.12). Therefore, using the identity det A = e Tr log A the effective action of the theory is ˜ S eff ( U ) = − N τ Tr ln (cid:16) h + 2 √ h + h (cid:17) , (B.17)where the trace is taken in the tensor product space of the lattice sites and the SU(N) group.Expanding the logarithm of the effective action (B.17) and staying with the terms of the firstorder in h , we obtain the approximation S eff ( U ) = − N τ Tr p + κ h o . (B.18)As shown in [6], this expression may be computed directly from (2.1). Expanding the righthand side in powers of κ we get S eff ( U ) = c o N τ − c N τ κ X µν Tr U µ U ν U ∗ µ U ∗ ν + O ( κ ) + h.c. , (B.19)where c o is real, c = 1 / . . Expanding directly (B.17) in κ one gets ˜ S eff ( U ) = ˜ c o N τ − ˜ c N τ κ X µν Tr U µ U ν U ∗ µ U ∗ ν + O ( κ ) + h.c. (B.20)with ˜ c = 3 / (8 √
2) = 0 . . . . . While both formulations have the same Wilson loop struc-ture, they differ in the coefficients multiplying the loops. The coefficient of the plaquettediffers by . Scaling the length of the extra dimension according to the relation c N τ = 1 κ δ (B.21)– 22 –e get the effective theory S eff ( U ) = c o N τ − κ δ X µν Tr U µ U ν U ∗ µ U ∗ ν + O ( κ − δ ) + h.c. , (B.22)which is the Wilson theory enlarged by larger loops. Since the mean value of the first termmay be written in the form (see eq. (4.10)) X µν h Tr U µ U ν U ∗ µ U ∗ ν i = w o − w κ + O ( κ ) , (B.23)where w o and w are positive, the action corresponds to the Yang-Mills action in the contin-uum limit if we set δ = 2 . The other terms vanish in the continuum limit.We derived the effective action for N τ even. For odd N τ one may follow a more generalapproach. Writing the action (B.5) in terms of the Fourier modes ω k = πN τ (2 k + 1) , k = 1 , , . . . , N τ , (B.24)which respect the boundary condition along the extra dimension and integrating Grassmannfields the partition function may be written as a product of determinants ˜ Z F ( U ) = N τ Y k =1 det ( h + i sin ω k ) . (B.25)If N τ is even and using (B.16) the product takes the form ˜ Z F ( U ) = N τ / Y k =1 det (cid:0) ω k + κ h o (cid:1) (B.26)with an extra factor det h in case N τ is odd. We neglect this factor since it gives a smallcontribution (see the last subsection). Therefore the effective action is ˜ S eff ( U ) = − N τ / X k =1 Tr ln (cid:18) κ ω k h o (cid:19) . (B.27)The Wilson loop expansion is derived by expanding the logarithm in κ ˜ S eff ( U ) ∝ − κ ˜ c N τ X µν Tr U µ U ν U ∗ µ U ∗ ν + O ( κ ) + h.c. , (B.28)where ˜ c is the definite integral ˜ c = 1 N τ N τ / X k =1 ω k ) ≈ Z π − π dω π ω ) = 38 √ . (B.29)In a similar fashion one may compute coefficients of larger loops. Both methods agree in thelarge N τ limit. Note that the effective Yang-Mills theory described here is valid in the caseof two or more dimensions. In the following we examine the one-dimensional case.– 23 – ne-dimensional case This case is special since the action is given in terms of the Polyakov loop S d =1 eff ( U ) ∼ ± N τ κ N d N d Tr U (1) U (2) · · · U ( N d ) + h.c. , (B.30)where the sign is determined by the parity of N d and boundary conditions. Scaling the lengthof the extra dimension according to the relation N τ ∼ κ N d +2 (B.31)the action is of the form S d =1 eff ( U ) ∼ N d κ Tr U (1) U (2) · · · U ( N d ) + h.c. . (B.32)In the following we give a precise meaning of a weakly coupled regularization of the Yang-Mills theory. A weakly coupled regularized theory
In this subsection we show that the regularization of the Yang-Mills theory considered inthis paper is valid at weak coupling only. In the case of strong coupling the theory differsconsiderably from the plaquette theory. If N τ is a small integer, which corresponds to alarge value of the coupling constant κ ∼ , the expression of the effective action (B.18) isno longer valid. In this case one has to use the general expression (B.25). For example, at N τ = 1 one has ˜ Z F ( U ) = det h (B.33)and therefore ˜ S eff ( U ) = − Tr ln (cid:0) + κ h o (cid:1) . (B.34)Since κ ∼ the logarithm may not be expanded in convergent power series of κ . However,it may be expanded in terms of convergent series of another parameter if one notices that thespectral radius of h o is bounded by d . Defining h = h o − d , κ = κ d κ (B.35)we have κ h o = κ κ (1 + κ h ) . (B.36)The spectral radius of κ h is bounded by one and we may expand ˜ S eff ( U ) ∝ − Tr ln (cid:0) + κ h (cid:1) (B.37)– 24 –n a convergent series of κ . In this case the effective coupling of the plaquette is κ eff ∼ κ ∼ (2 d ) . (B.38)Therefore, although the effective coupling of the plaquette is strong, we may not expand theeffective theory in terms of κ . The situation is similar with larger Wilson loops. Repeatingthe exercise in the weak coupling limit S eff ( U ) = − N τ κ κ Tr q + κ h (B.39)and expanding the right hand side in terms of κ the effective coupling of the plaquette is κ eff ∼ κ N τ κ ∼ (cid:18) d κ κ (cid:19) N τ κ . (B.40)For vanishing κ we get κ eff ∼ N τ κ ∼ κ , (B.41)as expected. In this sense, the regularization presented in this paper is a weakly coupledregularized Yang-Mills theory. References [1] D. J. Gross, Frank Wilczek,
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