The QCD beta-function from global solutions to Dyson-Schwinger equations
Guillaume van Baalen, Dirk Kreimer, David Uminsky, Karen Yeats
aa r X i v : . [ h e p - t h ] O c t The QCD β -function from global solutions toDyson-Schwinger equations Guillaume van Baalen ∗ , Dirk Kreimer † , David Uminsky ∗ , and Karen Yeats ∗ October 27, 2018
Abstract
We study quantum chromodynamics from the viewpoint of untruncated Dyson–Schwinger equa-tions turned to an ordinary differential equation for the gluon anomalous dimension. This non-linear equation is parameterized by a function P ( x ) which is unknown beyond perturbation theory.Still, very mild assumptions on P ( x ) lead to stringent restrictions for possible solutions to Dyson–Schwinger equations.We establish that the theory must have asymptotic freedom beyond perturbation theory andalso investigate the low energy regime and the possibility for a mass gap in the asymptotically freetheory. Acknowledgments
D.K. and K.Y. were supported by NSF grant DMS-0603781. D.U. and G.v.B. are supported by NSFgrant DMS-0405724 and thank C.E. Wayne for discussions. D.K. thanks Ivan Todorov for discussions.
We study non-perturbative aspects of quantum chromodynamics (QCD). We do so by investigatingDyson–Schwinger equations. Instead of solving a truncated version of these a priori very intricate ∗ Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston MA, 02215, USA † CNRS-IHES 91440 Bures sur Yvette, France; and Center for Mathematical Physics Boston University, 111 Cumming-ton Street, Boston MA, 02215, USA
We follow the methods employed in our work on QED [5], adopted to the study of QCD in the back-ground field approach as developed by Abbott [1, 2]. Let us first reconsider the situation for quantumelectrodynamics. There, thanks to the Ward identity, it suffices to consider the anomalous dimension γ ( x ) of the photon which is essentially the β -function, β ( x ) = xγ ( x ) .This anomalous dimension is obtained from the photon’s self-energy, a two-point function which isdetermined non-perturbatively by the knowledge of a single Lorentz scalar function G ( x, L ) = 1 − ∞ X k =1 γ k ( x ) L k , with x the fine structure constant and L = ln ( − q /µ ) , where the inverse photon-propagator is ( q g µν − q µ q ν ) G ( x, L ) .Combining this with the combinatorial Dyson-Schwinger equations and using an expansion into suit-able integral kernels which parametrize the corresponding integral equation, the Dyson-Schwingerequation combine with the renormalization group equation to givei) a recursion for the γ k : γ k ( x ) = − k γ ( x )(1 − x∂ x ) γ k − ( x ) , k ≥ , (1)ii) a differential equation for γ ( x ) : γ ( x )(1 − x∂ x ) γ ( x ) + γ ( x ) − P ( x ) = 0 . (2)Here, i) comes from the renormalization group, ii) from the DSE, and P ( x ) is a suitably constructedseries over residues. The appearance of the operator (1 ± x∂ x ) is typical for a gauge theory.We will now address a non-abelian gauge theory, resulting in a similar set-up (in particular, i) and ii)remain form-invariant) once we learned to make effective use of quantum gauge invariance to reduceagain to a single ODE. Obviously, however, P ( x ) will be a different function, in particular, it willchange sign. We study the consequences of this fact with minimal knowledge on the behavior of P ( x ) .Still, as in QED, we will see that we can learn quite a bit regarding the non-perturbative sector of QCD.To proceed, we turn to the background field method.2 .2 Background field method We first have to consider the set of vertices and propagators in the background field gauge. They definea set R given as follows: QQ ¯ ccdQQQdAQQQQQQAQQQAQAQ ¯ cQc ¯ cAc ¯ ccQA ¯ ccAAAA . Here, names attached to the vertices and propagators are short hand memos of the corresponding mono-mials in the Lagrangian.The field content is Q for the internal quantized gauge field, A for an external background gaugefield, ¯ c, c for the anti-ghost and ghost field. Coupling of Fermionic matter will not change the ensuingdiscussion in any way and is omitted for convenience. See Abbott [2] for details.With the set R comes an accompanying set of 1PI Feynman graphs naturally labeled by elements inthis set according to their type and number of external legs.We consider in particular Green functions for such graphs and adopt the results of [18], see also [19],which read as expected ∀ r ∈ R X r = I ± X k ≥ [ g ] k X | γ | = k γ ) B γ ; r + Q v ∈ γ [0] X v Q e ∈ γ [1] √ X e ! , with B γ + ( h ) = X Γ ∈ < Γ > bij ( γ, h, Γ) | h | ∨ maxf (Γ) 1( γ | h ) Γ , (3)3here maxf (Γ) is the number of maximal forests of Γ , | h | ∨ is the number of distinct graphs obtainableby permuting edges of h , bij ( γ, h, Γ) is the number of bijections of external edges of h with an insertionplace in γ such that the result is Γ , and finally ( γ | h ) is the number of insertion places for h in γ [18]. P Γ ∈ < Γ > indicates a sum over the linear span < Γ > of generators of H .Next, we divide by the ideal I which implements the Slavnov–Taylor identities which here is generatedorder in order in g by X ¯ cAc = X ¯ cQc = X ¯ cc , X dQQQ = X dAQQ = X QQ = X AA ,X AAQQ = X AQAQ = X AQQQ = X QQQQ = X dQQQ . On H/I we then get two independent Green functions which need renormalization X AA , X ¯ cc , corresponding to a mere two-element set R H/I = { AA, ¯ cc } . Also, we then find an combinatorial invariant charge uniquely defined as C = I / √ X AA = I / √ X QQ , so that, as in QED, the β -function is just half the negative anomalous dimension of the gauge field.Note that the addition of massless fermions would just add in H/I an element ¯ ψψ for the fermionself-energy but would not change the ideal or the invariant charge.The system of combinatorial Dyson Schwinger equations is then X AA = I − X k [ g ] k B k,AA + (cid:16) X AA [ C ] − k (cid:17) X ¯ cc = I − X k [ g ] k B k, ¯ cc + (cid:16) X ¯ cc [ C ] − k (cid:17) . Note that this determines X AA in terms of itself, while X ¯ cc is a function of itself and X QQ . We write X r = I − P k ≥ [ g ] k c rk , with c rk ∈ H/I the generators of a sub Hopf algebra [18, 9] given by allgraphs which contribute to an amplitude r at a chosen order k . Note that the resolution of a Greenfunctions into images of Hochschild one-cocycles B r + is the mathematical equivalent of a resolution ofall overlapping divergences into non-overlapping integral kernels. That this is possible in a non-abeliantheory was realized early by Baker and Lee [7].See [18, 19, 25, 26] for explicit examples how Hochschild cohomology and Hopf algebras relate.There hence is a quotient Hopf algebra H AA spanned by generators c AAk of X AA . Similarly, goingtemporarily to the quotient Hopf algebra H ¯ cc defined by X AA = I , c AAk = 0 , we find that this is a4ocommutative Hopf algebra (which is obvious from setting C = I ) and hence we find a factorizationof groups into an abelian subgroup Spec( H ¯ cc ) and a normal subgroup Spec( H AA ) , Spec(
H/I ) = Spec( H AA ) ⋊ Spec( H ¯ cc ) , corresponding to a short exact sequence which splits I → Spec( H AA ) → Spec(
H/I ) → Spec( H ¯ cc ) → I . It is this factorization which allows us to compute the β -function of QCD by an ODE for a singleequation below. Note that the situation is similar to QED: there, the Ward identity allows for a similarsemi-direct product structure between photon amplitudes and Fermionic matter. Gauge invariance thenallows to eliminate all short-distance singularities in the abelian subgroup thanks to the work of Baker,Johnson and Willey, and one is left with the photon propagation as the only source of renormalization.Here, we can compute the β -function from G AA ( x, L ) , but would have to consider the full coupledsystem to determine G ¯ cc ( x, L ) (and G ¯ ψψ ( x, L ) ), which we do not attempt here.Furthermore, note that the simplification at k = 1 B ,AA + (cid:0) X AA [ C ] − (cid:1) = B ,AA + ( I ) . This is typically for gauge theories and emphasizes that we are in a single equation situation with s = 1 [5]. Terms B r + ( I ) always deliver a pure residue from their short-distance singularities, and these termsare intimately connected to fermion determinants. We will not pursue this connection any further here.The background field method is then suited to our approach as it allows us to compute the QCD betafunction from a single ordinary differential equation.Indeed, a change of basis of primitives allows to reduce the application of Feynman rules to the study ofone-variable Mellin transforms for the integral kernels for the above primitives B k,r + ( I ) , and from therewe can strictly follow the techniques of [20, 21, 27] to get to an single ordinary differential equation: γ ( x ) + γ ( x ) − P ( x ) − xγ ( x ) γ ′ ( x ) = 0 . Here, P ( x ) is a suitable series over primitives. As always, we renormalize using a momentum schemewith subtractions at q = µ . That scheme is uniquely suited [14] to our gauge-invariant non-perturbativeapproach. From perturbative computations, asymptotic freedom is firmly established. We will establish it beyondperturbation theory below. Perturbatively, this is mainly a self-consistency statement: assuming thatthe QCD coupling constant is small, we approximate the theory by its loop expansion to a few orders.The resulting polynomial approximation to the beta function supports the claim of asymptotic freedom5n perturbation theory: β ( x ) < , < x < , hence at large momentum transfer lim − Q →∞ α s ( L ) → , L = ln( − Q /µ ) . The coupling indeed becomes small in that limit.As usual, perturbation theory agrees well with observations: asymptotic freedom is a well-establishedexperimental fact.Much more intricate is the study − Q → + . This is beyond the reach of perturbation theory. Nevertheless, different approaches point out that inthat limit, the gluon propagator might turn to a constant, confirming an old suggestion of Cornwall thatthe free gluon develops in the interacting theory a momentum-dependent mass which vanishes at highenergies, but turns to a non-vanishing constant in the limit − Q → + . We study this behavior fromour viewpoint in section 3.Results to this effect were already obtained byi) Lattice computations [4];ii) numerical study of Dyson Schwinger equations truncated in a gauge invariant way [13];iii) in the Gribov-Zwanziger formalism [17].Below, we reconsider the problem from a study of the possible structure of solutions of Dyson Schwingerequations. We want to establish asymptotic freedom beyond perturbation theory, and want to discussto what extent a solution which exhibits asymptotic freedom can also exhibit a mass gap.Again, as in the case of QED, we find the most interesting solution to be a separatrix. In the case ofQCD, that separatrix is the only solution which has asymptotic freedom. In QCD, the Dyson-Schwinger equation for γ is d γ ( x )d x = f ( γ ( x ) , x ) ≡ γ ( x ) + γ ( x ) − P ( x ) xγ ( x ) . (4)We will assume that the primitive skeleton function satisfies the following assumptions:H1: P is a twice differentiable function on R + , with P (0) = 0 , P ′ (0) < and P ′′ (0) < .H2: There exist x ⋆ such that P ( x ) > − and P ′′ ( x ) ≤ (i.e. P is concave) on [0 , x ⋆ ] .H3: The function P ( x ) satisfies P ( x ) < for all x > .6s in [5], we avoid the singularities of (4) at γ = 0 and x = 0 by specifying with an initial conditionat x ⋆ , namely, d γ ( x )d x = f ( γ ( x ) , x ) ≡ γ ( x ) + γ ( x ) − P ( x ) xγ ( x ) , γ ( x ⋆ ) = γ . (5)This ensures that solutions of (5) exist at least locally around x = x ⋆ . Though we will mainly look forsolutions or (4) with γ ( x ⋆ ) < and x ≥ , we will occasionally comment on the γ ( x ⋆ ) > case.In the QED case, we proved in [5] (see also Section 5 of the present paper) the existence of a uniquevalue γ ⋆ ( x ) (the separatrix) separating solutions that exist globally for all x ≥ x from those thatcan only be continued up to a finite x max > x . As shown in Section 5, all solutions in QED can becontinued as x → , differing there ‘only’ by a flat behavior ∼ e − x . In QCD the situation is reversed:we will prove that all solutions starting at some appropriate x ⋆ can be continued as x → ∞ , but thatthere is a unique value γ ⋆ ( x ⋆ ) that separates solutions that cannot be continued as x → from thosethat can, which either satisfy γ (0) = − if γ ( x ⋆ ) < γ ⋆ ( x ⋆ ) or γ (0) = 0 if γ ( x ⋆ ) = γ ⋆ ( x ⋆ ) . We callthe solution γ ⋆ ( x ) that satisfies γ ⋆ (0) = 0 the asymptotically free solution.We will also use more speculative hypotheses on P ( x ) :S1: There exists p > such that P ( x ) = − cx p + o ( x p ) as x → ∞ .S2: There exist x c ≥ − P ′ (0) such that P ′′ ( x ) ≤ for all x ∈ [0 , x c ] S3: There exist a (finite) interval [ x c , x d ] with x c > x ⋆ such that − Z x d x c P ( z )2 z d z ≥ . S4: There exist finite x l and x r such that P ( x l ) = P ( x r ) = − and P ( x ) > − for all ≤ x < x l and x > x r . The function P ( x ) does not satisfy S1, S2 and S3, but rather lim x →∞ ( P ( x ) − P ∞ ) =lim x →∞ xP ′ ( x ) = 0 for some P ∞ > − .Let us briefly comment on our logic here. The H1-H3 hypotheses are a bare minimum for the resultswe will present below. Within perturbation theory, we have P ( x ) = γ ( x ) + O ( x ) = − β x − β x + O ( x ) (6)as x → , where − β and − β are the and -loop coefficients of the β function, namely β = 9 and β = 64 for n f = 6 . As such, the hypotheses H1 and H2 are reasonable. While it also follows from(6) that P ( x ) < at least for small values of x , extending this to all values of x is somewhat morespeculative.As we will show below, if P ( x ) < − on a ‘sufficiently large interval’, for instance if either S1, S2 orS3 hold, all solutions of (5) satisfy γ ( x ) = − for some x ≥ x ⋆ . It then follows from H3 that theygrow linearly as x → ∞ if D ( P ) = − Z ∞ x ⋆ P ( z ) z d z < ∞ , D ( P ) = ∞ . Incidentally, we showed in [5] (see also Section 5 of the presentpaper), that the finiteness/infiniteness of D ( P ) was intimately linked with the existence/non-existenceas x → ∞ of solutions of the analogous of (5) for QED. It is striking to see that the same criterion distinguishes between different type of behavior in QCD as well .Note that an anomalous dimension growing at least linearly as x → ∞ leads to a Landau pole forthe running coupling, and hence a serious obstacle to studying the infrared behavior of the Gluonpropagator. Despite that, we will show in Section 3.1 that this pole (if present) can be removed usingan unsubtracted dispersion relation, see e.g. [24], and the Gluon propagator can still be studied in theinfrared limit.In contrast, if γ ( x ) is finite as x → ∞ , we avoid the Landau pole, and can study the Gluon propagatorwithout using dispersion relations. Such constant asymptotics for γ can only happen if P ( x ) tends toa constant as x → ∞ . This motivates (part of) the hypothesis S4. Under that hypothesis, we will showthat there is only one solution that satisfies lim x →∞ γ ( x ) = − √ P ∞ ≡ γ ∞ . If P ∞ = 0 , we call that solution the confinement solution γ c ( x ) , and if P ∞ > , we call it a strongconfinement solution . On physical grounds, the asymptotically free and (strong) confinement solutionsneed to be the same. Unfortunately, for generic P ( x ) satisfying H1, H2 and S4, these two solutions aredifferent. We did not succeed in finding a sufficient condition on P ( x ) that guarantees both solutionsare the same. Despite that, a necessary condition is certainly that P ( x ) makes at least one (small)excursion below − , while avoiding the S1-S3 conditions, see also figure 1 below or Section 4.4.By standard folklore and heuristics [3], a nowhere vanishing β -function, β ( x ) < , ∀ x > , whichwe will indeed establish below under assumption H1 above, implies a mass gap in QCD, and hence aconfinement scenario following old ideas of Cornwall [15]. We will come back to that in Section 3 ofthis paper. Remark 2.1 If P ∞ > in hypothesis S4, P ( x ) has a further zero for a finite x > , P ( x ) = 0 . Insuch a case, using the running coupling formulation of (4), one sees that some solutions spiral aroundthe zero of P ( x ) , themselves having infinitely many zeroes. The asymptotically free solution γ ⋆ ( x ) mayor may not spiral around x = x , depending on details of P . Should it be captured, we get a solution γ ( x ) which has a UV fix-point at zero and an infrared fixpoint at x . This is the Banks-Zaks scenario[8]. In the remainder of this section, we are going to state our main results. The proofs and technical detailsare postponed to Section 4 of this paper.Our first main result gives a complete characterization of the behavior of solutions of (5) for x < x ⋆ .In particular, it establishes the uniqueness of the solution exhibiting asymptotic freedom. though of course the primitive skeleton functions P are different in both cases heorem 2.2 Under the hypotheses H1 and H2 , there is a unique value γ ⋆ ( x ⋆ ) < such that thecorresponding solution γ ⋆ ( x ) of (5) exists for all x ∈ [0 , x ⋆ ] and satisfies lim x → γ ⋆ ( x ) = 0 . Additionally,that solution satisfies p P ( x ) − ≤ γ ⋆ ( x ) ≤ P ′ (0) x , (7) If γ ( x ⋆ ) < γ ⋆ ( x ⋆ ) , then the corresponding solution satisfies lim x → γ ( x ) = − . If γ ⋆ ( x ⋆ ) < γ ( x ⋆ ) < ,then there exist x min > such that the corresponding solution satisfies γ ( x min ) = 0 . Note that since √ P ( x ) − = P ′ (0) x + O ( x ) as x → , (7) shows that γ ⋆ ( x ) = P ′ (0) x + O ( x ) as x → .As we will prove in Proposition 4.8) of Section 4 below, all solutions of Theorem 2.2 can be continuedas x → ∞ by only adding the H3 assumptions. In Proposition 4.6, we will show that solutions thatsatisfy γ ( x min ) = 0 for some < x min < x ⋆ can be continued in the first quadrant (becoming double-valued) by reverting to the so-called ‘running coupling’ formulation of (5) (see also [5]). In particularthese solutions will satisfy γ ( x ) > for some x > x min .Our second main result concerns the asymptotic behavior as x → ∞ of solutions that enter the firstquadrant, or attain the value − somewhere. This last condition can be verified under additional as-sumptions on P such as S1, S2 or S3. Proposition 2.3
Assume P ( x ) satisfies H1-H3 and that one of the two following statements holds:1. − < γ ( x ⋆ ) < and P ( x ) satisfies S1 or S3,2. γ ( x ⋆ ) ≤ γ ⋆ ( x ⋆ ) and P ( x ) satisfies S2.Then there exists x > x ⋆ such that the corresponding solution γ ( x ) satisfies γ ( x ) = − . To be able to state our asymptotic result as x → ∞ , we need first to introduce the slope function S P ( x , x ) . This function is given by S P ( x , x ) = (cid:18) γ ( x ) x + 2 Z xx − P ( z ) z d z (cid:19) , Note that, if D ( P ) < ∞ , the slope function S P ( x , x ) goes to a finite value as x → ∞ for any x .We can now completely describe the asymptotic behavior as x → ∞ of solutions of (5):9 heorem 2.4 Assume P ( x ) satisfies H1-H3. If there exists x > such that either γ ( x ) > or γ ( x ) ≤ − then x S P ( x , x ) ≤ γ ( x ) ≤ x (cid:18) S P ( x , x ) + 1 x (cid:19) − if γ ( x ) > , − x S P ( x , x ) ≤ γ ( x ) ≤ − x (cid:18) S P ( x , x ) − x (cid:19) − if γ ( x ) ≤ − . Furthermore, if D ( P ) < ∞ and γ ( x ) ≤ − or γ ( x ) > , there exists s > such that lim x →∞ γ ( x ) x = ( − s < if γ ( x ) ≤ − s > if γ ( x ) > . If γ ( x ) ≤ − , the convergence towards the limit is given by (cid:12)(cid:12)(cid:12) γ ( x ) x + s (cid:12)(cid:12)(cid:12) ≤ C Z ∞ x − P ( z ) z d z . (8) If D ( P ) < ∞ and γ ( x ) > , then (8) also hold, with − s replaced by s . We want to stress here that the slope value s depends on the actual solution under consideration. Also,for solutions that eventually enter the first quadrant, the value of the slope s has no reason to be thesame along the two branches of the solution (the one in the first quadrant, and the one in the fourth).Also, note that the result depends only on the assumption γ ( x ) = − .We now state the existence and uniqueness of confinement solutions under hypothesis S4. Theorem 2.5
Assume P ( x ) satisfies H1, H2 and S4. Then there exist a unique solution γ c ( x ) of (4)such that lim x →∞ γ c ( x ) = lim x →∞ − p P ( x )2 ≡ γ ∞ . Furthermore, γ c ( x ) satisfies the usual trichotomy as x decreases: either γ c (0) = − , or γ c (0) = 0 , or γ c ( x ) cannot be continued for x < x min for some x min > where γ c ( x min ) = 0 . Finally, if there exists x max such that ′ P ( x ) > for all x > x max (hence P ( x ) is strictly increasing towards P ∞ ), then − √ P ∞ + 12 ≤ γ c ( x ) ≤ − p P ( x ) + 12 for all x > x max . As already noted above, we cannot show that γ ⋆ ( x ) = γ c ( x ) without additional hypotheses on P ( x ) .We can however note that the two types of solutions are compatible: the confinement solution satisfiesthe integral equation γ c ( x ) = − Z ∞ P ( xt ) t γ c ( xt ) d t , (9)10 ∞ = − γ xx l x r γ c , − ( x ) γ c , + ( x ) γ ⋆ , + ( x ) γ ⋆ , − ( x ) Figure 1: A ‘fragile’ solution: the solid bold curve is the solution combining asymptotic freedomand confinement for some artificial P ( x ) with P ∞ = 0 . The dashed curves are the nullclines γ + γ − P ( x ) = 0 . The solid curves γ c , ± and γ ⋆ , ± correspond to confinement and asymptotically freesolutions with P ( x ) perturbed so that the gap | x r − x l | is slightly larger (‘ + ’ subscripts) or smaller (‘ − ’subscripts). In both cases, the asymptotically free solution and the confinement solution do not match.whose r.h.s. converges to as x → if P ( x ) /γ c ( x ) → as x → . Characterizing the set of functions P ( x ) for which the confinement solution and the asymptotically free one are the same is a difficultproblem, see e.g. figure 1 for an example with an ‘artificial’ P ( x ) . A necessary condition is that P ( x ) makes an excursion below − on (at least) one interval so that the nullclines, i.e. the location in ( x, γ ) -plane where γ ′ ( x ) = 0 , γ ± ( x ) = ± p P ( x ) − show a gap as in figure 1. However if the gap is ‘too wide’, we have γ ⋆ ( x ) = γ ∞ for some finite x ,and γ c ( x min ) = 0 for some x min > , whereas if the gap is ‘too small’, γ ⋆ ( x ) intersects the nullclineat some x > x r , and hence cannot reach γ ∞ as x → ∞ , while γ c ( x ) intersects the nullcline at some x < x l , and hence cannot reach as x → . 11e conclude this section by explaining the terminology asymptotically free , confinement and strongconfinement for our solutions. These come from the study of the inverse Gluon propagator, P − ( x, Q ) = Q G ( x, L ) with L = ln( Q /µ ) . In Section (3.2) below, we solve the RGE equation expressing the scale invariance of G ( x, L ) . Inparticular, in Theorem 3.2, we show that if γ ( x ) → γ ∞ as x → ∞ , then G ( x, L ) = X ( L, x ) x , where X ( t, x ) is the running coupling, i.e. the solution of d X ( t, x )d t = X ( t, x ) γ ( X ( t, x )) with X ( t = 0 , x ) = x . The possible large | t | behavior of X ( t, x ) are given by X ( t, x ) ≃ x ∞ e γ ∞ t as t → −∞ if lim x →∞ γ ( x ) = γ ∞ , − P ′ (0) t as t → ∞ if lim x → γ ( x ) x = P ′ (0) ,x e − t as t → ∞ if lim x → γ ( x ) = − , where x ∞ and x are some positive functions of x . Thus, in all cases where γ ( x ) can be continued to x = 0 , G ( x, L ) → in the ultraviolet regime L → ∞ , and moreover P − ( x, Q ) → as L → ∞ if lim x → γ ( x ) x = P ′ (0) , + x as L → ∞ if lim x → γ ( x ) = − , hence the terminology asymptotic freedom for the solution that satisfies γ ( x ) = xP ′ (0) + O ( x ) as x → . In the infrared regime, however, G ( x, L ) ≃ x ∞ x e γ ∞ L = x ∞ x (cid:18) Q µ (cid:19) γ ∞ as L → −∞ , and hence we find lim L →−∞ P − ( x, Q ) = µ x ∞ x if lim x →∞ γ ( x ) = − ∞ if lim x →∞ γ ( x ) < − , (10)a finite mass gap if γ ∞ = − . 12 The Gluon propagator, confinement and mass gaps in QCD
In QED, the coupling is weak at low energy or momentum transfer. In our previous work [5], wecould hence define boundary conditions at low energy, and studied the behavior of P k γ k L k for L > .In particular, for γ = γ (¯ x ( L )) , a continuation to L < was never needed by the choice of ourrenormalization conditions. On the other hand, for large L ≫ we could establish a separatrix, whichpossibly avoids a Landau pole at any finite positive L . We conjectured that this might be the solutionchosen by Nature, and further detailed analysis of its properties awaits more analysis of the function P ( x ) . Should it turn out that P ( x ) is such that the separatrix will not avoid a Landau pole ( i.e. turnsto infinity at finite L ), we will have to turn to dispersion relations to understand the non-perturbativecorrections coming with such a pole, as recognized by Shirkov and collaborators early on [11].For QCD, we again fix a small coupling, but this time large momentum transfer, L ≫ for ourboundary conditions. We are now interested in a continuation to L ≪ , in particular we are interestedin L → −∞ . Under very mild assumptions on P ( x ) , and certainly by any experience from perturbativeapproximations of the theory, we expect the anomalous dimension γ to go below the value − at somefinite coupling x , and hence ¯ x ( L ) to turn to infinity at some finite negative L . Shifting that L to zeroessentially defines the scale Λ QCD , and we are interested for that shifted L Λ to study the regime L Λ < ,in particular L Λ → −∞ . To consider such a limit based from an approach formulated for L Λ > , wewill use dispersion relations. Our approach is motivated again by Shirkov and collaborators work [24].On general grounds, we know that ¯ x ( L Λ ) and G ( x, L Λ ) can be treated by an unsubtracted dispersionrelation [23]: f disp ( Q ) = Z ∞ ℑ ( f ( σ )) σ + Q − iη d σ . The inverse propagator needs a subtracted dispersion relation, which leads back to an unsubtracteddispersion relation for G [23].The Dyson–Schwinger equations themselves are supposed to hold for the whole theory regardless ofthe sign of L Λ . Similar, the renormalization group equations for the running coupling are supposed tohold. Our derivation which turned the Dyson–Schwinger equations into a ODE was valid for L > − L Λ ,hence remain valid, after shifting, for L Λ > .Continuing to L Λ < will generate non-perturbative corrections to γ ( x ) , and hence β ( x ) , determinedfrom the requirement that equations of motion, renormalization group flow and analyticity propertiesof field theory are what they are supposed to be.Any perturbative approximation is in accordance with these properties of field theory only up to theorder considered. When we study solutions of Dyson–Schwinger equations, we demand accord withthese properties as a guide to find the necessary non-perturbative corrections in the region L Λ < .13e hence will start by first applying a dispersion relation to analyze ¯ x ( L Λ ) . That leads to a corrected γ , disp due to the fact that at L Λ = 0 we find that ¯ x = ∞ . Combining this with the assumption that theuncorrected γ ( x ) is driven below -1 at finite x allows an easy estimate of the corrected γ , disp and alsoallows for consistency with the direct analysis of G by dispersion methods.Let us now start with a study of ¯ x ( L Λ ) . We start our considerations by boundary conditions such that G ( x, L ) = 1 at some very high momentum transfer µ ≫ , L = ln( Q /µ ) , which determines asuitably small x in agreement with say deep inelastic scattering experiments [10].We have d¯ x ( x, L )d L = ¯ xγ (¯ x ) ⇔ Z ¯ x ( x,L ) x uγ ( u ) du = L , and assume that the integral Z ∞ x uγ ( u ) d u < ∞ . In particular, we assume that γ ( x ) < − for some positive finite x , in accordance with our previousdiscussions and experimental evidence. We thus define Λ = µ + e nR ∞ x uγ u ) d u o . Setting L Λ = ln( Q / Λ ) to absorb the dependence on x, µ , we get − Z ∞ ¯ x ( L Λ ) uγ ( u ) d u = L Λ . (11)This equation defines L Λ (¯ x ) as well as the inverse function ¯ x ( L Λ ) . Since ¯ x (0) = + ∞ , we cannot trustour solution for Q < Λ . To remedy this, we use our previous result that for large x , γ ( x ) → − sx ,under our present assumptions.Following the conventions of Shirkov [24], we define a running coupling in accordance with the ex-pected analytic behavior of field theory using a dispersion relation: ¯ x disp ( Q ) = 1 π Z ∞ ℑ (¯ x (ln( σ/ Λ ))) σ + Q − iη d σ . (12)The pole at − Q gives us back the uncorrected ¯ x ( L Λ ) . But by assumption, there is a further pole in thecomplex σ -plane, located at L Λ ( ∞ ) = 0 . To study the contribution from that pole, we first note that as ¯ x → ∞ , we have L Λ = − Z ∞ ¯ x ( L Λ ) uγ ( u ) d u ≃ Z ∞ ¯ x ( L Λ ) su d u = 1 s ¯ x ( L Λ ) , and hence ¯ x ( L Λ ) ≃ sL Λ near L Λ = 0 . Feeding this relation into (12) gives (see [24]) ¯ x disp ( L Λ ) = ¯ x ( L Λ ) + 1 s (1 − e L Λ ) .
14f we were to identify s with the one-loop coefficient of the β -function, this would reproduce Shirkov’sanalysis for one-loop QCD, see [24], where Shirkov also notes that s seem not to vary much at low looporders. Note that the correction to ¯ x ( L Λ ) goes to the finite value /s in the infrared limit L Λ → −∞ .Now, γ also obeys an unsubtracted dispersion relation. As γ ( x ) is finite for all finite x and dependson L only through ¯ x , the dispersion integral will correct γ (¯ x ( L )) ≃ − s ¯ x ( L Λ ) by γ , disp (¯ x ( L Λ )) = γ (¯ x ( L Λ )) − − e L Λ . Note now that the correction to γ goes to − as L Λ → −∞ . Using (1) and xγ ( x ) ∂ x = ∂ L , we get γ k, disp (¯ x ( L Λ )) = γ k (¯ x ( L Λ )) + γ k, corr ( L Λ ) with γ k, corr ( L Λ ) → ( − k k ! as L Λ → −∞ . As such, the correction from the dispersion relation to the function G ( x, L Λ ) satisfies G corr ( x, L Λ ) → ∞ X k =0 ( − k k ! L k Λ = e − L Λ = Λ Q as L Λ → −∞ , which gives an inverse propagator satisfying lim L Λ →−∞ P − ( x, Q ) = − Λ . This gives a finite and renormalization group invariant positivity-violating mass gap. This comparesnicely with the results of Gracey et.al. [17].Note that we rely completely on the assumption that physical quantities in massless field theory haveneither poles nor branch cuts off the negative (in our conventions) real axis and the result that thesolutions we get from our ODEs for the uncorrected ¯ x, γ k , G ( x, L Λ ) are in accordance with these re-quirements but for the isolated pole at L Λ = 0 .If a future analysis of P ( x ) justifies the trust in field theory expressed in this section remains to beseen. We are content having identified a clean mechanism for the generation of a mass gap, based onthe consequences of the Hopf algebra structure underlying local field theory, and assuming analyticityproperties in accordance with the underlying axiomatic structure of local quantum fields.Let us add a few comments which put our results in context. We rely on the assumptions on the function P ( x ) outlined above. Note that the ODE for the anomalous dimension γ ( x ) takes into account all theiterations of superficially divergent graphs into each other, hence all renormalon ambiguities met in thecontext of resummation of a perturbative series are taken care of. We hope this will make P ( x ) , whichhas an asymptotic expansion related to a weighted skeleton expansion, amenable to more constructivemethods of analysis in the future, in contrast to γ ( x ) whose resummation as a perturbative series facessuch ambiguities.On the other hand, we emphasize that our analysis avoids any truncation of Dyson–Schwinger equa-tions, and any assumptions made on the infrared behaviour and infrared powercounting of QCD ampli-tudes. It thus complements the approaches available so far in the literature, see for example [16]. While15ther methods based on Dyson–Schwinger equations attempt to solve them in some justifiable limit ortruncation, we establish properties of the solution of the full equations depending on assumptions madefor P ( x ) . G ( x, L ) Our goal in this section is to analyze the dressing function G ( x, L ) that modulates the free inversepropagator. On general grounds, this function solves the Renormalization Group equation (cid:16) − ∂ L + xγ ( x ) ∂ x − sγ ( x ) (cid:17) G ( x, L ) = 0 and G ( x,
0) = 1 for x ≥ , (13)where s = ± distinguishes between writing the propagator as P ( x, Q ) = G ( x, ln( Q /µ )) s Q with s = 1 or s = − . Note that the difference of (13) from the usual RGE equation (cid:16) µ∂ µ + ˜ β ( g ) ∂ g + ˜ γ ( g ) (cid:17) G = 0 is merely a matter of convention on the definition of β , γ and x . In particular, (13) agrees with two-loopcomputation of the Gluon propagator.We will here make the following (minimal) hypothesis on γ ( x ) : Hypothesis 3.1
The function γ ( x ) is a negative C ([0 , ∞ ) , ( −∞ , function whose only possiblezero is at x = 0 , where γ ( x ) = − dx q + O ( x q +1 ) with q ≥ and d > . Note that the results of Section 2 give q = 1 and d = − P ′ (0) for the asymptotically free solution γ ( x ) , in accordance with perturbation theory. For the solutions satisfying γ (0) = − however, weget q = 0 and d = 1 .We first note that one can attempt to solve (13) by writing G ( x, L ) = 1 − s ∞ X k =1 γ k ( x ) L k with γ k ( x ) = γ ( x ) k (cid:16) x∂ x − s (cid:17) γ k − ( x ) k ≥ , (14)since one has (at least formally) that (cid:16) − ∂ L + xγ ( x ) ∂ x − sγ ( x ) (cid:17) G ( x, L ) = − sγ ( x ) + s ∞ X k =1 kγ k ( x ) L k − − γ ( x ) (cid:16) x∂ x − s (cid:17) γ k ( x ) L k = s ∞ X k =2 (cid:18) γ k ( x ) − γ ( x ) k (cid:16) x∂ x − s (cid:17) γ k − ( x ) (cid:19) kL k = 0 . (15)16his approach naturally raises the (difficult) question of convergence of the series in (14) and (15).We will use instead an alternative way of solving (13) that avoids these convergence problems. Inparticular, while the series (14) necessarily converge on a symmetric interval of the form ( − L , L ) forsome (possibly infinite) L ≥ , our approach will give a solution to (13) that is defined on an intervalof the form ( − L , ∞ ) for the same L . As such, the approach below shows that the limit L → ∞ of G ( x, L ) makes sense also if the series solution converges only on a finite interval.Our method is based on the fact that (13) can be transformed into a linear transport equation by ap-propriately factorizing G ( x, L ) into one part that cancels the term involving no derivatives and one partthat solves a genuine transport equation of the form ( − ∂ L + xγ ( x ) ∂ x ) H ( x, L ) = 0 . (16)As such, it is important at first to consider the characteristics curve of (16). For each fixed x > , wefirst define X ( t, x ) as the solution of the running coupling equation d X ( t, x )d t = X ( t, x ) γ ( X ( t, x )) with X ( t = 0 , x ) = x . (17)Since γ ( x ) is assumed to be C , solutions of this equation exist at least locally around t = 0 . Forfurther reference, we denote by D ( x ) the maximal interval of existence of the solution of (17) for afixed x .Then for each fixed ( x, L ) ∈ R + × R , we define the characteristic curve C ( x, L ) as C ( x, L ) = n ( X ( t, x ) , L − t ) with t ∈ D ( x ) o , = (cid:26) (cid:18) X, L − Z Xx d zzγ ( z ) (cid:19) with X ∈ R + (cid:27) . (18)Note that both above formulations of the characteristic curve C ( x, L ) are equivalent. The characteristicscorresponding to different values of L are vertical translations of the same curve in the ( x, L ) -plane.By hypothesis 3.1, we get that the characteristics are all asymptotically vertical as X → , and, as afunction of t , we have X ( t, x ) ≃ ( e − ct as t → ∞ if q = 0( ct ) − q as t → ∞ if q > . (19)The behavior of the characteristics as X → ∞ depends on the asymptotic behavior of γ ( x ) as x → ∞ .In all cases, C ( x, L ) approaches ( ∞ , L − L ∞ ( x )) as X → ∞ , where L ∞ ( x ) is the possibly infinite quantity defined by L ∞ ( x ) = Z ∞ x d zzγ ( z ) . This shows that the maximal interval of existence for solutions of (17) is D ( x ) = ( L ∞ ( x ) , ∞ ) . Theresults of Section 2 show that L ∞ ( x ) is finite if there exists x > where γ ( x ) = − , and infinite17 x L xL = L ∞ ( x ) Figure 2: Generic shape of characteristics curves in ( x, L ) -plane. The left panel shows the case where L ∞ ( x ) is finite, the right panel when it is infinite. In the left panel, the characteristics cross the line L =0 if and only if they are above the (bold) curve L = L ∞ ( x ) , and all characteristics are asymptoticallyhorizontal as x → ∞ . In the right panel, all characteristics cross the line L = 0 , as they link (0 , −∞ ) to ( ∞ , ∞ ) . 18therwise. If L ∞ ( x ) > −∞ , the characteristic C ( x, L ) intersects the line L = 0 if and only if L >L ∞ ( x ) . If − < γ ( x ) < for all x > , then L ∞ ( x ) = −∞ for all x > , and all characteristiccurves cross the line L = 0 . The generic shape of the characteristics curves is displayed in figure 2.We can now give the solution of (13) in accordance (for s = − , as expected) with Bogoliubov-Shirkov([12], App.IX, eq.(27)): Theorem 3.2
Assume γ ( x ) satisfies Hypothesis 3.1. Then the solution of (13) is given by G ( x, L ) = (cid:18) xX ( L, x ) (cid:19) s (20) for all ( x, L ) such that L ∞ ( x ) < L < ∞ and x > . We want to stress here that the relation (20) hold only for pair of values of ( x, L ) satisfying L ∞ ( x ) 0) = 1 if onespecifies γ ( x ) for x < as well.Assuming X ( t, x ) to be analytic for all t ∈ D ( x ) , it is a straightforward computation (see Section(3.3)) to show that (14) is the Taylor series expansion of (20) at L = 0 . As such, the series solution(14) is expected not to converge for | L | > L ∞ ( x ) . However, the function provided by (20) is definedfor unbounded positive L and do solve (13) for all such L . If L ∞ ( x ) > −∞ , Theorem 3.2 thus gives G ( x, L ) for values of ( x, L ) for which the series formulation fails. proof of Theorem 3.2. We first set G ( x, L ) = x s H ( x, L ) (21)for all x > . Substitution into (13) gives (cid:16) − ∂ L + xγ ( x ) ∂ x (cid:17) H ( x, L ) = 0 and H ( x, 0) = x − s if x > . (22)We then note that H ( x, L ) is constant along the characteristic curve C ( x, L ) , for we have dd t (cid:16) H ( X ( t, x ) , L − t ) (cid:17) = − D H ( X ( t, x ) , L − t ) + D H ( X ( t, x ) , L − t ) d X ( t, x )d t = h(cid:16) − ∂ L + xγ ( x ) ∂ x (cid:17) H ( x, L ) i x = X ( t,x ) ,L = L − t = 0 . If the curve C ( x, L ) crosses the line L = 0 in the ( x, L ) -plane, it does so at t = L , and we get H ( x, L ) = H ( X ( L, x ) , 0) = X ( L, x ) − s , which completes the proof. 19 .3 The series solution for G ( x, L ) Throughout this section, we consider γ to be a fixed solution of (5) that exists for all x ≥ . We firstpick x > and t ∈ R . These values are arbitrary. We then introduce T ( x ) = t + Z xx d zγ ( z ) z . (23)We then note that lim x →∞ T ( x ) = ( T ∞ = −∞ if − < γ ( x ) ≤ ∀ x ≥ ,T ∞ > −∞ if ∃ x > with γ ( x ) = − . (24)Since T ′ ( x ) = γ ( x ) x < , x T ( x ) is an invertible map from R + to [ T ∞ , ∞ ) . Moreover, we have T (0) = ∞ and T ( ∞ ) = T ∞ . The inverse map ˜ x ( t ) (the solution of t = T (˜ x ( t )) ) satisfies the ‘runningcoupling equations’ d˜ γ ( t )d t = ˜ γ ( t ) + ˜ γ ( t ) − P (˜ x ( t )) and d˜ x ( t )d t = ˜ x ( t )˜ γ ( t ) , (25)where ˜ γ ( t ) = γ (˜ x ( t )) . Since T ( ∞ ) = T ∞ , ˜ x ( t ) diverges as t → T ∞ (and so does ˜ γ ( t ) if γ ( x ) = − for some x > ).Consider now the series solution (14). We first introduce the functions S k such that γ k ( x ) = 1 k ! x s S k ( T ( x )) for k ≥ . Substitution into (14) gives S k ( T ( x )) = S ′ k − ( T ( x )) = (cid:18) dd t S k − ( t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = T ( x ) . Since s = ± , we find from (25) that S ( t ) = ˜ γ ( t )˜ x ( t ) s = − s dd t (˜ x ( t ) − s ) . Using s = 1 , we find − sγ k ( x ) = x s k ! (cid:18) d k d t k (cid:16) x ( t ) s (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = T ( x ) (26)for all k ≥ . Since ˜ x ( T ( x )) = x , the r.h.s. of (26) is equal to when k = 0 , and so G ( x, L ) = 1 + ∞ X k =1 − sγ k ( x ) L k = x s ∞ X k =0 L k k ! (cid:18) d k d t k (cid:16) x ( t ) s (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = T ( x ) . (27)We now fix < x < ∞ . Since x is finite, T ( x ) > T ∞ , and since the above series is a Taylor seriesof ˜ x ( t ) − s at t = T ( x ) , the series converges and takes the value ˜ x ( t + L ) − s for small values of L if weassume z ( t ) to be analytic at t = T ( x ) . We thus find G ( x, L ) = (cid:18) x ˜ x ( T ( x ) + L ) (cid:19) s , (28)20t least for sufficiently small L . This formula is the same as the one of Theorem 3.2: ˜ x ( T ( x ) + t ) solves d˜ x ( T ( x ) + t )d t = ˜ x ( T ( x ) + t ) γ (˜ x ( T ( x ) + t )) , (29)with initial condition ˜ x ( T ( x )) = x at t = 0 , and so ˜ x ( T ( x ) + t ) = X ( t, x ) by uniqueness of solutionsof (17) and (29).The question of convergence of (27) for large L depends on the (in-)finiteness of T ∞ . If T ∞ is finite, ˜ x ( t ) has a pole as t → T ∞ , and (27) must diverge for L ≤ T ∞ − T ( x ) = Z ∞ x d zzγ ( z ) = L ∞ ( x ) as in Section 3.2. However, since the series (27) can only converge for L on symmetric intervals, it isalso divergent for L ≥ − L ∞ ( x ) > . In that case, we have to revert to Section 3.2 for the validity of(28). In this section, we present the technical details proving the statements of Section 2 on QCD. In ouranalysis of (5), we use mainly two tools: integral representations of the solutions and null-clines. Ourstrategy is directly inspired from [6] and our previous work [5] on QED.There are two types of integral equations one can write for (5). The first one (see also [5]) reads γ ( x ) = x (1 + γ ( x ⋆ )) x ⋆ − − x Z xx ⋆ P ( z ) z γ ( z ) d z . (30)For the second one, we let γ ( x ) be a solution of (5) and γ ( x ) be any function. Then for all x , x ∈ I ,where I is the common interval where γ and γ are defined, we have γ ( x ) − γ ( x ) = (cid:16) γ ( x ) − γ ( x ) (cid:17) K [ γ , γ ]( x , x ) + Z x x R [ γ ]( y ) K [ γ , γ ]( y, x ) d y , (31)where R [ γ ]( x ) ≡ d γ ( x )d x − γ ( x ) + γ ( x ) − P ( x ) xγ ( x ) ,K [ γ , γ ]( x , x ) ≡ (cid:18) xx (cid:19) exp (cid:18) − Z x x P ( z ) zγ ( z ) γ ( z ) d z (cid:19) . The null-clines are defined as the locations in ( x, γ ) -plane where solutions satisfy γ ′ ( x ) = 0 ). Theseare given by the graph of the two functions γ ± c ( x ) = ± p P ( x ) − . 21n particular, as P (0) = 0 by hypothesis H1, and P ( x ) > − for x ∈ [0 , x ⋆ ] , the null-clines extend atleast up to the line x = x ⋆ in the ( x, γ ) -plane. On the null-clines, the second derivative of γ is givenby γ ′′ ( x ) = dd x (cid:0) f ( γ ± c ( x ) , x ) (cid:1) = P ′ ( x ) | xγ ± c ( x ) | . (32)By hypotheses H1 and H2, we have P ′ ( x ) = P ′ (0) + Z x P ′′ ( z )d z ≤ P ′ (0) < ∀ x ∈ [0 , x ⋆ ] . (33)Hence by (32), solutions of (5) can have at most one local maximum in the interval x ∈ [0 , x ⋆ ] , and nolocal minimum. For further reference, we also note that by hypotheses H1 and H2, P ( x ) = P ′ (0) x + Z x (cid:18)Z y P ′′ ( z )d z (cid:19) d y ≤ P ′ (0) x ∀ x ∈ [0 , x ⋆ ] . (34)In particular, we have min x ∈ [0 ,x ⋆ ] − P ( x ) x = − P ′ (0) = | P ′ (0) | > , (35)and since P ( x ⋆ ) > − by hypothesis on x ⋆ , we have x ⋆ < − P ′ (0) < ∞ . We can now establish the existence, uniqueness and properties of the solution with asymptotic freedom. Theorem 4.1 Under the hypotheses H1 and H2 , there exists a unique value γ ⋆ ( x ⋆ ) such that the cor-responding solution γ ⋆ ( x ) of (5) exists for all x ∈ [0 , x ⋆ ] and satisfies lim x → γ ⋆ ( x ) = 0 .Proof. We first prove that the solution is unique. Namely, assume ab absurdum that γ and γ are twosolutions of (5) on [0 , x ⋆ ] satisfying lim x → γ i ( x ) = 0 . Since γ i ( x ⋆ ) < and lim x → γ i ( x ) = 0 , we necessarilyhave γ i ( x ) ≥ γ + c ( x ) for all x ∈ [0 , x ⋆ ] . Since γ is a solution, we can apply (31) with R [ γ ] = 0 , andusing (35), we get | γ ( x ) − γ ( x ) | ≥ | γ ( x ⋆ ) − γ ( x ⋆ ) | xx ⋆ exp (cid:18)Z x ⋆ x | P ′ (0) | γ + c ( z ) d z (cid:19) . (36)Since γ + c ( z ) = P ′ (0) z + O ( z ) as z → , the r.h.s. of (36) diverges as x → , which contradicts lim x → γ i ( x ) = 0 . This shows that there can be at most one solution of (5) satisfying lim x → γ ( x ) = 0 .22o prove the existence of this solution, we define the two sets I = { γ ( x ⋆ ) ∈ ] γ + c ( x ⋆ ) , s.t. ∃ x min ∈ ]0 , x ⋆ [ with γ ( x min ) = 0 } , I = { γ ( x ⋆ ) ∈ ] γ + c ( x ⋆ ) , s.t. ∃ x ∈ ]0 , x ⋆ [ with γ ( x ) = γ + c ( x ) } . We will prove in Propositions 4.2 and 4.4 below that these sets are non-empty. Continuity of solutionsw.r.t. initial conditions imply that they are open, while (31) with R [ γ ] = 0 shows that solutions are ordered , which imply that each I i is a single interval. Now since I and I are disjoint open intervals,there exist at least one initial condition γ ⋆ ( x ⋆ ) that is in neither sets, and hence the correspondingsolution γ ⋆ ( x ) satisfies lim x → γ ⋆ ( x ) = 0 .To establish that I is non-empty, we show that initial conditions at x = x ⋆ sufficiently close to the x -axis necessary cross it at some x min < x ⋆ . Proposition 4.2 For all γ ∈ [ P ′ (0) x ⋆ , , there exists x min ∈ ]0 , x ⋆ [ such that the solution of (5) onlyexists on x ∈ [ x min , x ⋆ ] and γ ( x min ) = 0 .Proof. Pick γ ( x ) = P ′ (0) x . Then by (34), we have R [ γ ]( x ) = 1 x P ′ (0) Z x (cid:18)Z y P ′′ ( z )d z (cid:19) d y > ∀ x ∈ [0 , x ⋆ ] . Applying (31), we get γ ( x ) = P ′ (0) x + Z x ⋆ x R [ γ ]( y ) K [ γ , γ ]( y, x ) d y > P ′ (0) x ∀ x < x ⋆ . (37)In particular, γ cannot cease to exist by diverging to −∞ at a finite x < x ⋆ . We also see from (37) thatthere exists x < x ⋆ with γ ( x ) > P ′ (0) x . Using R [ γ ] > , (35) and γ ( x ) ≥ P ′ (0) x in (31), wefind γ ( x ) ≥ P ′ (0) x + ( γ ( x ) − P ′ (0) x ) xx exp (cid:18)Z x x − P ′ (0) z d z (cid:19) ≥ xx (cid:16) P ′ (0) x + ( γ ( x ) − P ′ (0) x )e | P ′ (0) | x − | P ′ (0) | x (cid:17) . The proof is completed since γ ( x ) − P ′ (0) x > .Before proving that the interval I is non-empty, we can establish an additional property of the asymp-totically free solution γ ⋆ : Corollary 4.3 The solution γ ⋆ of Theorem 4.1 satisfies γ + c ( x ) < γ ( x ) < P ′ (0) x . (38) for all x ∈ ]0 , x ⋆ ] . roof. To get the lower bound in (38), we recall that solutions can have at most one local maximum in ]0 , x ⋆ ] and no local minimum. Since lim x → γ ⋆ ( x ) = 0 , γ ( x ) > γ + c ( x ) for all x ∈ ]0 , x ⋆ ] . The upper boundfollows immediately from Proposition 4.2.We now show that initial conditions sufficiently close (but above) the null-cline γ + c ( x ⋆ ) necessarilycross it at some x < x ⋆ . Proposition 4.4 There exist ǫ ≪ sufficiently small such that the solution of (5) with γ ( x ⋆ ) = γ + c ( x ⋆ )(1 − ǫ ) satisfies γ ( x ) = γ c ( x ) for some < x < x ⋆ .Proof. Let < ǫ < and γ ( x ⋆ ) = γ + c ( x ⋆ )(1 − ǫ ) . By continuity of solutions and since γ + c ( x ⋆ ) <γ ( x ⋆ ) < γ ( x ⋆ )(1 − ǫ ) , there exists < x < x ⋆ such that ( γ + c ( x ⋆ ) < γ ( x ) < γ + c ( x ⋆ )(1 − ǫ ) ∀ x ∈ ] x , x ⋆ ] and γ ( x ) = γ + c ( x ⋆ ) or γ ( x ) = γ + c ( x ⋆ )(1 − ǫ ) . (39)Using these inequalities and hypothesis H2, we get d γ d x ≥ − ǫ − ǫ P ( x ⋆ ) γ + c ( x ⋆ ) ∀ x ∈ [ x , x ⋆ ] , which, upon integration, gives γ ( x ) ≤ γ + c ( x ⋆ )(1 − ǫ ) + 2 ǫ − ǫ P ( x ⋆ ) γ + c ( x ⋆ ) ln (cid:18) x ⋆ x (cid:19) ∀ x ∈ [ x , x ⋆ ] . (40)Let now x be the value at which the r.h.s. of (40) attains γ + c ( x ⋆ )(1 − ǫ ) , namely x = x ⋆ exp (cid:18) − γ + c ( x ⋆ ) (1 − ǫ )2 | P ( x ⋆ ) | (cid:19) < x ⋆ . We now consider the two alternatives x < x and x ≥ x .Assume first that x < x . Then (40) shows that γ ( x ) < γ + c ( x ⋆ )(1 − ǫ ) , and thus by definition of x (see (39)), we have γ ( x ) = γ + c ( x ⋆ ) and since by H2, γ + c ( x ) increases as x → , there exists an x ∈ [ x , x ⋆ ] with γ ( x ) = γ + c ( x ) .Consider now the other possible case, namely x ≥ x . Since now γ + c ( x ⋆ ) ≤ γ ( x ) ≤ γ + c ( x ⋆ )(1 − ǫ ) for all x ∈ [ x , x ⋆ ] , and we conclude from (40) that γ ( x ) − γ + c ( x ) ≤ γ + c ( x ⋆ ) − γ + c ( x ) − ǫ γ + c ( x ⋆ ) + 2 ǫ − ǫ P ( x ⋆ ) γ + c ( x ⋆ ) ln (cid:18) x ⋆ x (cid:19) . (41)The proof is completed by noting that for x very close to (but strictly less than) x ⋆ , γ + c ( x ⋆ ) − γ + c ( x ) isnegative, while the last two (positive terms) in (41) can be made arbitrarily small by picking ǫ smallenough. 24 .2 Behavior towards x = 0 of non-asymptotically free solutions We first consider solutions of (5) corresponding to initial conditions γ ( x ⋆ ) < γ ⋆ ( x ⋆ ) . We have thefollowing result. Proposition 4.5 Let γ < γ ⋆ ( x ⋆ ) . The corresponding solution γ ( x ) of (5) exists for all x ∈ [0 , x ⋆ ] andsatisfies γ ( x ) = − O ( x ln( x )) as x → .Proof. Note first that since γ < γ ⋆ ( x ⋆ ) , there always exists x ≤ x ⋆ such that γ ( x ) ≤ γ + c ( x ) .Namely, if this does not already hold at x ⋆ , the proof of Theorem 4.1 shows that it will eventually holdat some smaller value of x . Since solutions can have at most one local maximum and no local minimumin [0 , x ⋆ ] , we then have min( − , γ ( x )) ≤ γ ( x ) ≤ γ + c ( x ) ∀ x ∈ [0 , x ] . In particular, these solutions exist for all values of x ∈ [0 , x ⋆ ] . We then apply (30) and get − Cx + x | γ + c ( x ) | Z x x − P ( z ) z d z ≤ γ ( x ) ≤ − Cx + x max(1 , | γ + c ( x ) | ) Z x x − P ( z ) z d z , where C = γ ( x ) x . By hypotheses H1 and H2 (see also (34), P ( x ) = P ′ (0) x + O ( x ) it is thenstraightforward to prove that γ ( x ) = − O ( x (1 + ln( x ))) as x → , which completes the proof.We now consider solutions of (5) corresponding to initial conditions γ ( x ⋆ ) in the interval ] γ ⋆ ( x ⋆ ) , .We have the following result. Proposition 4.6 Let γ ∈ ] γ ⋆ ( x ⋆ ) , . The corresponding solution γ ( x ) of (5) satisfies γ ( x min ) = 0 forsome x min ∈ ]0 , x ⋆ [ . It can then be continued and enters the first quadrant (becoming double-valued),and thus satisfies γ ( x ) > for x > x min .Proof. Let x min ≥ be the minimal value such that γ ( x ) exists ∀ x ∈ [ x min , x ⋆ ] . We claim that x min > and γ ( x min ) = 0 . Namely, since γ ( x ⋆ ) > γ ⋆ ( x ⋆ ) , we have by (31) with R [ γ ] = 0 that γ ( x ) = γ ⋆ ( x ) + ( γ ( x ⋆ ) − γ ⋆ ( x ⋆ )) xx ⋆ exp (cid:18)Z x ⋆ x − P ( z ) zγ ⋆ ( z ) γ ( z ) d z (cid:19) > γ ⋆ ( x ) (42)for all x ∈ [ x min , x ⋆ ] . Assuming ab absurdum that x min = 0 and γ ( x ) < for all x ∈ [0 , x ⋆ ] leads toa contradiction, for then we would have γ ( x ) > γ ⋆ ( x ) ≥ γ + c ( x ) for all x ∈ [0 , x ⋆ ] , and using (35) and(42), we get γ ( x ) ≥ γ ⋆ ( x ) + ( γ ( x ⋆ ) − γ ⋆ ( x ⋆ )) xx ⋆ exp (cid:18)Z x ⋆ x | P ′ (0) | γ + c ( z ) d z (cid:19) + ∞ as x → . So x min > and γ ( x min ) = 0 . Although (5) is singular at γ ( x min ) = 0 ,these solutions can be continued in the first quadrant by reverting to the so-called ‘running coupling’formulation of (5) (see also (25) and [5]). Namely, we introduce a new independent variable t , andwrite x = X ( t ) and γ ( X ( t )) = ˜ γ ( t ) , getting d˜ γ ( t )d t = ˜ γ ( t ) + ˜ γ ( t ) − P ( X ( t )) ˜ γ ( t ) = 0 , d X ( t )d t = X ( t )˜ γ ( t ) X ( t ) = x min . These equations are not singular at ˜ γ = 0 , and thus solutions will exist (at least locally around t = t ).Since P ( x ) < , the solution to these equations will satisfy ˜ γ ( t ) = γ > and X ( t ) = x > x min forsome finite t > t . x → ∞ We first show that solutions in the first quadrant are global, and satisfy appropriate estimates as x → ∞ . Proposition 4.7 Let γ ( x ) > , and assume P ( x ) satisfies H3 . The corresponding solution γ ( x ) exists for all x ≥ x , and satisfies < xS P ( x , x ) ≤ γ ( x ) ≤ xS P ( x , x ) + xx − for all x ≥ x .Proof. We first note that 12 dd x (cid:16) γ ( x ) (cid:17) = γ ( x ) d γ d x = γ ( x ) x + γ ( x ) − P ( x ) x ≥ γ ( x ) − P ( x ) x . By integration, we find γ ( x ) ≥ x s γ ( x ) x + 2 Z xx − P ( z ) z d z = xS P ( x , x ) > . (43)This shows that solutions cannot cease to exist by reaching γ ( x ) = 0 at some x > x . Inserting (43)into (30) gives γ ( x ) ≤ x (1 + γ ( x )) x − x Z xx − P ( z ) z S P ( x , z ) d z = xS P ( x , x ) + xx − , (44)since d S P ( x , x )d x = − P ( x ) x S P ( x , x ) . ∞ at afinite x ≥ x either.We now turn to the fate of any type of solutions of (5) with γ ( x ⋆ ) < as x → ∞ . Our first result isthat these solutions are global, i.e. , they can be extended as x → ∞ . In particular, the asymptoticallyfree γ ⋆ ( x ) is global. Proposition 4.8 Let γ ( x ⋆ ) < and assume P ( x ) satisfies H3 . The corresponding solution of (5)exists for all x ≥ x ⋆ , and satisfies − xS P ( x ⋆ , x ) ≤ γ ( x ) ≤ max γ ( x ⋆ ) , sup z ∈ [ x ⋆ ,x ] p P ( z ) − ! < for all x ≥ x ⋆ . In particular, if D ( P ) < ∞ , solutions grow at most linearly as x → ∞ .Proof. For the lower bound, note first that, as in the proof of Proposition 4.6, we have 12 dd x (cid:16) γ ( x ) (cid:17) = γ ( x ) d γ d x = γ ( x ) x + γ ( x ) − P ( x ) x ≤ γ ( x ) − P ( x ) x , which gives γ ( x ) ≥ − xS P ( x ⋆ , x ) upon integration. This shows that solutions cannot diverge to −∞ at a finite x > x ⋆ . Now suppose ab absurdum that there exists x max < ∞ such that γ ( x max ) = 0 . Byhypothesis H1-H3, we have − < sup x ∈ [ x ⋆ ,x max ] P ( x ) < , and thus γ max = max γ ( x ⋆ ) , sup z ∈ [ x ⋆ ,x max ] p P ( z ) − ! satisfies − < γ max < . Note then that f ( γ , x ) is strictly negative along the γ = γ max line since sup x ∈ [ x ⋆ ,x max ] xf ( γ max , x ) ≤ (cid:18) γ max − δγ max (cid:19) ≤ − . Since γ ( x ⋆ ) ≤ γ max , this shows that γ ( x ) ≤ γ max for all x ∈ [ x ⋆ , x max ] , contradicting the ab absurdum assumption. Hence solutions exist globally as x → ∞ .Our second result concern the asymptotics of some of these solutions as x → ∞ . Namely, we canestimate the growth of solutions that are somewhere less than − . Proposition 4.9 Assume P ( x ) satisfies H3 and γ ( x ⋆ ) < . If the corresponding solution of (5) satis-fies γ ( x ) ≤ − for some x ≥ x ⋆ , then − xS P ( x , x ) ≤ γ ( x ) ≤ − xS P ( x , x ) + xx − < for all x ≥ x . roof. The lower bound is already contained in Proposition 4.8. The upper bound then follows imme-diately from the lower bound and the integral formulation (30).The condition γ ( x ) ≤ − is essential in Proposition 4.9 to guarantee that the upper bound is indeednegative. We now give possible scenarios that guarantee solutions indeed reach γ = − . Proposition 4.10 Assume one of the two following statements holds:1. − < γ ( x ⋆ ) < and P ( x ) satisfies S1 or S3,2. − < γ ( x ⋆ ) ≤ γ ⋆ ( x ⋆ ) and P ( x ) satisfies S2.Then there exists x > x ⋆ such that the corresponding solution γ ( x ) satisfies γ ( x ) = − .Proof. We consider the alternative 1. first. Note that S1 implies S3, so we can use hypothesis S3 only.As shown in Proposition 4.8, any solution starting at γ ( x ⋆ ) < exists for all values of x ≥ x ⋆ . Inparticular, γ ( x c ) = γ min < . If γ min ≤ − , the proof is completed. If − < γ min < , we assume ab absurdum that γ ( x ) > − for all x ∈ [ x c , x d ] . Note that there cannot be an x ∈ [ x c , x d ] such that γ ( x ) = 0 , hence − < γ ( x ) < for all x ∈ [ x c , x d ] , and we have dd x ( γ ( x ) ) = 2 γ ( x ) + γ ( x ) + x − P ( x ) + 12 x ≥ − P ( x ) + 12 x for all x ∈ [ x c , x d ] . Upon integration, we thus find that γ ( x ) ≤ − s γ − Z xx c P ( z )2 z d z ≤ − q γ < − by hypothesis S3, which is a contradiction.Consider then the alternative 2. Under hypothesis S2, we can extend (35) to get P ( x ) ≤ P ′ (0) x forall x ∈ [0 , x c ] . Consider now γ ( x ) = P ′ (0) x . We have R [ γ ]( x ) > for all x ∈ [0 , x c ] . Thus, since γ ( x ⋆ ) ≤ γ ⋆ ( x ⋆ ) and γ ⋆ ( x ⋆ ) < P ′ (0) x ⋆ by Corollary 4.3, we find from (31) that γ ( x ) = P ′ (0) x − (cid:12)(cid:12)(cid:12) γ ( x ⋆ ) − γ ( x ⋆ ) (cid:12)(cid:12)(cid:12) K [ γ , γ ]( x ⋆ , x ) − Z xx ⋆ R [ γ ]( y ) K [ γ , γ ]( y, x ) d y ≤ P ′ (0) x for all x ∈ [ x ⋆ , x c ] . The proof is completed since x c > − P ′ (0) by hypothesis S2, and thus γ ( x c ) ≤ P ′ (0) x c ≤ − .We conclude this section by showing that D ( P ) < ∞ implies that all solutions that are either positiveor go below γ = − have a finite slope as x → ∞ .28 roposition 4.11 Assume D ( P ) < ∞ and γ ( x ) ≤ − or γ ( x ) > , there exists s > such that lim x →∞ γ ( x ) x = ( − s < if γ ( x ) ≤ − s > if γ ( x ) > . If γ ( x ) ≤ − , the convergence towards the limit is given by (cid:12)(cid:12)(cid:12) γ ( x ) x + s (cid:12)(cid:12)(cid:12) ≤ C Z ∞ x − P ( z ) z d z . (45) If D ( P ) < ∞ and γ ( x ) > , then (8) also hold, with − s replaced by s .Proof. Note that from Proposition 4.7 and 4.9, the hypothesis D ( P ) < ∞ implies that all solutionsunder consideration here satisfy c x ≤ | γ ( x ) | ≤ c x (46)for some c , c > and all x ≥ x . The integral formulation (30) then gives γ ( x ) x = 1 + γ ( x ) x − x − Z xx P ( z ) z γ ( z ) d z , (47)from which the proof follows immediately, since the r.h.s. of (47) converges by (46) and the hypothesis D ( P ) < ∞ . In this section, we consider P ( x ) satisfying the hypotheses H1,H2 and S4. In particular, recall that weassume the existence of x r > such that P ( x r ) = − and P ( x ) > − for all x > x r and P ( x ) tendingto a finite limit as x → ∞ . Any solution of (5) that satisfies lim x →∞ γ ( x ) = γ ∞ ≡ − √ P ∞ , (48)needs to solve the integral equation obtained by taking the (formal) limit x ⋆ → ∞ in (30), namely γ c ( x ) = − x Z ∞ x P ( z ) z γ c ( z ) d z = − Z ∞ P ( xt ) t γ c ( xt ) d t . (49)Defining T [ h ]( x ) = Z ∞ P ( xt ) − P ∞ t ( γ ∞ + h ( xt )) d t − P ∞ γ ∞ Z ∞ h ( xt ) t ( γ ∞ + h ( xt )) d t , 29e see that any confinement solution can be written as γ c ( x ) = γ ∞ + h ( x ) where h ( x ) satisfies h ( x ) = T [ h ]( x ) . Consider then B x the Banach space obtained by completing the space of C ∞ ([ x , ∞ ) , R ) functions under the norm k f k x ≡ sup x ≥ x | f ( x ) | + x | f ′ ( x ) | . Since lim x →∞ P ( x ) − P ∞ = lim x →∞ xP ′ ( x ) = 0 , k P − P ∞ k x can be made as small as one likes by taking x > x r large enough. Standard arguments then show that T is a contraction in a ball of positiveradius ρ < centered at in B , which shows there exists a unique h ∈ B solving h = T [ h ] . Since h ( x ) is regular, γ c ( x ) = γ ∞ + h ( x ) solves (5) for all x ≥ x . We now remark that γ c ( x ) will satisfy(49) as long as it exists when x decreases below x . However, it can only cease to exist if it satisfies γ ( x min ) = 0 for some x min > , for the r.h.s. of (5) is negative for large negative γ . Assuming it canbe continued up to x = 0 , we can conclude from Theorem (2.2) that either γ c (0) = 0 or γ c (0) = − .Finally, if P ′ ( x ) > for all x > x max , then the lower nullcline γ − ( x ) = − p P ( x ) + 12 is decreasing towards γ ∞ . If there was an x > x max such that γ c ( x ) = γ − ( x ) , then γ c ( x ) wouldenter a region of strictly positive derivatives w.r.t. x , and hence we would get γ c ( x ) > γ − ( x ) > γ ∞ ,contradicting (48). Similarly, if there was an x > x max such that γ c ( x ) = γ ∞ , then γ c ( x ) would entera region of strictly negative derivative w.r.t. x , and hence would satisfy γ c ( x ) < γ ∞ for all x > x ,contradicting (48) again. We thus find that under the monotonicity assumption on P ( x ) , we have γ ∞ ≡ − √ P ∞ < γ ( x ) < − p P ( x )2 for all (finite) x > x max . This concludes the proof of Theorem 2.5. In a previous publication (see [5]), we have considered the initial value problem d γ ( x )d x = f s ( γ ( x ) , x ) ≡ γ ( x ) + γ ( x ) − P ( x ) sxγ ( x ) , γ ( x ) = γ > . (50)with < x < fixed and P a C function on [0 , ∞ ) , positive on (0 , ∞ ) , with P (0) = 0 and P ′ (0) = 0 .Defining the following (possibly infinite) quantity D ( P ) = Z ∞ x − P ( z ) z d z , we showed that D ( P ) was intimately linked to the behavior/existence as x → ∞ of solutions startingwith γ ( x ) > . Namely, if D ( P ) < ∞ , we showed that there was a smallest (non-zero) value γ ⋆ ( x ) γ ( x ) ≥ γ ⋆ ( x ) ) from solutions that cannot be continued to x = ∞ .We also showed that despite the singular nature of (50) as x → and/or γ → , all solutions of (50)could be continued for all x ∈ [0 , x ] , and would approach (0 , while satisfying for all x ∈ [0 , x ] thebound γ ( x ) ≤ C b x if s < C b x | ln( x ) | if s = 1 C b x /s if s > , (51)for some constant C b = C b ( s, γ ) . In Lemma 5.2 below, we will improve the above to get linear boundsin all cases (i.e. for s ≥ as well). In the mean time, we define, for any (finite) integer p ≥ , thetruncation of the (divergent) series solution: γ ,p ( x ) = P ′ (0) x + p X n =2 a n x n . The coefficients { a n } n =2 ...p can be found recursively by imposing | R [ γ ,p ]( x ) | ≤ Cx p − as x → ,where the remainder map R is defined by R [ γ ]( x ) ≡ d γ ( x )d x − γ ( x ) + γ ( x ) − P ( x ) sxγ ( x ) for γ ( x ) any function on [0 , x ] . Note that by choosing x sufficiently small, we can ensure that γ ,p ( x ) > for all x ∈ [0 , x ] . Also, since P ′ (0) = 0 and γ ,p is continuous, there exists a constant C > such that γ ( x ) ≤ Cx for all x ∈ [0 , x ] .Finally, we also note that if γ solves (50) and γ ( x ) is any positive function on [0 , x ] , then for all x ∈ [0 , x ] , we have γ ( x ) − γ ( x ) = (cid:16) γ ( x ) − γ ( x ) (cid:17) K [ γ , γ ]( x , x ) + Z x x R [ γ ]( y ) K [ γ , γ ]( y, x ) d y , (52)where K [ γ , γ ]( x , x ) ≡ exp (cid:18)Z xx sz + P ( z ) szγ ( z ) γ ( z ) d z (cid:19) = (cid:18) xx (cid:19) s exp (cid:18) − Z x x P ( z ) szγ ( z ) γ ( z ) d z (cid:19) . We are now in position to show that all solutions to (50) agree to all orders in perturbation theory withthe series solution. Theorem 5.1 Let γ and γ be two solutions of (50). Fix p ≥ and let γ ,p as above. Then there existconstants C and C p such that (cid:12)(cid:12) γ ( x ) − γ ,p ( x ) (cid:12)(cid:12) ≤ C p x p , (53) (cid:12)(cid:12)(cid:12) γ ( x ) − γ ( x ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) γ ( x ) − γ ( x ) (cid:12)(cid:12)(cid:12) (cid:18) xx (cid:19) s exp (cid:18) C (cid:16) x − x (cid:17)(cid:19) , (54) where (53) holds for all x ∈ [0 , x / and (54) for all x ∈ [0 , x ] . 31n other words, for any solution of (50), any derivative (of finite order) converges as x → to thecorresponding derivative of γ ,p . Hence a (truncated) power series expansion at x = 0 of any solutionagrees to any (finite) order with the truncated divergent series of the same order. Also, the differencebetween any two solutions of (50) decays faster than e − C/x as x → . Proof. In Lemma 5.2 below, we prove that there exist constants c and c such that γ ( x ) ≤ c x and γ ( x ) ≤ c x for all x ∈ [0 , x ] if γ and γ are solutions of (50). In particular, there exist a constant C > such that K [ γ , γ ]( y, x ) ≤ (cid:18) xy (cid:19) s exp (cid:18) − Z yx Cz d z (cid:19) = (cid:18) xy (cid:19) s exp (cid:18) C (cid:16) y − x (cid:17)(cid:19) (55)for all ≤ x ≤ y ≤ x . Since R [ γ ]( x ) = 0 if γ is a solution of (50), we have γ ( x ) − γ ( x ) = (cid:16) γ ( x ) − γ ( x ) (cid:17) K [ γ , γ ]( x , x ) , from which we get (54) immediately.On the other hand, γ ,p also satisfies γ ,p ( x ) ≤ Cx for all x ∈ [0 , x ] , and so K [ γ , γ ,p ] also satisfies K [ γ , γ ,p ]( y, x ) ≤ (cid:18) xy (cid:19) s exp (cid:18) C (cid:16) y − x (cid:17)(cid:19) for all ≤ x ≤ y ≤ x . In particular, for fixed x , K [ γ , γ ,p ]( x , x ) decays faster than any power lawas x → . Assume now that x ≤ x / , and note that (cid:12)(cid:12) γ ( x ) − γ ,p ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) γ ( x ) − γ ,p ( x ) (cid:12)(cid:12) K [ γ , γ ,p ]( x , x ) + C Z x x y p − K [ γ , γ ,p ]( y, x ) d y . Splitting the integral over [ x, x ] into [ x, x ] and [2 x, x ] , and using that K [ γ , γ ,p ]( y, x ) is a decreasingfunction of y , we find Z xx y p − K [ γ , γ ,p ]( y, x ) d y ≤ Z xx y p − K [ γ , γ ,p ]( x, x ) d y = x p (cid:0) p − p (cid:1)Z x x y p − K [ γ , γ ,p ]( y, x ) d y ≤ K [ γ , γ ,p ](2 x, x ) Z x x y p − d y = K [ γ , γ ,p ](2 x, x ) (cid:0) x p − x p p (cid:1) . The proof of (53) is completed by noting that by (55), K [ γ , γ ,p ](2 x, x ) and K [ γ , γ ,p ]( x , x ) decayfaster than any power law as x → . Lemma 5.2 For any γ > , the solution of (50) exists for all x ∈ [0 , x ] and γ ( x ) ≤ C x ∀ x ∈ [0 , x ] for some C = C ( x , γ ) > . roof. We first note for future reference that P ( x ) x is continuous for all x ∈ [0 , x ] , and that there existsconstants C ± > such that C − x ≤ P ( x ) ≤ C + x for all x ∈ [0 , x ] . By (51), we only have to consider s ≥ . We first choose α > such that α ≥ (cid:26) max(2 C + , C b | ln( x ) | ) if s = 1max (cid:16) C + , x ( s − , C sb s − ( s − s − (cid:17) if s > . We then note that f s ( α x, x ) − α = 1 sx − P ( x ) sx α + α (cid:0) s − (cid:1) ≥ sx − C + sxα + α (cid:0) s − (cid:1) ≥ sx + α (cid:0) s − (cid:1) , from which it follows that f s ( α x, x ) > α ∀ x ∈ [0 , X ( s, α )] where X ( s, α ) = (cid:26) x if s = 1 α ( s − if s > . (56)We then note that C b X (1 , α ) | ln( X (1 , α )) | = C b x | ln( x ) | ≤ αx = αX (1 , α ) ,C b X ( s, α ) /s = C b (4 α ( s − /s ≤ s − = αX ( s, α ) if s > . (57)In other words, (51) and (57) imply that the solution of (50) satisfies γ ( X ( s, α )) ≤ αX ( s, α ) . Since X ( s, α ) ≤ x , this means that as x decreases, solutions enter the triangle ∆ s,α = (cid:8) ( x, γ ) | x ∈ [0 , X ( s, α )] and ≤ γ ≤ α x (cid:9) through its right boundary (i.e. at x = X ( s, α ) ).Solutions that enter ∆ s,α at x = X ( s, α ) cannot satisfy γ ( x ⋆ ) = αx ⋆ at some x ⋆ < X ( s, α ) , forby (56), we would have γ ( x ) > αx for all x ∈ ( x ⋆ , X ( s, α )] , a contradiction with γ ( X ( s, α )) ≤ αX ( s, α ) . Hence γ ( x ) ≤ αx for all x ∈ [0 , X ( s, α )] , and the proof is completed by noting that thebound (51) is stronger than γ ( x ) ≤ αx for x ∈ ( X ( s, α ) , x ] . References [1] L. F. Abbott, Introduction To The Background Field Method, Acta Phys. Polon. B (1982) 33.[2] L. F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B (1981) 189.[3] R. Acharya, Mass gap in quantum chromodynamics, arXiv:hep-th/0611220.[4] A. C. Aguilar, D. Binosi and J. 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