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Dipartimento di Fisica, Universit`a di Parma and

arXiv:hep-ph/9203222v1 27 Mar 1992UPRF–91–316CPT–91/P.2629INFRARED FACTORIZATION,WILSON LINESAND THE HEAVY QUARK LIMITG. P. Korchemsky∗†Dipartimento di Fisica, Universit`a di Parma andINFN, Gruppo Collegato di Parma, I–43100 Parma, Italye-mail:korchemsky@vaxpr.cineca.itandA.

V. Radyushkin†Centre de Physique Theorique, CNRS - Luminy,F 13288 Marseille, Francee-mail:radyush@cebafAbstractIt is shown that, in QCD, the same universal function Γcusp(ϑ, αS) determines the in-frared behaviour of the on-shell quark form factor, the velocity-dependent anomalous dimen-sion in the heavy quark effective field theory (HQET) and the renormalization properties ofthe vacuum averaged Wilson lines with a cusp. It is demonstrated that a combined use ofthe methods developed in the relevant different branches of quantum field theory essentiallyfacilitates the all-order study of the asymptotic and analytic properties of this function.∗INFN Fellow†On leave from the Laboratory of Theoretical Physics, JINR – Dubna

1.IntroductionA considerable effort is being made now to construct a new approach to the heavy-quark physics. The basicidea [1, 2] is to use the fact that the masses of the heavy quarks c and b are essentially larger than the QCDscale Λ, and to start (when appropriate) with an effective field theory [1, 2], in which the heavy quark massesare infinite: mQ →∞.In the infinite mass limit, the components of the heavy quark momentum pµ tend to infinity together with thequark mass: p ∼mQ, the ratio pµ/mQ approaching a fixed value vµ.

The 4-velocity vµ = (1/√1 −v2, v/√1 −v2)just specifies the direction in which the infinitely heavy quark is moving, and the heavy quark effective theory(HQET) [3, 4] has a peculiar property that each effective quark field hv(x) is characterized by a 4-velocity vectorvµ.The electroweak interactions can change the velocity of the heavy quark v1 →v2 and, as noted by Isgur andWise [2], it is the 4-velocity change rather than the momentum transfer that is an important dynamic variablefor the heavy meson form factors, both for elastic (like γ∗D →D and γ∗B →B) and for weak (B →Deν-type)transition form factors. In other words, the QCD currents ¯Q2ΓQ1 are substituted in the HQET by the effectivecurrents ¯hv2Γhv1 containing fields related to the two velocities v1 and v2.An important observation made in [5] is that the effective currents ¯hv2Γhv1 have a universal (the same forall Γ-matrices) velocity-dependent anomalous dimension γw(αS).

The velocity change is characterized by thescalar product w ≡(v1v2) or, since the velocity vectors have a unit “length”, v21 = v22 = 1, by the angle ϑbetween them: w = cosh ϑ.As a matter of fact, the authors of [5] are not the first who encountered an angle-dependent anomalousdimension in their calculation. Similar things happened before in two other and, at first sight, entirely differentbranches of quantum field theory.First, more than 10 years ago, Polyakov [6], trying to reformulate the dynamics of gauge fields in termsof string operators, has observed that the vacuum average of a Wilson loop having a cusp, possesses extraultraviolet divergences.

These specific divergences, “generated” by the cusp, are multiplicatively renormalized[7], and the relevant anomalous dimension Γcusp(ϑ, αS) depends on the cusp angle ϑ.The second example is related to the infrared (IR) behaviour of perturbative QCD which is known to becontrolled by the subprocesses involving soft gluons. Since the soft gluons cannot change the momenta pi of theincoming and outgoing particles, one faces here a situation completely similar to that in the heavy-quark limit:the particles are just moving in fixed directions (specified by pi) through the gluonic cloud.

The study of theIR behavior of perturbative QCD has not a purely academic interest: even if the IR singularities cancel, thereremain finite IR-induced contributions. In our papers [8, 9], we developed an approach that enables one to studythese contributions using a renormalization group equation.

The relevant anomalous dimensions ΓIR(ϑij, αS)depend just on the angles ϑij between the relevant momenta: cosh ϑij = (pipj)/qp2i p2j.The IR-induced terms are especially important for high momentum transfer processes, when Q2ij ≡−(pi −pj)2 ≫p2i ∼p2j ≡p2, i.e., when some angles ϑij are large. We established that, in this limit, the anomalousdimension ΓIR(ϑij, αS), to all orders in αS, is linear in ϑij or, what is the same, in log(Q2ij/p2):ΓIR(ϑij, αS) = K(αS) log(Q2ij/p2) + A(αS).

(1)The function K(αS) plays an important role in the IR-induced and the double logarithmic contributions 1 todifferent hard processes: the same combinationK(αS) = αSπ CF +αSπ2CFCA6736 −π212−Nf518(2)(where CF = 4/3 and CA = 3 are quark and gluon color factors and Nf = 3 is the number of light quarkflavours) governs the x →1 asymptotics of the deep inelastic structure functions [9] and the large perturbativecorrections to the Drell-Yan cross section [11] and the e+e−−annihilation [12] .One might ask: are all these anomalous dimensions independent or there exists a connection between allof them?It is our goal in the present paper to study the interrelation between the infrared behaviour ofperturbative QCD and the renormalization properties both of the Wilson loops (lines) and of the effective1A function equivalent to K(αS) is the basic object in another RG-type approach to the double-logarithmiceffects developed by Collins [10]1

heavy quark currents in the HQET, to demonstrate that the apparently unrelated functions, each playing afundamental role in its field, are precisely identical and correspond to the QCD analogue of the bremsstrahlungfunction known from the early QED years.2.Infrared singularities and the heavy quark limitA simple but essential fact is that there are no velocity- or angle-dependent ultraviolet divergences in anystandard QCD diagram. This means that the HQET has extra UV divergences: the heavy quark limit mQ →∞is singular.

In particular, a logarithmic divergence in the HQET may result from a log(mQ/λ)-contribution inQCD, with λ remaining finite in the heavy quark limit. However, since all the momenta pµ scale as mQ in thislimit: pµ →mQvµ, the only combination that might be much smaller than m2Q is (p2 −m2Q), the heavy quarkoff-shellness, and the singularity may result only from a term like log(m2Q/(p2−m2Q)).

Such a term is a standardinfrared logarithm, with (p2 −m2Q) serving as an infrared cut-off. This indicates that the velocity-dependentdivergences should be closely related to the IR behaviour of perturbative QCD.

Indeed, studying the hadronscontaining a single heavy quark, one is interested in the kinematic situation when the heavy quark energy differsfrom its mass by some finite amount E, which has a meaning of the average energy of the light quark(s), bindingenergy, etc. Thus, the virtuality of the heavy quark (p2 −m2Q) ≈2mQE is much smaller than its mass squared:∆=p2 −m2Qm2Q≪1.In the heavy quark limit, mQ →∞, the parameter ∆tends to zero and, in this sense, the heavy quark is almoston-shell.It is well-known that, in gauge theories, the on-shell form factors have infrared (IR) singularities.

The classicexample is the elastic electron on-shell (p21 = p22 = m2) form factor in QED. At the one-loop level one has, forthe Dirac form factorF(Q2, m2) = 1 −α2π logm2λ2B Q2m2+ (IR regular terms) ,where Q2 = −(p1 −p2)2 is the transferred momentum, λ is the (fictitious) photon mass serving as the IR cut-offparameter and BQ2m2is the bremsstrahlung functionB Q2m2=1 + 2m2Q2q1 + 4m2Q2logq1 + 4m2Q2 + 1q1 + 4m2Q2 −1−1 .

(3)Since λ2 is treated as a scale much smaller than m2, we have the same kinematic situation as in the heavy quarklimit.The IR logarithms in QED are known to exponentiate when summed over all orders [13]F(Q2, m2) = exp−α2π logm2λ2B Q2m2× (IR regular terms). (4)In this expression, one can easily factorize the “long distance” contributionµ2/λ2−α2π B absorbing all the IRdivergences, and the “short-distance” factorm2/µ2−α2π B .

As a result, one can rewrite eq. (4) asF(Q2, m2) =µ2λ2−α2π BCm2µ2 , Q2m2(5)where the IR regular factor C( m2µ2 , Q2m2 ) now has an explicit dependence on the IR factorization scale µ.The derivation of this factorized representation has a striking resemblance to the standard factorizationprocedure for the collinear mass singularities applied to deep inelastic scattering (in the form of the operatorproduct expansion, OPE) and to other hard processes.

Using it, one can factorize the original function, e.g.,Wn(Q2, p2), the n-th moment of the deep inelastic structure function, into the short-distance coefficient function2

Cn(Q2/µ2) and the long-distance sensitive matrix element fn(µ2/p2) absorbing all the collinear logarithms log p2singular in the p2 →0 limit:Wn(Q2, p2) = fn(µ2/p2) Cn(Q2/µ2).The factorization scale µ separates the high- and low-momentum regions: the essential momenta k arek > µ for the C-function and k < µ for the f-function. An important fact is that the matrix elements of thelocal composite twist-2 operators On, producing fn, possess extra UV divergences absent in the original theory(QCD).

Of course, the factorization procedure automatically imposes the UV cut-offk < µ, and there are noactual infinities. But now one can make use of the fact that the µ-dependence of the fn-function is just thedependence on the UV cut-offparameter and incorporate the renormalization group to study it.

That is howthe n-dependent anomalous dimensions γn(αS) of the composite operators On and/or z-dependent anomalousdimensions (evolution kernels) P(z, g) of the parton distribution functions f(x, µ2) come into play.The analogy between the factorization properties of the QED formula (5) and the OPE-type factorizationhas a very deep reason [8]: in QCD (and in any similar gauge theory) one can factorize out the IR-singularterms of the on-shell quark form factorF(Q2, m2) = FIR × (IR regular terms).The IR-sensitive factor FIR accumulates all the effects due to interactions of the massive particle (heavy quark,if one studies the heavy quark limit) with soft quanta - gluons and massless quarks. The explicit expression forFIR is given by the vacuum average of the Wilson line operator W(C)FIR = ⟨0|T expigZ +∞0dt vµ2 Aµ(v2t)expigZ 0−∞dt vµ1 Aµ(v1t)|0⟩≡⟨0|W(C)|0⟩(6)calculated along the classic path C of the massive (“heavy”) particle.

It goes from −∞along the direction ofthe initial momentum p1 = mv1 to the interaction point 0 and then along the direction of the final momentump2 = mv2 to +∞. The soft gluons coupling directly to the heavy particle are described by the gauge potentialsAµ(x).To summarize, in this limit (one can treat it either as the IR limit λ →0 or as the heavy quark limitm →∞), all the effects due to interactions of the massive particle with the gluons can be described by a Wilsonline.

This result has a natural interpretation: the wave function of a very massive particle propagating throughthe gluonic cloud acquires only a phase factor equal to the Wilson line. All other effects are suppressed bypowers of λ/m.

In particular, they cannot produce contributions singular in the IR limit λ →0.Just like in the OPE case, the factorization scale µ works as an ultraviolet cut-offfor FIR, since only smallmomenta k < µ contribute to it. And such a cut-offis really necessary because, as established by Polyakov [6],any Wilson loop with a cusp has extra UV divergences.3.Wilson lines and the heavy quark effective field theoryThe properties of the Wilson lines and loops where studied in detail at the beginning of the 80’s.

A veryimportant result [14] is that one can interpret the vacuum average of a Wilson line W(C) as the propagator ofone-dimensional fermions “living” on the integration path C:Tr W(C) = ⟨ψa(+∞) ¯ψa(−∞)⟩≡ZD ¯ψ(t)Dψ(t) ψa(+∞) ¯ψa(−∞) exp(−iS)(7)where the action S, in our case, is defined byS = iZ +∞−∞dt ¯ψ(t) ∂∂tψ(t) −gZ 0−∞dt ¯ψ(t)vµ1 Aµ(v1t)ψ(t) −gZ +∞0dt ¯ψ(t)vµ2 Aµ(v2t)ψ(t). (8)It is easy to show that the one-dimensional fermions defined in this way exactly coincide with the effective heavyquark field of the HQET:⟨0|hv1(v1t)|h, v1⟩= ⟨ψ(t) ¯ψ(−∞)⟩= T expigZ t−∞dσ vµ1 Aµ(v1σ),t < 0(9)3

and⟨h, v2|¯hv2(v2t)|0⟩= ⟨ψ(+∞) ¯ψ(t)⟩= T expigZ +∞tdσ vµ2 Aµ(v2σ),t > 0 . (10)The representation (7) is very convenient to analyze the general renormalization properties of the Wilson lines.In particular, it was found [7, 14] that for a smooth integration path all the UV divergences can be compensatedby a renormalization of fields and vertices.The Wilson lines in the r.h.s.

of eqs. (9) and (10) are not closed, and their end-points produce extra UVdivergences.

They can be multiplicatively renormalized, and the relevant anomalous dimension γend(αS) wascalculated, in the Feynman gauge, to one loop in [7, 14] and to two loops in [15]γend(αS) = −αSπ CF −αSπ2CF1912CA −13Nf.From the relations (9) and (10) it follows that γend(αS) coincides with the anomalous dimension of the effectivequark field in the HQET and with that of the one-dimensional fermions in the theory defined by (8). TheHQET calculation was performed in [16].As mentioned earlier, each cusp of the integration path also generates extra UV divergences governed by theanomalous dimension Γcusp that depends on the cusp angle ϑ.

In the theory of one-dimensional fermions (8) oneobtains Γcusp as the anomalous dimension of the composite operator ¯ψ(+0)ψ(−0). Using the relations (9) and(10) one can identify this operator with a heavy quark current ¯hv2(0)hv1(0).

Thus, the anomalous dimension ofthe heavy quark current in the HQET coincides with the cusp anomalous dimension of the relevant Wilson lineΓcusp(ϑ, αS) = γw(αS),(11)provided, of course, that w = cosh ϑ.In general, it is a difficult problem to calculate Wilson loops, even in perturbation theory. However, it ispossible to evaluate ⟨0|W(C)|0⟩= FIR using the explicit form of the contour C defined by eq.(6).

The pathC has an infinite length and, as a consequence, ⟨0|W(C)|0⟩has infrared divergences. Moreover, ⟨0|W(C)|0⟩depends on the directions of the rays forming C. Note now that the directions defining the line C are specifiedby the dimensionless ratios vµ1 = pµ1/m and vµ2 = pµ2/m .

Hence, the dimensionful parameters that explicitlyappear in the vacuum average (6) are the IR cut-offλ and the factorization scale µ, the latter, as emphasizedearlier, working as an ultraviolet cut-offfor the new divergences generated by the cusp of the path C at thepoint 0. This means that the UV scale µ and the IR scale λ should basically appear in the ratio.

There is alsothe ordinary renormalization scale µR fixing the definition of the running coupling constant g(µR):FIR = FIRµλ,µµR, (v1v2), g(µR)= FIRµλ,µµR, (p1p2)m2, g(µR).After taking µ = µR, the IR factor FIR, being equal to the vacuum average of a Wilson loop, obeys, in general,the RG equationµ ∂∂µ + β(g) ∂∂g + ΓIR(p1p2)m2, gFIRµλ, 1, (p1p2)m2, g= 0 ,where the anomalous dimension ΓIR controls the IR induced logarithms log(µ/λ). Since the path C has a cusp,one identifies ΓIR with the cusp anomalous dimension:ΓIR(p1p2)m2, αS= Γcusp(ϑ, αS).

(12)Furthermore, since the IR singularities of FIR coincide with those of the original amplitude (e.g., withthose present in the on-shell form factor), the cusp anomalous dimension Γcusp(ϑ, αS) must reproduce thebremsstrahlung function (3):Γcusp(ϑ, αS) = αSπ CF B Q2m2+ O(α2S)(13)The one-loop result for the cusp anomalous dimension can be taken from the pioneering paper by Polyakov [6],where it was calculated in the Euclidean space. In Minkowski case, one should change ϑ →iϑ to getΓ1−loopcusp(ϑ, αS) = αSπ CF (ϑ coth ϑ −1).4

To check the equivalence relation (13), one should just use that Q2 = 2m2(cosh ϑ −1). Taking the 1-loopexpression for γw(αS) from [5],γw(αS) = αSπ CFw√w2 −1log(w +pw2 −1) −1(14)one can also verify, at the one-loop level, the relation (11).The equivalence relations (11) and (13) manifest a close connection between the HQET and the theory ofWilson lines.

The cusp anomalous dimension appears here as a new universal function controlling the IR-inducedproperties of perturbative QCD. All the specifics of the process under study is contained in the dependence ofΓcusp(ϑ, αS) on the cusp angle ϑ determined entirely by the kinematics of the process.4.Non-abelian exponentiation theoremUsing effective fields, the one-dimensional fermions or the effective heavy-quark fields, one can essentially simplifya general study of the renormalization properties of the Wilson lines.

However, working within the effective fieldtheory approach, one can easily miss some properties of the relevant anomalous dimensions which are evidentif one treats them in the context of the Wilson line formalism.For example, in QED, it is possible to get an exact expression for Γcusp(ϑ, α).The reason is that, inan abelian gauge theory without massless fermions (electrons in QED are treated as massive particles! ), thevacuum average of a Wilson exponential is an exponential of the vacuum average corresponding to the photonpropagator:⟨0|T expigZCdzµ Aµ(z)|0⟩= exp(ig)22ZCdzµ1ZCdzν2⟨0|T Aµ(z1)Aν(z2)|0⟩.As a result, the cusp anomalous dimension in QED is completely determined by the first loop:ΓQEDcusp (ϑ, α) = απ B Q2m2.There are no higher order corrections.

From the IR side, this property is well-known: all the IR singularitiesare given in QED by the exponential of the one-loop result [13].In QCD, the situation is more complicated. First, there are light quarks u, d and s which can be treatedas massless at a typical hadronic scale ∼1 GeV .Second, the gluons are described by non-abelian gaugefields.

Nevertheless, it is possible to prove a non-abelian exponentiation theorem [17]: the vacuum average ofa Wilson line operator in QCD is an exponential of some expression, to which, of course, not only the firstloop contributes. Still, the property observed in QED is partially preserved: the higher-order corrections toΓcusp(ϑ, αS) do not repeat information contained in the one-loop result or, in general, in all the previous loops:calculating Γcusp(ϑ, αS), it is sufficient to consider only the diagrams which do not contain the lower-orderones as subgraphs, i.e., the essential diagrams are two-particle irreducible in the current channel.

As a result,the higher-order contributions, in the gluonic sector, are essentially non-abelian. In particular, the second-loop term should not contain terms proportional to C2F , the square of the one-loop color weight CF .

All suchterms, present on diagram-by-diagram level, should eventually cancel, and only the terms proportional to theessentially non-abelian color weight CF CA might remain. This observation can be used to check the results ofthe higher-order calculations for the relevant anomalous dimensions.The two-loop contribution to Γcusp in pure gluodynamics (QCD without massless quarks) was calculated inour paper [18].2 The massless quark contribution can be also added [9] and the total result isΓ2−loopcusp(ϑ, αS)=αSπ2CF−Nf518(ϑ coth ϑ −1) + CA12 +6736 −π224(ϑ coth ϑ −1)−coth ϑZ ϑ0dψψ coth ψ + coth2 ϑZ ϑ0dψψ(ϑ −ψ) coth ψ(15)−12 sinh 2ϑZ ϑ0dψψ coth ψ −1sinh2 ϑ −sinh2 ψ log sinh ϑsinh ψ.2Independently, a similar calculation was performed by Knauss and Scharnhorst [15].

They failed, however,to get rid of all double integrations in their result.5

It explicitly satisfies the requirement of the non-abelian exponentiation theorem: there are no C2F -terms in it.In the framework of the HQET, the two-loop calculation of the velocity-dependent anomalous dimension wasperformed in a recent paper by Ji [19]. The author presents his result diagram by diagram, in terms of 7 integralsI1, .

. .

, I7, two of which (I1 and I7) are calculated explicitly and 5 others are claimed to be not calculable interms of the elementary functions. The integral I6 even contains a double integration.3From our formula(15), it is clear that there should be really only three independent integrals containing no double integrations.The most worrying is the fact that the C2F -term in [19] is not explicitly equal to zero: it is proportional to(I1I3 + I4 −I5).

Checking the results of [19], we managed to show that I3 = 0. This observation allowed usto simplify another integral, I4.

Then, performing a straightforward integration by parts, we obtained thatI5 = I4. Thus, the total C2F coefficient in [19] vanishes, as it should.The same statement is valid for the end-point anomalous dimension γend(αS) or, equivalently, for the anoma-lous dimension of the effective heavy field in the HQET.

In particular, no C2F -terms really appear at the two-looplevel. The authors of [16], apparently, have not realized that their two-loop results have this property.5.Cusp anomalous dimension for small and large anglesIncorporating the exponentiation theorem, one can easily obtain an important result mentioned in the introduc-tion that the anomalous dimension Γcusp(ϑ, αS), for large angles ϑ, is linear in ϑ (or log(Q2/m2)) to all ordersof perturbation theory:Γcusp(ϑ, αS)|ϑ→∞= ϑK(αS) + O(ϑ0) .

(16)This fact has a simple explanation. In this limit, log(Q2/m2) is a typical collinear logarithm resulting from inte-gration over the regions where the gluons are almost collinear to v1 or v2, i.e., when they are emitted/absorbedat small angle.

However, in the two-particle irreducible diagrams (only these really contribute to Γcusp(ϑ, αS))there is just one angular integration producing a singularity in the m →0 limit. Hence, just a single logarithmemerges, and we get eqs.

(1), (16).Using the explicit expression (15), one can check that our Γcusp(ϑ, αS) is linear in ϑ for large ϑ. However,this linearity requirement is not satisfied by the result [19] given by Ji, since it contains an exponentially risingcontribution O(exp(ϑ)) in the term proportional to CF Nf. In general, we disagree with his results for the termshaving this color factor.From the expression (2), it follows that the first two coefficients of the perturbative expansion for theasymptotic slope K(αS) are positive.There are good reasons to expect that K(αS) should be positive toall orders.

In particular, it can be shown [9] that K(αS) determines the asymptotic behavior of the partondistribution function f(x, Q2) as x →1 for large Q2. More specifically, the exponent a(Q2) in the asymptoticrepresentation f(x, Q2) ∝(1 −x)a(Q2) obeys the evolution equation [9]ddQ2 a(Q2) = K(αS(Q2)).If K(αS) is a positive function, then a(Q2) is an increasing function of Q2, and, for x ∼1, the parton distributionstend to zero faster and faster with increasing Q2.

And that is just what one is expecting within the QCD partonpicture: if one probes the structure of a hadron at smaller and smaller distances, the probability to find a partoncarrying almost all momentum of the hadron should become smaller and smaller.In the opposite limit of small cusp angles, eq. (15) givesΓcusp(ϑ, αS)|ϑ→0 = ϑ2αS3π CF +αSπ2CFCA4754 −π218−Nf554+ O(ϑ4)(17)Apparently, the expansion of the two-loop cusp anomalous dimension (15) contains only even powers of the angleϑ.

In view of the equivalence relation (11) this seems natural: Γcusp(ϑ, αS) depends on ϑ through w = cosh ϑ,which is an even function of ϑ. However, this simple reasoning is true only if Γcusp(ϑ, αS) is analytic at ϑ = 0or, equivalently, if γw(αS) is an analytic function of w in the vicinity of the point w = 1.

This fact is notimmediately obvious, since the explicit 1-loop expression (14) for γw(αS) contains the square root√w2 −1capable of generating a cut starting just at w = 1.3We failed to check this integral, since the explicit expression presented in [19] contains a variable u (or afunction u) not defined in the paper.6

To study the analyticity properties of γw(αS), it is useful to use its relation to the anomalous dimension ofthe IR factor FIR (12) and the fact that FIR enters into the factorized expression for the elastic quark formfactor F(Q2, m2). Hence, γw(αS) possesses only those singularities which the quark form factor has as a functionof Q2 = 2m2(w −1).

It is well known, that the nearest singularity of the form factor is due to the pair creationin the annihilation channel and it corresponds to the point Q2 = −4m2. This implies that γw(αS) is an analyticfunction at w = 1, with the nearest singularity at w = −1.

Indeed, using the explicit expressions (3) and (15),one can easily check that the bremsstrahlung function has a cut starting at Q2 = −4m2 while the two-loop cuspanomalous dimension has the only pole singularity at cosh ϑ = −1.46.ConclusionsThus, we demonstrated that three important functions, that appear in different branches of the quantum theoryof gauge fields, coincide. The same function, the cusp anomalous dimension of a Wilson line, appears whenone studies the IR singularities of the on-shell form factors or the velocity-dependent anomalous dimensionin the effective heavy quark field theory.This is because in both cases, all the IR induced terms can befactorized from the original amplitudes into the universal IR factor given by the vacuum average of a Wilsonline.

This object, though it is a nonlocal functional of the gauge fields, has rather simple (and well studied)renormalization properties. As a consequence, the anomalous dimensions of the Wilson lines govern the IRlogarithms in the same manner as the anomalous dimensions of the local composite twist-2 operators controlthe collinear singularities one encounters studying the structure functions of deep inelastic scattering.In practical aspect, the well-developed machinery of the Wilson loop formalism can provide a good supportfor the heavy quark effective field theory.

In particular, we established the equivalence of some recent HQETcalculations [5, 16] with the earlier results for the Wilson lines [6, 7, 15].We also corrected the two-loopcalculation [19] of the velocity-dependent anomalous dimension.Acknowledgment.One of us (A.V.R.) is grateful to E. De Rafael and J. Soffer for the warm hospitality atCPT (Marseille).

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