Z. Phys. C 76, 523–535 (1997) ZEITSCHRIFTFUR PHYSIK Cc© Springer-Verlag 1997

The high-energy quark–quark scattering:from Minkowskian to Euclidean theoryEnrico Meggiolaro

Dipartimento di Fisica, Universita di Pisa, I-56100 Pisa, Italy

Received: 3 January 1997

Abstract. In this paper we consider some analytic prop-erties of the high-energy quark–quark scattering amplitude,which, as is well known, can be described by the expectationvalue of two lightlike Wilson lines, running along the classi-cal trajectories of the two colliding particles. We will showthat the expectation value of two infinite Wilson lines, form-ing a certain hyperbolic angle in Minkowski space-time, andthe expectation value of two infinite Euclidean Wilson lines,forming a certain angle in Euclidean four-space, are con-nected by an analytic continuation in the angular variables:the proof is given for an Abelian gauge theory (QED) inthe so-calledquenchedapproximation and for a non-Abeliangauge theory (QCD) up to the fourth order in the renormal-ized coupling constant in perturbation theory. This couldopen the possibility of evaluating the high-energy scatter-ing amplitude directly on the lattice or using the stochasticvacuum model.

1 Introduction

There is a class ofsofthigh-energy scattering processes, i.e.,elastic scattering processes at high squared energiess in thecenter of mass and small squared transferred momentumt(that is s → ∞ and |t| � s, let us say|t| ≤ 1 GeV2), forwhich QCD perturbation theory cannot be safely applied,sincet is too small. Elaborate procedures for summing per-turbative contributions have been developed [1, 2], even ifthe results are not able to explain the most relevant phenom-ena.

A non-perturbative analysis, based on QCD, of thesehigh-energy scattering processes was performed by Nacht-mann in [3]. He studied thes dependence of the quark–quark (and quark-antiquark) scattering amplitude by ana-lytical means, using a functional integral approach and aneikonal approximation to the solution of the Dirac equationin the presence of a non-Abelian external gluon field.

In a previous paper [4] we proposed an approach to high-energy quark–quark (and quark-antiquark) scattering, basedon a first-quantized path-integral description of quantumfield theory developed by Fradkin in the early 1960’s [5].

In this approach one obtains convenient expressions for thefull and truncated-connected scalar propagators in an exter-nal (gravitational, electromagnetic, etc.) field and the eikonalapproximation can be easily recovered in the relevant limit.Knowing the truncated-connected propagators, one can thenextract, in the manner of Lehmann, Symanzik, and Zimmer-mann (LSZ), the scattering matrix elements in the frame-work of a functional integral approach. This method wasoriginally adopted in [6] in order to study Planckian-energygravitational scattering.

The high-energy quark–quark scattering amplitude comesout to be described by the expectation value of two lightlikeWilson lines, running along the classical trajectories of thetwo colliding particles.

In the center-of-mass reference system (c.m.s.), takingthe initial trajectories of the two quarks along thex1-axis,the initial four-momentap1, p2 and the final four-momentap′1 andp′2 are given, in the first approximation (eikonal ap-proximation), by

p1 ' p′1 ' (E,E, 0t) , p2 ' p′2 ' (E,−E, 0t) . (1)

Let us indicate withxµ1 (τ ) andxµ2 (τ ) the classical trajectoriesof the two colliding particles in Minkowski space-time:

xµ1 (τ ) = zµt + pµ1 τ , xµ2 (τ ) = pµ2 τ , (2)

wherezt = (0, 0, zt), with zt = (z2, z3), is the distance be-tween the two trajectories in thetransverseplane (the co-ordinates (x0, x1) are often calledlongitudinal coordinates).The high-energy (s → ∞ and |t| � s) quark–quark scat-tering amplitude turns out to be controlled by the Fouriertransform, with respect to the transverse coordinateszt, ofthe expectation value of the two lightlike Wilson lines run-ning alongxµ1 (τ ) andxµ2 (τ ):

W1(zt) = P exp

[−ig

∫ +∞

−∞Aµ(zt + p1τ )pµ1dτ

];

W2(0) = P exp

[−ig

∫ +∞

−∞Aµ(p2τ )pµ2dτ

], (3)

whereP stands for “path ordering” and Aµ = AaµT

a. Thespace-time configuration of these two Wilson lines is shownin Fig. 1.

524

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W2 W1

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Fig. 1. The space-time configuration of the two lightlike Wilson linesW1and W2 entering in the expression (4) for the high-energy quark–quarkelastic scattering amplitude

Explicitly indicating the color indices (i, j, . . .) and thespin indices (α, β, . . .) of the quarks, the scattering amplitudecan be written as

Mfi = 〈ψiα(p′1)ψkγ(p′2)|M |ψjβ(p1)ψlδ(p2)〉∼

s→∞−i

Z2ψ

· δαβδγδ · 2s∫d2zteiq·zt

×〈[W1(zt)− 1]ij [W2(0)− 1]kl〉A , (4)

where〈. . .〉A is the average, in the sense of the functionalintegration, over the gluon fieldAµ. Zψ is the fermion-fieldrenormalization constant, which can be written in the eikonalapproximation as [3]

Zψ ' 1Nc〈Tr[W1(zt)]〉A =

1Nc〈Tr[W1(0)]〉A

=1Nc〈Tr[W2(0)]〉A . (5)

(The two last equalities come from the Poincare invarianceof the theory.)

In a perfectly analogous way one can also derive thehigh-energy scattering amplitude in the case of the AbeliangroupU (1) (QED). The resulting amplitude is equal to (4),with the only obvious difference being that now the lightlikeWilson linesW1 andW2 are functionals of the Abelian fieldAµ (so they are not matrices). Thanks to the simple formof the Abelian theory (in particular to the absence of self-interactions among the vector fields), it turns out that it ispossible to explicitly evaluate (at least in thequenchedap-proximation) the expectation value of the two Wilson lines:the details of the calculation are reported in the Appendixof [4] and one finally recovers the well-known result forthe eikonal amplitude of the high-energy scattering in QED[7–9].

From (4) it seems that thes dependence of the scatteringamplitude is all contained in the kinematic factor 2s in frontof the integral. In fact we can write

Mfi = 〈ψiα(p′1)ψkγ(p′2)|M |ψjβ(p1)ψlδ(p2)〉∼

s→∞−i · 2s · δαβδγδ · gM (ij,kl)(t, s) , (6)

where, apparently, the quantity

gM (ij,kl)(t, s) ≡ 1Z2ψ

∫d2zteiq·zt

×〈[W1(zt)− 1]ij [W2(0)− 1]kl〉A (7)

only depends ont = −q2. Yet, as was pointed out by Ver-linde and Verlinde in [10], this is not true: in fact, one caneasily be convinced (for example by making a perturbativeexpansion) that it is a singular limit to take the Wilson linesin (7) exactly lightlike. As suggested in [10], one can regu-larize this sort of “infrared” divergence by letting each linehave a small timelike component, so that they coincide withthe classical trajectories for quarks with a finite massm.Therefore, one first has to evaluate the quantity

gM (ij,kl)(t, β) (8)

for two Wilson lines along the trajectories of two quarksmoving with velocityβ and−β (0 < β < 1) along thex1-axis. In other words, one first considers two infinite Wilsonlines forming a certain hyperbolic angleχ in Minkowskispace-time. Then, to obtain the correct high-energy scatter-ing amplitude, one has to perform the limitβ → 1, that isχ→∞, into the expression (8):

Mfi = 〈ψiα(p′1)ψkγ(p′2)|M |ψjβ(p1)ψlδ(p2)〉∼

s→∞−i · 2s · δαβδγδ · gM (ij,kl)(t, β → 1) . (9)

In this way one obtains a lns dependence of the amplitude,as expected from ordinary perturbation theory [1, 2] and asconfirmed by the experiments on hadron–hadron scatteringprocesses. In Sect. 3 we shall see how this explicitly worksby evaluating the amplitude (7) for QCD up to the orderO(g4

R) in the perturbative expansion (gR being the renor-malized coupling constant).

The direct evaluation of the expectation value (7) is ahighly non-trivial matter, as it is also strictly connected withthe ultraviolet properties of Wilson-line operators [11]. Re-cently, in [12], it has been found that there is a correspon-dence between high-energy asymptotics in QCD and renor-malization properties of the so-called cross singularities ofWilson lines. The asymptotic behavior of the quark–quarkscattering amplitude turns out to be controlled by a 2×2 ma-trix of the cross anomalous dimensions of Wilson lines. Analternative non-perturbative approach for the calculation ofthe expectation value (7) has been proposed in [13]. It con-sists in studying the Regge regime of large energies and fixedmomentum transfers as a special regime of lattice gauge the-ory on an asymmetric lattice, with a spacinga0 in the longi-tudinal direction and a spacingat in the transverse direction,in the limit a0/at → 0.

At the moment, the only non-perturbative numerical es-timate of (7), which can be found in the literature, is that of[14] (where it has been generalized to the case of hadron–hadron scattering): it has been obtained in the framework ofthe model of the stochastic vacuum (SVM). Before its appli-cation to high-energy scattering, the SVM must be translatedfrom Euclidean space-time, in which it is naturally formu-lated, to the Minkowski continuum. As is claimed in [14],the more safe way (from the point of view of the func-tional integration) would be the other way, i.e., to continuethe scattering amplitude from the Minkowski world to theEuclidean world.

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In this paper we try to go just that way and adapt thescattering amplitude to the Euclidean world. More explicitly,we shall prove that the expectation value of two infinite Wil-son lines, forming a certain hyperbolic angle in Minkowskispace-time, and the expectation value of two infinite Eu-clidean Wilson lines, forming a certain angle in Euclideanfour-space, are connected by an analytic continuation in theangular variables. In Sect. 2 we shall first prove this for theAbelian case (QED), by explicitly evaluating the correlationof two infinite Wilson lines both in Minkowski space-timeand in Euclidean four-space, using the so-calledquenchedapproximation. Then, in Sect. 3, we shall prove that this re-sult can be extended also to non-Abelian gauge theories.Finally, in the last section, we shall discuss some interestingconsequences (such as the re-derivation of theRegge polemodel [15]) and some possible direct applications, mostlyfor lattice gauge theories (LGT) and the stochastic vacuummodel, of this relationship of analytic continuation.

2 The Abelian case

In this section we shall discuss the Abelian case (See also[12] and [13]). The fermion–fermion electromagnetic scat-tering amplitude, in the high-energy limits → ∞ and|t| � s, can be derived following the same procedure usedin [4]. The resulting amplitude is formally identical to (4),with the only obvious difference being that now the Wilsonlightlike linesW1 andW2 are functions of the Abelian fieldAµ (so they are not matrices):

Mfi = 〈ψα(p′1)ψγ(p′2)|M |ψβ(p1)ψδ(p2)〉∼

s→∞−i

Z2ψ

· δαβδγδ · 2s∫d2zteiq·zt

×〈[W1(zt)− 1][W2(0)− 1]〉A . (10)

The electromagnetic lightlike Wilson linesW1 andW2 aredefined as in (3), after replacingg with e, the electriccoupling-constant (electric charge), and the gluon field withthe photon field. Thanks to the simple form of the Abeliantheory (in particular to the absence of self-interactions amongthe vector fields), it turns out that it is possible to explic-itly evaluate the expectation value of the two Wilson lines,thus finally recovering the well-known result for the eikonalamplitude of the high-energy scattering in QED (see [7–9]).The details of the calculation are reported in [4].

The Wilson lines in (10) are taken exactly lightlike. Weshall now let each line to have a small timelike compo-nent, so that they coincide with the classical trajectories forfermions with a finite massm. The electromagnetic WilsonlinesW1 andW2 are now defined as

W1(zt) = exp

[−ie

∫ +∞

−∞Aµ(zt +

p1

mτ )pµ1mdτ

],

W2(0) = exp

[−ie

∫ +∞

−∞Aµ(

p2

mτ )pµ2mdτ

], (11)

wherem is the mass of the fermions andpµ1 andpµ2 are thetwo four-momenta defining the two trajectories 1 and 2 inMinkowski space-time:

Xµ(1)(τ ) = zµt +

pµ1mτ ,

Xµ(2)(τ ) =

pµ2mτ . (12)

In the c.m.s. of the two particles, taking the spatial mo-menta p1 and p2 = −p1 along thex1-direction, the twofour-momentap1 andp2 are

pµ1 = E(1, β, 0t) ,pµ2 = E(1,−β, 0t) , (13)

whereβ is the velocity (in the units withc = 1) andE =m/√

1− β2 is the energy of each particle (so that:s = 4E2).We shall evaluate the expectation value〈W1(zt)W2(0)〉A

in the so-calledquenchedapproximation, where vacuum po-larization effects, arising from the presence of loops of dy-namical fermions, are neglected. This amounts to settingdet(K[A]) = 1, whereK[A] = iγµDµ −m is the fermionmatrix. Thus we can write that

〈W1(zt)W2(0)〉A ' 1Z

∫[dAµ]eiSAW1(zt)W2(0) , (14)

whereSA = − 14

∫d4xFµνF

µν is the action of the electro-

magnetic field andZ =∫

[dAµ]eiSA is the pure-gauge par-

tition function. We then add to the pure-gauge LagrangianLA = − 1

4FµνFµν a gauge-fixing termLGF = − 1

2α (∂µAµ)2

(covariant or Lorentz gauge). The expectation value (14)becomes, denotingLF0 = LA +LGF ,

〈W1(zt)W2(0)〉A' 1Z ′

∫[dA] exp

[i

∫d4x(LF0 +AµJ

µ)

], (15)

whereZ ′ =∫

[dA] exp

(i

∫d4xLF0

)andJµ(x) is a four-

vector source defined as

Jµ(x) = Jµ(1)(x) + Jµ(2)(x)

= −e[εµ(1)δ(x− βt)δ(xt − zt)

+εµ(2)δ(x + βt)δ(xt)] , (16)

with εµ(1) = (1, β, 0, 0) andεµ(2) = (1,−β, 0, 0). The functionalintegral

Z0[J ] ≡∫

[dA]

× exp

[i

∫d4x(LF0 +AµJ

µ)

](17)

can be evaluated with standard methods (completing thequadratic form in the exponent). One thus obtains

Z0[J ] = Z ′ · exp

[i

2

∫d4x

∫d4yJµ(x)

×Dµν(x− y)Jν(y)

], (18)

whereDµν is the free photon propagator (apart from a factor−i):

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Dµν(x− y) =∫

d4k

(2π)4

e−ik(x−y)

k2 + iε

×(gµν − (1− α)

kµkνk2 + iε

). (19)

In the following we shall choose the gauge-fixing parameterα equal to 1 (Feynmangauge). From (15)÷ (19), we derivethe following expression for the expectation value (14) ofthe two Wilson lines:

〈W1(zt)W2(0)〉A' 〈W1(zt)〉A〈W2(0)〉A× exp

[i

∫d4x

∫d4yJµ(1)(x)J(2)µ(y)

×∫

d4k

(2π)4

e−ik(x−y)

k2 + iε

]. (20)

Therefore, using the explicit form (16) of the four-vectorsourceJµ(x) to evaluate the double integral in (20), onefinds that (in thequenchedapproximation)

〈W1(zt)W2(0)〉A〈W1〉2A

' exp

[−ie2

(1 +β2

2β

)∫d2kt(2π)2

e−ikt·zt

k2t − iε

]. (21)

We have made use of the Poincare invariance of the theoryto write: 〈W1〉A ≡ 〈W1(zt)〉A = 〈W1(0)〉A = 〈W2(0)〉A. Wenow introduce the hyperbolic angleψ [in the plane (x0, x1)]of the trajectory 1 in (12), i.e.,∆X1

(1) = β∆X0(1), so that,

if ∆l is the line-distance (∆l2 = (∆X0(1))

2 − (∆X1(1))

2 =(∆X0

(1))2(1− β2)) we have

∆X0(1) = ∆l coshψ ,

∆X1(1) = ∆l sinhψ , (22)

and, therefore,β = tanhψ. With this notation, it is immediateto recognize that theβ-dependent factor in front of (21) isequal to coth(2ψ); so that

〈W1(zt)W2(0)〉A〈W1〉2A

' exp

[−ie2 cothχ ·

∫d2kt(2π)2

e−ikt·zt

k2t − iε

], (23)

whereχ = 2ψ is hyperbolic angle [in the plane (x0, x1)]between the two trajectories 1 and 2 of the two collidingparticles. The exponential in the last equation turns out tobe equal to

e−ie2Λ cothχ · exp

(ie2

2πcothχ · ln |z|

), (24)

whereΛ is an infinite constant phase and is therefore physi-cally unobservable. The origin of this infinite constant phaseresides in the fact that the fermion–fermion scattering am-plitude in QED has infrared (IR) divergences, due to theemission of low-energy massless vector mesons. The tradi-tional way to handle these IR divergences is to introduce anIR cutoff in the form of a vector meson massλ. In this waythe integral overkt in the exponent of (23) is substituted bythe expression

∫d2kt(2π)2

e−ikt·zt

k2t + λ2

≡ 12π

K0(λ|zt|) , (25)

whereK0 is the modified Bessel function. In the limit ofsmallλ this last expression can be replaced by

12π

K0(λ|z|) ∼λ→0

− 12π

ln

(12eγλ|zt|

). (26)

Absorbing 12e

γ in λ and puttingΛ = −(1/2π) ln( 12e

γλ) (sothat Λ → ∞ when λ → 0), we just obtain the expression(24) for the exponential in (23).

We can now repeat the above procedure and evaluate thequantity (5) in the Euclidean theory. The electromagneticEuclidean Wilson linesWE1 andWE2 are defined as in (2):

WE1(zt) = exp

[−ie

∫ +∞

−∞A(E)µ (zt + v1τ )v1µdτ

],

WE2(0) = exp

[−ie

∫ +∞

−∞A(E)µ (v2τ )v2µdτ

], (27)

where nowv1µ andv2µ are the Euclidean four-vectors [lyingin the plane (x1, x4)] defining the two trajectories 1 and 2in Euclidean four-space:

X (1)Eµ(τ ) = ztµ + v1µτ ,

X (2)Eµ(τ ) = v2µτ , (28)

and ztµ = (z1, z2, z3, z4) = (0, zt, 0). We can choosev1 andv2 normalized to 1:v2

1 = v22 = 1. Moreover, due to theO(4)

symmetry of the theory, we can choose the c.m.s. of thetwo particles, taking the spatial momentav1 and v2 = −v1along thex1-direction. The two four-momentav1 andv2 are,therefore,

v1µ = (sinφ, 0t, cosφ) ,

v2µ = (− sinφ, 0t, cosφ) , (29)

where φ is the angle formed by each trajectory with thex4-axis. As before, we can evaluate the expectation value〈WE1(zt)WE2(0)〉A (where now〈. . .〉A is the functional in-tegral with respect to the gluon fieldA(E)

µ in the Euclideantheory) in thequenchedapproximation. Thus we have

〈WE1(zt)WE2(0)〉A' 1ZE

∫[dA(E)

µ ]e−S(E)A WE1(zt)WE2(0) , (30)

whereS(E)A = 1

4

∫d4xEF

(E)µν F

(E)µν is the Euclidean action

of the electromagnetic field andZE =∫

[dA(E)µ ]e−S

(E)A is

the corresponding pure-gauge partition function. As usually,we add to the pure-gauge LagrangianL(E)

A = 14F

(E)µν F

(E)µν

a gauge-fixing termL(E)GF = 1

2α (∂µA(E)µ )2 (covariant or

Lorentz gauge). The expectation value (30) becomes, de-notingL(E)

0 = L(E)A +L(E)

GF ,

〈WE1(zt)WE2(0)〉A' 1Z ′E

∫[dA(E)]

× exp

[−∫d4xE(L(E)

0 + iA(E)µ JEµ)

], (31)

527

whereZ ′E =∫

[dA(E)] exp

(−∫d4xEL

(E)0

)andJEµ(xE)

is a four-vector source defined as

JEµ(xE) = J (1)Eµ(xE) + J (2)

Eµ(xE)

= e[v1µδ(xE1 cosφ− xE4 sinφ)δ(xEt − zt)+v2µδ(xE1 cosφ + xE4 sinφ)δ(xEt)] . (32)

The functional integral

Z (E)0 [JE ] ≡

∫[dA(E)]

× exp

[−∫d4xE(L(E)

0 + iA(E)µ JEµ)

](33)

can be evaluated with standard methods (completing thequadratic form in the exponent). One thus obtains

Z (E)0 [JE ] = Z ′E · exp

[− i

2

∫d4xE

∫d4yEJEµ(xE)

×D(E)µν (xE − yE)JEν(yE)

], (34)

whereD(E)µν is the free photon Euclidean propagator, i.e.,

D(E)µν (xE − yE) =

∫d4kE(2π)4

e−ikE (xE−yE )

k2E

×(δµν − (1− α)

kEµkEνk2E

). (35)

In the following we shall choose the gauge-fixing parameterα equal to 1 (Feynmangauge). From (31)÷ (35), we derivethe following expression for the expectation value (30) ofthe two Euclidean Wilson lines (including the regulating IRcutoff in the form of a small photon massλ, which must beput equal to zero at the end of the calculation):

〈WE1(zt)WE2(0)〉A' 〈WE1(zt)〉A〈WE2(0)〉A× exp

[−∫d4xE

∫d4yEJ

(1)Eµ(xE)J (2)

Eµ(yE)

×∫

d4kE(2π)4

e−ikE (xE−yE )

k2E + λ2

]. (36)

Finally, making use of the explicit form (32) of the four-vector sourceJEµ(xE) to evaluate the double integral in(36), and using also theO(4) plus translation invariance ofthe Euclidean theory to write:〈WE1〉A ≡ 〈WE1(zt)〉A =〈WE1(0)〉A = 〈WE2(0)〉A, one finds the result

〈WE1(zt)WE2(0)〉A〈WE1〉2A

' exp

[−e2 cotθ ·

∫d2kt(2π)2

e−ikt·zt

k2t + λ2

]. (37)

We have indicated withθ ≡ 2φ the angle [in the plane(x1, x4)] between the two trajectories in Euclidean four-space. The angleθ in (37) is taken in the interval [0, π]:it is always possible to make such a choice by virtue of theO(4) symmetry of the Euclidean theory. When comparingthe two expressions (23) and (37), we immediately recognizethat they are linked by the following analytic continuationin the angular variables:

〈WE1(zt)WE2(0)〉A〈WE1〉2A

−→θ→−iχ

〈W1(zt)W2(0)〉A〈W1〉2A

;

or :〈W1(zt)W2(0)〉A

〈W1〉2A−→χ→iθ

〈WE1(zt)WE2(0)〉A〈WE1〉2A

. (38)

This allows to reconstruct the high-energy scattering ampli-tude by evaluating a correlation of infinite Wilson lines inthe Euclidean world, then by continuing this quantity in theangular variable,θ → −iχ, and finally by performing thelimit χ→∞ (i.e., β → 1). In fact, from (10) we can write

Mfi = 〈ψα(p′1)ψγ(p′2)|M |ψβ(p1)ψδ(p2)〉∼

s→∞−i · 2s · δαβδγδ · gM (t, χ→∞) , (39)

where the quantitygM (t, χ) is defined as

gM (t, χ) =∫d2zteiq·zt

〈[W1(zt)− 1][W2(0)− 1]〉A〈W1〉2A

. (40)

It was shown in [3] that, in the eikonal approximation, onecan approximate the fermion-field renormalization constantas follows:

Zψ ' 〈W1〉A . (41)

ThereforegM (t, χ → ∞) is exactly the Abelian versionof the asymptotic amplitude (7). Moreover, whenevert isnot exactly equal to zero, i.e.,q /= 0 (t = −q2 < 0), theexpression (40) reduces to

gM (t, χ) =∫d2zteiq·zt

〈W1(zt)W2(0)〉A〈W1〉2A

. (42)

Thereforeg(t, χ) turns out to be the Fourier transform, withrespect to the transverse coordinateszt, of the quantity (23),which we have evaluated in thequenchedapproximation.In the Euclidean theory we can define the correspondingquantity

gE(t, θ) =∫d2zteiq·zt

〈WE1(zt)WE2(0)〉A〈WE1〉2A

. (43)

This is just the Fourier transform, with respect to the trans-verse coordinateszt, of the quantity (37), which we haveevaluated in thequenchedapproximation. Using the relation(38), we can derive that

gE(t, θ) −→θ→−iχ

gE(t,−iχ) = gM (t, χ) ;

or : gM (t, χ) −→χ→iθ

gM (t, iθ) = gE(t, θ) . (44)

3 The case of QCD at orderO(g4R)

In this section we shall see that the same relationship ofanalytic continuation appears to be valid also for the case of anon-Abelian gauge theory: we shall prove this up to the orderO(g4

R) in perturbation theory (gR being the renormalizedcoupling constant). Let us consider the following quantity,defined in Minkowski space-time:

gM (t, p1 · p2) =1Z2W

∫d2zteiq·zt

×〈[W1(zt)− 1]ij [W2(0)− 1]kl〉A , (45)

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Fig. 2. The contributions of the type (1, 1)|g2R

[a], (2, 2)|g4R

[b and c],

(3, 1)|g4R

[from d to f] and (1, 3)|g4R

[from g to i] to the amplitudes (52)

and (94). The notation is explained in the text

Fig. 3. The contributions of the type (2, 1)|g4R

[j ], (1, 2)|g4R

[k], (1, 1)

of order O(g4R) [from l to p], plus the counterterms [q and r ], for the

amplitudes (52) and (94). The notation is explained in the text

wherep1 andp2 are the four-momenta [lying (for example)in the plane (x0, x1)], which define the two Wilson linesW1andW2. By virtue of the Lorentz symmetry, we can definep1andp2 as in (13): that is, we choose, as the reference frame,the c.m.s. of the two particles, moving with speedβ and−β along thex1-direction. Of course,gM can only dependon the scalar quantities constructed with the vectorsp1, p2andq = (0, 0, q): the only possibilities areq2 = −q2 = t andp1 ·p2, becausep1 · q = p2 · q = 0 andp2

1 = p22 = m2 are fixed.

In such a reference frame, we can writep1 · p2 = m2 coshχ,whereχ = 2ψ (with β = tanhψ) is the hyperbolic angle [inthe plane (x0, x1)] between the two Wilson linesW1 andW2. The Wilson lines are defined as

W1(zt) ≡ P exp

[−ig

∫ +∞

−∞Aµ(zt +

p1

mτ )pµ1mdτ

];

W2(0) ≡ P exp

[−ig

∫ +∞

−∞Aµ(

p2

mτ )pµ2mdτ

]. (46)

Moreover, we have put, in (45),

ZW ≡ 1Nc〈Tr[W1(zt)]〉A

=1Nc〈Tr[W1(0)]〉A =

1Nc〈Tr[W2(0)]〉A . (47)

(The two last equalities come from the Poincare invariance.)This is a sort of Wilson-line’s renormalization constant: asshown in [3],ZW coincides with the fermion renormaliza-tion constantZψ in the eikonal approximation. We want toexplicitly evaluate the quantity (45) up to the orderO(g4

R) inperturbation theory. This corresponds to evaluate the Feyn-man diagrams in Figs. 2 and 3, where the two horizontaloriented lines represent the Wilson linesW1 andW2. Firstof all we need to expandZW up to the orderO(g2

R) inperturbation theory:

ZW = 1 +Z (2)W g2

R +O(g4R) . (48)

Clearly, we do not need to consider also theO(g4R) piece, of

the formZ (4)W g4

R, since the expectation value〈. . .〉A in (45)is an object of orderO(g2

R). As will become clear in thefollowing, we do not need to know the explicit expressionfor the coefficientZ (2)

W . Since we are interested in evaluatingthe quantity (45) up to the orderO(g4

R), we need to consideralso the effects of the renormalizations of the fields andthe coupling constantg, up to the orderO(g2

R). Using theconventional notation, we write

Aaµ = Z

1/23 Aa

Rµ ; g = ZggR , (49)

where the suffix “R” denotes the renormalized quantities.Therefore, we have that

Wi(zt)=P exp

[−iZ1W gR

∫ +∞

−∞ARµ(zt +

p1

mτ )pµ1mdτ

],

(50)

where the renormalization constantZ1W is defined as

Z1W ≡ ZgZ1/23 = 1 +Z (2)

1W g2R +O(g4

R) . (51)

Since we are interested in evaluating the amplitude

M (t, χ)=∫d2zteiq·zt〈[W1(zt)− 1]ij [W2(0)− 1]kl〉A ,

(52)

up to the orderO(g4R), the effects ofZ1W are visible when

we expand the Wilson linesWi only up to the first order ing:

Wi(zt)=1− iZ1W gR

∫ +∞

−∞ARµ(zt +

p1

mτ )pµ1mdτ + . . . .

(53)

529

This corresponds to consider only the diagrams of the one-gluon-exchanged type, having the following amplitude:

M(1,1) = Z21WMR(1,1) = MR(1,1) + (Z2

1W − 1)MR(1,1) , (54)

whereMR(1,1) is the “renormalized” one-gluon-exchangedamplitude, obtained fromM(1,1) by substituting the couplingconstantg in front and the gluon fieldAµ with the corre-sponding renormalized quantities:

MR(1,1) = −g2R(T a)ij(T

b)klpµ1p

ν2

m2

∫d2zteiq·zt

×∫dτ

∫dω〈Aa

Rµ(zt +p1

mτ )Ab

Rν(p2

mω)〉A . (55)

In our notation,M(i,j) denotes the contribution to the ampli-tude M, defined in (52), obtained after expanding the WilsonlinesW1 up to the orderO(gi) (i.e., up to the term contain-ing i gluon fields) and expanding the other Wilson lineW2up to the orderO(gj) (i.e., up to the term containingj gluonfields). Moreover, we define:MR(i,j) ≡ Z−(i+j)

1W M(i,j). If onewants to computeM(1,1) up to the orderO(g4

R), one mustproceed as follows, by virtue of (54) and (48):

M(1,1)|g4R

= MR(1,1)|g4R

+ 2Z (2)1W g2

R ·MR(1,1)|g2R. (56)

The expression forMR(1,1)|g2R can be immediately derived:

it corresponds to the diagram shown in Fig. 2a. In a givenLorentz gauge with a (bare) gauge parameterα, the freegluon-field propagator is given by

Gabµν(x− y) = −iδab

∫d4k

(2π)4

1k2 + iε

×[gµν − (1− α)

kµkνk2 + iε

]e−ik(x−y) . (57)

One thus finds that

M (a)(t, χ) = MR(1,1)(t, χ)|g2R

= g2R

1ti cothχ · (G1)ij,kl , (58)

whereG1 is the color factor for the one-gluon-exchangedprocess:

G1 ≡ T c(1) ⊗ T c

(2) ,

(G1)ij,kl ≡ (T c)ij(Tc)kl . (59)

Let us observe thatMR(1,1)|g2R

is gauge-independent, sincethe gauge parameterα does not appear at the right-hand-side of (58). The last term of (56) can be represented bythe diagrams in Figs. 3q and 3r, in which we have put acounterterm of the form−iZ (2)

1W g3R(T a)ij in one of the two

vertices between the gluon line and a Wilson line. So wehave that

M (q)(t, χ) = M (r)(t, χ)

= Z (2)1W g2

R ·MR(1,1)(t, χ)|g2R

= Z (2)1W g4

R ·1ti cothχ · (G1)ij,kl . (60)

The expression for the one-gluon-exchanged renormalizedamplitude up to the orderO(g4

R), i.e., MR(1,1)|g4R

, is givenby the sum of the contributions from the diagrams shown inFigs. 2a, 3l to 3p: this last one represents the insertion of acounterterm (Z3−1)δab(kµkν−gµνk2) into the gluon line. In

other words, one has to compute the quantity (55), using therenormalized gluon propagator up to the orderO(g2

R) whenevaluating the expectation value〈Aa

Rµ(zt+p1mτ )Ab

Rν(p2mω)〉A.

That is, in a given Lorentz gauge with a renormalized gaugeparameterαR = Z−1

3 α:

〈AaRµ(x)Ab

Rν(y)〉A = −i∫

d4k

(2π)4e−ik(x−y)Dab

Rµν(k) , (61)

whereDabRµν(k) is given by

DabRµν(k) = Z−1

3 Dabµν(k)

=δab

k2 + iε

[gµν − kµkν

k2+iε

1 +ΠR(k2)+ αR

kµkνk2 + iε

]. (62)

ΠR(k2) is a finite function of orderO(g2R), whose precise

form depends on the renormalization scheme which has beenadopted:

ΠR(k2) = g2RF

(2)(k2) +O(g4R) . (63)

At this point one can derive the full expression for the am-plitudeMR(1,1):

MR(1,1)(t, χ) =MR(1,1)(t, χ)|g2

R

1 +ΠR(t)

= g2R

1t[1 +ΠR(t)]

i cothχ · (G1)ij,kl . (64)

Let us observe that, differently fromMR(1,1)|g2R

, the valueof MR(1,1) is gauge-dependent, since the gauge parameterαR does appear insideΠR at the right-hand-side of (64).This is also generally true for the other diagrams in Figs. 2and 3. Therefore, for the following calculations, we shall fixthe gauge parameterαR to 1 (the so-calledFeynmangauge).Equation (64) is the full expression forMR(1,1), not truncatedat any perturbative order. Yet, we are only interested in theespression forMR(1,1) up to the orderO(g4

R):

MR(1,1)(t, χ)|g4R

=g2R

1t

[1− g2RF

(2)(t)]i cothχ · (G1)ij,kl .

(65)

Therefore, the contribution coming from theO(g4R) diagrams

shown in Figs. 3l to 3p is given by

M (l)(t, χ) + . . . +M (p)(t, χ)

= −g4R

F (2)(t)t

i cothχ · (G1)ij,kl . (66)

The contribution of orderO(g4R) coming from the two Feyn-

man diagrams shown in Figs. 2b and 2c is obtained by mul-tiplying the two pieces of orderO(g2) for each of the twotermsWi−1 in (52): i.e., we must evaluate the contributionM(2,2)|g4

R. The contributionM (b) coming from the graph in

Fig. 2b is conventionally calledladder term, while the othercontributionM (c), coming from the graph in Fig. 2c, will becalledcrossterm. These two contributions can be evaluatedin perturbation theory and the final result is (in theFeynmangauge, whereαR = 1)

M (b)(t, χ) +M (c)(t, χ)

= M(2,2)(t, χ)|g4R

= M (G1)(t, χ) · (G1)ij,kl +M (G2)(t, χ) · (G2)ij,kl , (67)

530

whereG1 has been already defined in (59) andG2 is thecolor factor for theladder process in Fig. 2b, i.e.,

G2 ≡ (T a(1)T

b(1))⊗ (T a

(2)Tb(2)) ,

(G2)ij,kl ≡ (T aT b)ij(TaT b)kl . (68)

In writing (67), we have made use of the following relationfor the color factors:

(T aT b)ij(TbT a)kl

= (T aT b)ij(TaT b)kl +

Nc

2(T c)ij(T

c)kl

≡ (G2)ij,kl +Nc

2(G1)ij,kl , (69)

which can be easily recovered using the algebra of the gen-eratorsT a, i.e., [T a, T b] = ifabcT

c; Nc is the number ofcolors of the theory (the gauge group isSU (Nc)). The co-efficientsM (G1)(t, χ) andM (G2)(t, χ) in front of the colorfactors in (67) are found to be

M (G1)(t, χ) = iNcg

4R

4πI(t)χ coth2χ ;

M (G2)(t, χ) = −12g4RI(t) coth2χ . (70)

In the previous expression we have adopted the conventionalnotation

I(t) =∫

d2kt(2π)2

1k2t + λ2

1(q− kt)2 + λ2

=∫d2zteiq·zt

(d2kt(2π)2

e−ikt·zt

k2t + λ2

)2

. (71)

(Remember that:t = −q2). The quantityλ is the usual reg-ularizing gluon mass, used as an IRcutoff: it must be putequal to zero at the end of the calculation.

We shall now compute the contribution from the di-agrams in Figs. 2d to 2i. They areO(g4

R) diagrams, ob-tained after expanding one of the two Wilson lines up tothe orderO(g3), and the remaining one up the first orderin g. We shall denote the sum of all these contributions byM(3,1)|g4

R+ M(1,3)|g4

R, in agreement with the notation intro-

duced above. One thus finds that

M(3,1)|g4R

= Z (2)W g2

R ·MR(1,1)|g2R

+∆M(3,1) ,

M(1,3)|g4R

= Z (2)W g2

R ·MR(1,1)|g2R

+∆M(1,3) , (72)

where∆M(3,1) and∆M(1,3) are divergent quantities, whoseregularized expressions depend on the adopted renormaliza-tion scheme. In theminimal subtraction(MS) renormaliza-tion scheme one finds that

∆M(3,1) = ∆M(1,3) = MR(1,1)|g2R· g2

R

(4π)2Nc

[1ε

+B

], (73)

where ε = (4− D)/2, D being the number of space-timedimensions, andB is a finite number (asε goes to zero).In the same renormalization scheme (MS), one also has that(always in theFeynmangaugeαR = 1):

Z1W = 1 +Z (2)1W g2

R +O(g4R) ,

with :Z (2)1W = − g2

R

(4π)2Nc

1ε. (74)

Therefore, from (73) and (74), one immediately concludesthat

∆M(3,1) +∆M(1,3) + 2Z (2)1W g2

R ·MR(1,1)|g2R

= MR(1,1)|g2R· g2

R

(4π)22NcB . (75)

In other words, the divergence contained in∆M(3,1) +∆M(1,3) is exactly calcelled out by the two diagrams withthe countertermZ (2)

1W , represented in Figs. 3q and 3r. Fi-nally, we have to evaluate the two diagrams in Figs. 3j and3k: in agreement with the notation introduced above, weshall denote their contribution byM(2,1)|g4

RandM(1,2)|g4

R,

respectively. However, explicit calculations show that theircontribution vanishes:

M(2,1)|g4R

= M(1,2)|g4R

= 0 . (76)

At this point we can sum up all the contributions previouslyevaluated in order to find the complete expression for theamplitudeM , defined by (8), up to the orderO(g4

R). Wefind that

M (t, χ)|g4R

=

[1 +

(2Z (2)

W − F (2)(t) +2NcB

(4π)2

)g2R

]·MR(1,1)(t, χ)|g2

R+MR(2,2)(t, χ) . (77)

Introducing here the expressions found above forMR(1,1)|g2R

[see (58)] and forMR(2,2)|g4R

[see (67) and (70)], we finallyfind the following expression forM (t, χ)|g4

R:

M (t, χ)|g4R

= g2R

1ti cothχ

×[

1 +

(2Z (2)

W − F (2)(t) +2NcB

(4π)2

+Nc

4πtI(t)χ cothχ

)g2R

]· (G1)ij,kl

−12g4RI(t) coth2χ · (G2)ij,kl . (78)

This expression allows us to immediately derive the quan-tity gM (t, χ), defined by (45), up to the orderO(g4

R). In fact,making use also of the expansion (74) for the renormaliza-tion constantZ1W , one finds that:

gM (t, χ)|g4R

=M (t, χ)Z2W

|g4R

= M (t, χ)|g4R− 2Z (2)

W g2R ·M (t, χ)|g2

R

= g2R

1ti cothχ

[1−

(F (2)(t) +

2NcB

(4π)2

+Nc

4πtI(t)χ cothχ

)g2R

]· (G1)ij,kl

−12g4RI(t) coth2χ · (G2)ij,kl . (79)

The quark–quark scattering amplitude in the high-energylimit turns out to be, up to the orderO(g4

R),

fM (s, t)|g4R

∼s→∞−i · δαβδγδ · 2s · gM (t, χ→∞)|g4

R

= δαβδγδ

[g2R

2st

[1− α(t) ln s]

531

·(G1)ij,kl + ig4RsI(t) · (G2)ij,kl

], (80)

where we have used the notation

α(t) ≡ −Ncg2R

4πtI(t) =

Ncg2R

4πq2I(t) . (81)

We have used the fact that bothβ andψ (or equivalentlyχ) are dependent ons. In fact, fromE = m/

√1− β2 and

from s = 4E2, one immediately finds that

β =

√1− 4m2

s. (82)

By inverting this equation and using the relationβ = tanhψ,we derive that

s = 4m2 cosh2ψ = 2m2(coshχ + 1) . (83)

Therefore, in the high-energy limits→∞ (or β → 1), thehyperbolic angleχ = 2ψ is essentially equal to the logarithmof s:

χ = 2ψ ∼s→∞ ln s . (84)

Moreover, cothχ ∼ 1 in this limit. This is why we havebeen able to approximate theO(g4

R) term which multiplies(G1)ij,kl as reported in (80). The result (80) is exactly whatcan be found by applying ordinary perturbation theory toevaluate the scattering amplitude up to the orderO(g4

R) [1,2]. In particular, as was pointed out in the Introduction, theln s factor in (80) comes from the fact that it is really asingular limit to take the Wilson lines in (45) exactly on thelight cone. As first predicted in [10], a proper regularizationof these singularities give rise to the lns dependence ofthe amplitude, as confirmed by the experiments on hadron–hadron scattering processes.

We want now to repeat the analogous calculation for theEuclidean theory. Let us consider, therefore, the followingquantity, defined in Euclidean space-time:

gE(t, v1 · v2) =1

Z2EW

∫d2zteiq·zt

×〈[WE1(zt)− 1]ij [WE2(0)− 1]kl〉A , (85)

where ¯zt = (0, zt, 0) and the expectation value〈. . .〉A mustbe intended now as a functional integration with respect tothe gauge variableA(E)

µ in the Euclidean theory. The Eu-clidean four-vectorsv1 and v2 [lying (for example) in theplane (x1, x4)] define the two Wilson linesWE1 andWE2:we can takev1 andv2 normalized to 1, with respect to theEuclidean scalar product (that is,v2

1 = v22 = 1). Clearly,gE

can only depend on the scalar variables constructed usingthe vectorsv1, v2 andqE = (0, q, 0): they areq2

E = q2 = −tand v1 · v2, sinceqE · v1 = qE · v2 = 0 andv2

1 = v22 = 1

are fixed. By virtue of theO(4) symmetry of the Euclideantheory, we can choose a reference frame in whichv1 andv2have the following values:

v1 = (sinφ, 0t, cosφ) ;

v2 = (− sinφ, 0t, cosφ) , (86)

with a value ofφ between 0 andπ/2 (so that the angle 2φbetween the two trajectories is in the interval [0, π]). In sucha reference frame, we can writev1 ·v2 = cosθ, whereθ = 2φ

is the angle [in the plane (x1, x4)] between the two EuclideanWilson linesWE1 andWE2. These last are defined as

WE1(zt) ≡ P exp

[−ig

∫ +∞

−∞A(E)µ (zt + v1τ )v1µdτ

];

WE2(0) ≡ P exp

[−ig

∫ +∞

−∞A(E)µ (v2τ )v2µdτ

], (87)

whereA(E)µ = A(E)a

µ T a. Moreover, we have put

ZEW ≡ 1Nc〈Tr[WE1(zt)]〉

=1Nc〈Tr[WE1(0)]〉 =

1Nc〈Tr[WE2(0)]〉 . (88)

(The two last equalities come from theO(4) plus translationinvariance.) We want to explicitly evaluate the quantity (85)up to the orderO(g4

R) in perturbation theory. Therefore, wehave to evaluate the Feynman diagrams in Figs. 2 and 3,where the two horizontal oriented lines now represent theEuclidean Wilson linesWE1 andWE2. As before, we needto expandZEW only up to the orderO(g2

R) in perturbationtheory:

ZEW = 1 +Z (2)EW g2

R +O(g4R) , (89)

even if, as will become clear in the following, we shallnot need to know the explicit expression for the coefficientZ (2)EW . As in the previous case, we need to consider also the

effect of the renormalizations of the fields and the couplingconstantg, up to the orderO(g2

R), that is:

A(E)aµ = Z

1/23 A(E)a

Rµ ; g = ZggR , (90)

where the suffix “R” denotes the renormalized quantities.Therefore we have that

WEi(zt)

= P exp

[−iZ1W gR

∫ +∞

−∞A(E)Rµ(zt + v1τ )v1µdτ

], (91)

the renormalization constantZ1W being defined by (51). Therenormalization constants in (90) are the same as those in(49) for the Minkowski world, if we adopt the same renor-malization scheme (for example, the MS scheme) for theEuclidean theory and the Minkowskian one. In such a case,the correspondence

A0(x) → iA(E)4 (xE) ,

Ak(x) → A(E)k (xE) (with : x0 → −ixE4) (92)

between the bare gluon fields in the two theories, turns intothe same correspondence for the renormalized gluon fields:

AR0(x) → iA(E)R4 (xE) ,

ARk(x) → A(E)Rk (xE) (with : x0 → −ixE4) . (93)

[As a matter of fact, the renormalization constantsZ3, Zg,etc., are always evaluated in the Euclidean world, also whenthey refer to the Minkowskian one: when evaluating theseconstants in Minkowski (momentum) four-space, one alwaysperforms a Wick rotation to Euclidean (momentum) four-space.]

We shall evaluate the amplitude

532

E(t, θ) =∫d2zteiq·zt

×〈[WE1(zt)− 1]ij [WE2(0)− 1]kl〉A , (94)

up to the orderO(g4R). As in the previous case, the effects of

Z1W are visible when we consider only the diagrams of theone-gluon-exchanged type, having the following amplitude:

E(1,1) = Z21WER(1,1) = ER(1,1) + (Z2

1W − 1)ER(1,1) , (95)

ER(1,1) being the “renormalized” one-gluon-exchanged am-plitude:

ER(1,1) =−g2R(T a)ij(T

b)klv1µv2ν

∫d2zteiq·zt

∫dτ

×∫dω〈A(E)a

Rµ (zt + v1τ )A(E)bRν (v2ω)〉A . (96)

In our notation,E(i,j) denotes the contribution to the am-plitude E, defined in (94), obtained after expanding the Eu-clidean Wilson lineWE1 up to the orderO(gi) (i.e., up tothe term containingi gluon fields) and expanding the otherEuclidean Wilson lineWE2 up to the orderO(gj) (i.e., upto the term containingj gluon fields). Moreover, we defineER(i,j) ≡ Z−(i+j)

1W E(i,j). We have to computeE(1,1) up to theorderO(g4

R), which, by virtue of (95) and (51), is given by

E(1,1)|g4R

= ER(1,1)|g4R

+ 2Z (2)1W g2

R · ER(1,1)|g2R. (97)

The expression forER(1,1)|g2R, corresponding to the diagram

shown in Fig. 2a, can be derived using in the calculation theEuclidean free gluon-field propagator. This last, in any givenLorentz gauge with a (bare) gauge parameterα, is given by

G(E)abµν (xE − yE)

= δab

∫d4kE(2π)4

1k2E

×[δµν − (1− α)

kEµkEνk2E

]e−ikE (xE−yE ) . (98)

The contribution coming from the one-gluon-exchange pro-cess pictorially represented in Fig. 2a, comes out to be, withthe notation already introduced for the color factor,

E(a)(t, θ) = ER(1,1)(t, θ)|g2R

= g2R

1t

cotθ · (G1)ij,kl . (99)

The last term of (97), represented by the diagrams in Figs. 3qand 3r, is given by

E(q)(t, θ) = E(r)(t, θ) = Z (2)1W g2

R · ER(1,1)(t, θ)|g2R

= Z (2)1W g4

R ·1t

cotθ · (G1)ij,kl . (100)

The first term in (97), i.e.,ER(1,1)|g4R

, is the one-gluon-

exchanged renormalized amplitude up to the orderO(g4R). It

is given by the sum of the contributions from the diagramsshown in Figs. 2a, 3l to 3p. [This last one represents theinsertion of a counterterm (Z3 − 1)δab(kEµkEν − δµνk

2E)

into the gluon line.] Therefore, one has to compute thequantity (96) using the renormalized gluon propagator upto the orderO(g2

R) when evaluating the expectation value〈A(E)a

Rµ (zt + v1τ )A(E)bRν (v2ω)〉A. That is:

〈A(E)aRµ (xE)A(E)b

Rν (yE)〉A

=∫

d4kE(2π)4

e−ikE (xE−yE )D(E)abRµν (kE) . (101)

The expression forD(E)abRµν (kE) can be derived from the cor-

responding expression (62) forDabRµν(k), making use of the

correspondence law (93) between the (renormalized) gluonfield in Minkowski four-space and the (renormalized) gluonfield in Euclidean four-space.D(E)ab

Rµν (kE) is obtained from

DabRµν(k) by making the replacements

k2 → −k2E (i.e., k0 → ikE4 , k → kE) ;

gµν → −δµν ; kµkν → kEµkEν . (102)

Therefore, in a Lorentz gauge with a renormalized gaugeparameterαR = Z−1

3 α, D(E)abRµν (kE) is given by

D(E)abRµν (kE) = Z−1

3 D(E)abµν (kE)

=δabk2E

δµν − kEµkEνk2E

1 +ΠR(−k2E)

+ αRkEµkEνk2E

, (103)

whereΠR is exactly the same finite function of orderO(g2R),

appearing in the expression (62) for the gluon propagator inMinkowski space-time:

ΠR(−k2E) = g2

RF(2)(−k2

E) +O(g4R) . (104)

As said before, the precise form ofΠR depends on the renor-malization scheme which has been adopted. At this point thederivation of the full expression for the amplitudeER(1,1) israther immediate and gives

ER(1,1)(t, θ) =ER(1,1)(t, θ)|g2

R

1 +ΠR(t)

= g2R

1t[1 +ΠR(t)]

cotθ · (G1)ij,kl . (105)

The value ofER(1,1) is gauge-dependent (asMR(1,1) was,too), since the gauge parameterαR does appear insideΠR

at the right-hand-side of (105). For the following calculationswe shall fix the gauge parameterαR to 1 (Feynmangauge),in conformity with the choice we have made in the previouscase. Equation (105) is the full expression forER(1,1), nottruncated at any perturbative order. Yet, we only need theespression forER(1,1) up to the orderO(g4

R):

ER(1,1)(t, θ)|g4R

= g2R

1t

[1− g2RF

(2)(t)]

× cotθ · (G1)ij,kl . (106)

Therefore, in the Euclidean theory, the contribution comingfrom theO(g4

R) diagrams shown in Figs. 3l to 3p is givenby

E(l)(t, θ) + . . . +E(p)(t, θ)

= −g4R

F (2)(t)t

cotθ · (G1)ij,kl . (107)

The contribution of orderO(g4R) coming from the two Feyn-

man diagrams shown in Fig. 2b (theladder term) and Fig. 2c(the crossterm), i.e., the contributionE(2,2)|g4

R, turns out to

be (in theFeynmangaugeαR = 1)

533

E(b)(t, θ) +E(c)(t, θ)

= E(2,2)(t, θ)|g4R

= E(G1)(t, θ) · (G1)ij,kl +E(G2)(t, θ) · (G2)ij,kl , (108)

whereG1 andG2 are two color factors defined in (59) and(68). The coefficientsE(G1)(t, θ) andE(G2)(t, θ) in front ofthe color factors in (108) are found to be

E(G1)(t, θ) =Ncg

4R

4πI(t)θ cot2 θ ;

E(G2)(t, θ) =12g4RI(t) cot2 θ . (109)

The contributionE(3,1)|g4R

+ E(1,3)|g4R

from the diagrams inFigs. 2d to 2i, obtained after expanding one of the two Wil-son lines up to the orderO(g3), and the remaining one upto the first order ing, can be written as

E(3,1)|g4R

= Z (2)EW g2

R · ER(1,1)|g2R

+∆E(3,1) ,

E(1,3)|g4R

= Z (2)EW g2

R · ER(1,1)|g2R

+∆E(1,3) , (110)

where∆E(3,1) and∆E(1,3) are divergent quantities, whoseregularized expressions depend on the adopted renormaliza-tion scheme. In the MS renormalization scheme one findsthat

∆E(3,1) = ∆E(1,3) = ER(1,1)|g2R· g2

R

(4π)2Nc

[1ε

+B

], (111)

where B is the same finite number (asε goes to zero),which appears in the corresponding expression (73) for theMinkowskian case. From (111) and from the expression (74)of Z1W up to the orderO(g2

R), one immediately derives that

∆E(3,1) +∆E(1,3) + 2Z (2)1W g2

R · ER(1,1)|g2R

= ER(1,1)|g2R· g2

R

(4π)22NcB . (112)

As in the Minkowskian case, the divergence contained in∆E(3,1)+∆E(1,3) is exactly cancelled out by the two diagramswith the countertermZ (2)

1W , represented in Figs. 3q and 3r.Finally, one has to evaluate the contributionsE(2,1)|g4

Rand

E(1,2)|g4R

, represented by the two diagrams in Figs. 3j and3k, respectively. Again, explicit calculations show that theircontribution vanishes:

E(2,1)|g4R

= E(1,2)|g4R

= 0 . (113)

We can now sum up all the contributions previously evalu-ated in order to find the complete expression for the ampli-tudeE, defined by (94), up to the orderO(g4

R):

E(t, θ)|g4R

=

[1 +

(2Z (2)

EW − F (2)(t) +2NcB

(4π)2

)g2R

]·ER(1,1)(t, θ)|g2

R+ER(2,2)(t, θ) . (114)

Introducing here the expressions found above forER(1,1)|g2R

[see (99)] and forER(2,2)|g4R

[see (108) and (109)], we finallyfind the following expression forE(t, θ)|g4

R:

E(t, θ)|g4R

= g2R

1t

cotθ

[1 +

(2Z (2)

EW − F (2)(t)

+2NcB

(4π)2+Nc

4πtI(t)θ cotθ

)g2R

]· (G1)ij,kl

+12g4RI(t) cot2 θ · (G2)ij,kl . (115)

The quantitygE(t, θ), defined by (85), can be immediatelyderived up to the orderO(g4

R), making use also of the ex-pansion (89) for the renormalization constantZEW :

gE(t, θ)|g4R

=E(t, θ)Z2EW

|g4R

= E(t, θ)|g4R− 2Z (2)

EW g2R · E(t, θ)|g2

R

= g2R

1t

cotθ

[1−

(F (2)(t) +

2NcB

(4π)2

+Nc

4πtI(t)θ cotθ

)g2R

]· (G1)ij,kl

+12g4RI(t) cot2 θ · (G2)ij,kl . (116)

After comparing the two expressions (79) and (116) forgM (t, χ)|g4

RandgE(t, θ)|g4

R, we immediately recognize that

they are linked by the following analytic continuation in theangular variable:

gE(t, θ)|g4R−→

θ→−iχgE(t,−iχ)|g4

R= gM (t, χ)|g4

R;

or : gM (t, χ)|g4R−→χ→iθ

gM (t, iθ)|g4R

= gE(t, θ)|g4R. (117)

This is the same relation we have already found in thepreceding section for the corresponding quantities in theAbelian case. It appears to be an absolutely “natural” corre-spondence law. There is apparently no reason why it shouldnot be true at higher perturbative orders: we shall thereforeassume that it is valid for the “full” amplitudes, i.e., not trun-cated at any perturbative order. In the next section we shalldiscuss some interesting consequences and some possibleapplications of this relationship of analytic continuation.

4 Concluding remarks and prospects

In the preceding section we have seen that also for a non-Abelian gauge theory it is possible to reconstruct the high-energy scattering amplitude by evaluating a correlation ofinfinite Wilson lines forming a certain angleθ in Euclideanfour-space, then by continuing this quantity in the angularvariable,θ → −iχ, whereχ is the hyperbolic angle betweenthe two Wilson lines in Minkowski space-time, and finallyby performing the limitχ → ∞ (i.e., β → 1). In fact, thehigh-energy scattering amplitude is given by

Mfi = 〈ψiα(p′1)ψkγ(p′2)|M |ψjβ(p1)ψlδ(p2)〉∼

s→∞−i · 2s · δαβδγδ · gM (t, χ→∞) . (118)

The quantitygM (t, χ), defined by (45) in the Minkowskiworld, is linked to the corresponding quantitygE(t, θ), de-fined by (85) in the Euclidean world, by the following ana-lytic continuation in the angular variables:

gE(t, θ) −→θ→−iχ

gE(t,−iχ) = gM (t, χ) ;

or : gM (t, χ) −→χ→iθ

gM (t, iθ) = gE(t, θ) . (119)

534

The important thing to note here is that the quantitygE(t, θ),defined in the Euclidean world, may be computed non per-turbatively by well-known and well-established techniques:first of all, of course, by means of the formulation of the the-ory on the lattice. Also the stochastic vacuum model [14],which is naturally defined for the Euclidean theory, may pro-vide a suitable instrument to evaluate the quantity (85). Inall cases, once one has obtained the quantitygE(t, θ), onestill has to perform an analytic continuation in the angularvariableθ → −iχ, and finally one has to extrapolate to thelimit χ→∞ (i.e.,β → 1). We are fully aware that this maynot be an easy way. Nevertheless, interesting new results areexpected along this direction. As an example, we shall showhow, using this approach, one can re-derive the well-knownRegge Pole Model[15], but in a different way, with respectto the original derivation. First of all, we writegE(t, θ) inthe partial-wave expansion:

gE(t, θ) =∞∑l=0

Al(t)Pl(cosθ) . (120)

If Al(t) can be analytically continued to complex values ofl, then we can re-write (120) in the following way:

gE(t, θ) =12i

∫C

Al(t)Pl(− cosθ)sin(πl)

dl , (121)

whereC is a contour in the complexl-plane, running anti-clockwise around the real positive l-axis and enclosing allnon-negative integers, while excluding all the singularitiesof Al. Equation (121) can be verified after recognizing thatPl(− cosθ) is an integer function ofl and that the singu-larities enclosed by the contourC of the expression underintegration in the (121) are simple poles at the non-negativeinteger values ofl. So the right-hand side of (121) is equalto the sum of the residues of the integrand in these poles andthis gives exactly the right-hand side of (120). The “minus”sign in the argument of the Legendre functionPl into (121)is due to the following relation, valid for integer values ofl:

Pl(− cosθ) = (−1)lPl(cosθ) . (122)

Then, we can reshape the contourC into the straight line<(l) = − 1

2. Equation (121) then becomes

gE(t, θ) = −∑

<(αn)>− 12

πrn(t)Pαn(t)(− cosθ)sin(παn(t))

− 12i

∫ − 12 +i∞

− 12−i∞

Al(t)Pl(− cosθ)sin(πl)

dl , (123)

whereαn(t) is a pole ofAl(t) in the complexl-plane andrn(t) is the corresponding residue. We have assumed thatAl

vanishes enough rapidly as|l| → ∞ in the right half-plane,so that the contribution from the infinite contour is zero.Equation (123) immediately leads to the asymptotic behaviorof the scattering amplitude in the limits→∞, with a fixedt (|t| � s). In fact, making use of the analytic extension(119) when continuing the angular variable,θ → −iχ, wederive that

gM (t, χ) = gE(t,−iχ)

= −∑

<(αn)>− 12

πrn(t)Pαn(t)(− coshχ)sin(παn(t))

− 12i

∫ − 12 +i∞

− 12−i∞

Al(t)Pl(− coshχ)sin(πl)

dl . (124)

The hyperbolic angleχ is linked to s by the relation (83).Therefore we can re-express coshχ in terms of s in thefollowing way:

coshχ =s

2m2− 1 . (125)

The asymptotic form ofPα(z) when|z| → ∞ is well known.It is a linear combination ofzα and ofz−α−1. When<(α) >−1/2, this last term can be neglected. Therefore, in the limits → ∞, with a fixed t (|t| � s), we are left with thefollowing expression:

gM (t, χ→∞) ∼∑

<(αn)>− 12

βn(t)sαn(t)

sin(παn(t)), (126)

whereβn(t) is independent ons (it only depends ont). Theintegral in (124), usually called thebackground term, van-ishes at least ass−1/2. Equation (126) allows to immediatelyextract the scattering amplitude according to (118):

Mfi ∼s→∞−i · 2s · δαβδγδ · gM (t, χ→∞)

∼ δαβδγδ∑

<(αn)>− 12

βn(t)s1+αn(t)

sin(παn(t)). (127)

This equation gives the explicit dependence of the scatteringamplitude at very high energys→∞ and a fixed transferredmomentumt (|t| � s). As we can see, this amplitude comesout to be a sum of powers ofs. If we put α(t) = 1 +αn(t),whereαn(t) is the pole with the largest real part (at thatgiven t), we can also write

Mfi ∼ δαβδγδ · β(t)sα(t) . (128)

This sort of behavior for the scattering amplitude was firstproposed by Regge in [15] and ¯α(t) is often called a “Reggepole”. In the original derivation [15], the asymptotic behav-ior (128) was recovered by analytically continuing to verylarge imaginary values the angle between the trajectories ofthe two exiting particles in thet-channel process. Instead, inour derivation, we have analytically continued the quantity(85), defined in the Euclidean theory, to very large (nega-tive) imaginary values of the angleθ between the two Eu-clidean Wilson lines. As in the original derivation, we haveassumed that the singularities ofAl are simple poles. If thereare other kinds of singularities, different from simple poles,their contribution will be of a different type and, in general,also logarithmic terms (ofs) may appear in the amplitude.Only a precise evaluation ofgE(t, θ) can reveal such behav-iors (after the analytic continuation). In the preceding sectionthis was done up to the orderO(g4

R) in perturbation theory.New interesting results are expected from a non perturbativeapproach, for example by directly computinggE(t, θ) on thelattice or by means of the stochastic vacuum model.

Acknowledgements.I would like to thank Adriano Di Giacomo and GuntherDosch for many useful discussions.

535

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