CAN THE POMERON BE DERIVED FROM A EUCLIDEAN–MINKOWSKIAN DUALITY?

After a brief review, in the first part, of some relevant analyticity and crossing-symmetry properties of the correlation functions of two Wilson loops in QCD, when going from Euclidean to Minkowskian theory, in the second part we shall see how these properties can be related to the still unsolved problem of the asymptotic s-dependence of the hadron-hadron total cross sections. In particular, we critically discuss the question if (and how) a pomeron-like behaviour can be derived from this Euclidean-Minkowskian duality.


Quark-quark scattering amplitudes
The quark-quark scattering amplitude, at high squared energies s in the center of mass and small squared transferred momentum t (that is to say: |t| ≤ 1 GeV 2 s), can be described by the expectation value of two infinite lightlike Wilson lines, running along the classical trajectories of the colliding particles [Nachtmann, 1991;EM1996;EM2001].
=⇒ there are infrared (IR) divergences, which are typical of 3+1 dimensional gauge theories ! → One can regularize this IR problem by letting the Wilson lines coincide with the classical trajectories for quarks with a non-zero mass m (so forming a certain finite hyperbolic angle χ in Minkowskian space-time: of course, χ → +∞ when s → ∞), and, in addition, by considering finite Wilson lines, extending in proper time from −T to T (and eventually letting T → +∞) [Verlinde, 1993;EM2002]: M qq (s; t) α α;β β i i;j j ∼ s→∞ −i 2s δ α α δ β β g qq M (χ → +∞; T → ∞; t) i i;j j , → Similarly g M , as a function of the complex variable χ, is the analytic extension of g M from the positive real axis (Reχ > 0, Imχ = 0+) to a domain D M = {χ ∈ C | − iχ ∈ D E } which also includes the imaginary segment (Reχ = 0, 0 < Imχ < π).
→ This analytic continuation (assuming certain analyticity hypotheses) is an exact, i.e., nonperturbative result, valid both for the Abelian and the non-Abelian case.
From Wilson lines to Wilson loops → The quantities g M (χ; T ; t) and g E (θ; T ; t), while being finite at any given value of T , are divergent in the limit T → ∞ (even if in some cases this IR divergence can be factorized out . . .).
→ A way to get rid of the problem of the IR-cutoff dependence is to consider an IR-finite physical quantity, like the elastic scattering amplitude of two colourless states in gauge theories, e.g., two qq meson states [Balitsky & Lipatov, 1978;1979].
→ The high-energy meson-meson elastic scattering amplitude can be approximately reconstructed by first evaluating, in the functional-integral approach, the high-energy elastic scattering amplitude of two qq pairs (usually called dipoles), of given transverse sizes R 1⊥ and R 2⊥ and given longitudinal-momentum fractions f 1 and f 2 of the two quarks in the two dipoles respectively [Nachtmann, 1997;Dosch et al., 1994]: → W 1 and W 2 are two Wilson loops, defined as: → Crossing symmetry relates the amplitude of this process to the amplitude of the "crossed process", defined as: q(p 1 , α, i)+q(−p 2 −p 2 , β , j ) → q(p 1 p 1 , α , i )+q(−p 2 , β, j).
Using the fact that the generators T a are hermitian and the variables A a µ are real, we can also write: And, therefore: That is, reminding the definition of the quark-quark correlator: Butp 2 = −p 2 is unphysical! Observe that: → To determine unambiguously which complex values of χ this substitution corresponds to, we will make use of the analyticcontinuation relation between the Minkowskian and the Euclidean theory and of the O(4) symmetry of the latter.
Similarly, for Euclidean Wilson lines: and so: That is to say: using the invariance under the O(4) 90 • "rotation" R 3 in the (x E1 , x E4 ) plane and under the O(4) "time-reversal" transf. R 2 : ) ) ) ) ) ) ) ) ) ) 1 0 ) ) ) ) ) ) ) ) ) ) ) @ ¤ ¥ 3 2 2 5 4¦ 6 7 ¤ ¥ " 2 2 8 4¦ $ % → Therefore we have obtained: Suppose now that this relation can be analytically extended to values of θ in a common analyticity domain D E for g qq E and g qq E : → Using the analytic-continuation relations, we thus obtain: where D M = {χ ∈ C| − iχ ∈ D E } is the common analyticity domain of g qq M and g qq M : Crossing relations for loop-loop correlators → Let us consider a certain Wilson loop: → Let us define the corresponding antiloop W by exchanging the quark and the antiquark trajectories: Consequently: → Going on as for the line-line case, from C(p, b, R, f ) = C(−p, b, R, f ) we immediately obtain: → Therefore we find the following crossing-symmetry relations: → And also, removing the IR cutoff T (T → ∞):

Perturbative expansion of the loop-loop correlators
Perturbation theory is the only calculation technique from first principles available both in the Minkowskian and in the Euclidean theory, and it can surely give us some useful insights about the analytic structure of the real (nonperturbative) correlation functions.
→ As a pedagogic example to illustrate these considerations, we first consider the simple case of QED, in the so-called quenched approximation, where fermion loops are neglected, i.e.: → In such an approximation the functional integrals become simple Gaussian integrals and one finds (taking where the coupling constant is now the electric charge e and: → The analytic extension C M of the Minkowskian correlator from the positive real axis χ ∈ R + and the analytic extension C E of the Euclidean correlator from the real segment θ ∈ (0, π) are given by: with the following analyticity domains: → Both the analytic-continuation relations and the crossing-symmetry relations are trivially satisfied ! → The loop-loop correlators (both in the Minkowskian and in the Euclidean theory) have been also computed up to the order O(g 6 ) of QCD perturbation theory [Babansky & Balitsky, 2003].

Angular singularities vs. bound states
It is well known that the loop-loop Euclidean correlation function G E (θ, T, z ⊥ ; 1, 2) in the case θ = 0 and T → ∞ is related to the van der Waals potential V dd ( z ⊥ ; 1, 2) between two static fermionantifermion dipoles: → As a pedagogic example, in quenched QED this quantity can be easily calculated from the expressions reported in [EM2005]: → The above-written relation tells us that the correlator G E when T → ∞ has a singularity in θ = 0. The use of the crossingsymmetry relation then immediately tells us that G E when T → ∞ has also a singularity in θ = π and, by virtue of the periodicity in θ, a singularity is expected in each point θ = kπ, k ∈ Z.
→ A similar result is expected to hold also for the line-line Euclidean correlation functions . . .

From Wilson loops to hadrons
→ From the loop-loop scattering amplitude . . .
→ . . . to the hadron-hadron elastic scattering amplitude: → It can be derived from the analytic continuation θ → −iχ of the corresponding Euclidean quantity: which can be evaluated non-perturbatively by well-known and well-established techniques available in the Euclidean theory. • C E (θ; . . .) is an analytic function of cot θ with analyticity domain D E = {θ ∈ C|θ = kπ, k ∈ Z}.
=⇒ By virtue of the optical theorem: all these results seem to imply (apart from possible s-dependences in the hadron wave functions!) s-independent hadron-hadron total cross sections in the asymptotic high-energy limit . . .
→ . . . in apparent contradiction to the experimental observations, which seem to be well described by a pomeron-like high-energy behaviour: , with : P 0.08.
=⇒ A behaviour like this seems to emerge directly when applying the Euclidean-to-Minkowskian analytic-continuation approach in these two cases: • Analyticity of the loop-loop correlation function in the angle.
• The first iteration of the BFKL kernel in the leading-log approximation, the so-called BFKL-pomeron behaviour, is reproduced [BFKL, 1975[BFKL, -1978:

How a pomeron-like behaviour can be derived
We start by writing the Euclidean hadronic correlation function in a partial-wave expansion: → Because of the crossing-symmetry relations, it is natural to decompose our hadronic correlation function C (hh) E (θ, t) as a sum of a crossing-symmetric function C + E (θ, t) and of a crossing- → The partial-wave expansions of these two functions are: Because of the relation P l (− cos θ) = (−1) l P l (cos θ), ∀l ∈ N, we can replace A l (t) respectively with A ± l (t) ≡ 1 2 [1 ± (−1) l ]A l (t): A l (t) , for even l 0 , for odd l ; A − l (t) = 0 , for even l A l (t) , for odd l .
The functions C ± E (θ, t) can also be called even-signatured and odd-signatured correlation functions respectively. =⇒ If we remember that: and we make use i) of the crossing-symmetry relations: and ii) of the rotational-and C-invariance of the squared hadron wave functions: → then we immediately conclude that the hadronic correlation function is automatically crossing symmetric (θ ↔π − θ): → Upon Euclidean-to-Minkowskian analytic continuation: → and therefore also the high-energy meson-meson elastic scattering amplitude M (hh) turns out to be automatically crossing symmetric (χ ↔iπ − χ) [=⇒ Pomeranchuk theorem].
=⇒ Let us therefore proceed by considering our crossing-symmetric Euclidean correlation function: We can now use Cauchy's theorem to rewrite this partial-wave expansion as an integral over l, the so-called Sommerfeld-Watson transform: [see, e.g., "Pomeron Physics and QCD", by Donnachie, Dosch, Landshoff & Nachtmann, 2002]. As in the original derivation . . .

C (b) (a)
. . . we make the fundamental assumption that the singularities of A l (t) in the complex l-plane (at a given t) are only simple poles.
→ Then we can use again Cauchy's theorem to reshape the contour C into the straight line Re(l) = − 1 2 and rewrite the integral as: C where σ + n (t) is a pole of A + l (t) in the complex l-plane and r + n (t) is the corresponding residue. We have also assumed that the large-= l behaviour of A + l is such that the integrand function vanishes enough rapidly (faster than 1/l) as |l| → ∞ in the right halfplane, so that the contribution from the infinite contour is zero. (Then A + l is unique, by virtue of Carlson's theorem !) =⇒ Making use of the Euclidean-to-Minkowskian analytic continuation θ → −iχ in the angular variable, we derive that, ∀χ ∈ R + : → Now we take the large-χ (large-s) limit of this expression, with: • Indeed, by virtue of the usual iε-prescription m 2 → m 2 − iε, with ε → 0+, (i.e., s → s + iε, with ε → 0+, as is well known), we have that: χ → χ + iε, with χ ∈ R + , ε → 0+; that is to say: Im(z ≡ cosh χ) > 0.
→ The asymptotic form of P ν (z) when z → ∞ is known to be: • We thus obtain, for each term in the sum [Re(σ + n ) > − 1 2 ]: • The integral, instead, usually called the background term, vanishes at least as 1/ √ s and thus can be neglected. =⇒ Therefore, in the limit s → ∞, with a fixed t (|t| s), we are left with the following expression: → . . . and, for the scattering amplitude: → This sort of behaviour for the scattering amplitude is known in the literature as a Regge behaviour, . . . and 1 + σ + n (t) ≡ α + n (t) is the so-called Regge trajectory. → Denoting with σ P (t) the pole with the largest real part (at that given t) and with β P (t) the corresponding coefficient β + n (t): where α P (t) ≡ 1 + σ P (t) is the pomeron trajectory.
→ Therefore, by virtue of the optical theorem: → In the original derivation the asymptotic behaviour is recovered by analytically continuing the t-channel scattering amplitude to very large imaginary values of the angle between the trajectories of the two exiting particles in the t-channel scattering process.
=⇒ Two important issues: • NO energy dependence of hadron wave functions =⇒ A + l , β + n and σ + n do NOT depend on s, but only on t ! • However, this is not enough to guarantee the experimentallyobserved universality of the pomeron trajectory α P (t) ! → We may start from the partial-wave expansion of the fundamental loop-loop Euclidean correlation function: (2l + 1)A l (t; 1, 2)P l (cos θ), → . . . and we arrive at the following Regge expansion for the (even-signatured) loop-loop Minkowskian correlator: → If we assume that (at least) the location of the pole σ P (t) ≡ a + n (t; 1, 2) with the largest real part does NOT depend on R 1⊥ , f 1 and R 2⊥ , f 2 , but only depends on t, we then find: with a universal pomeron trajectory α P (t) = 1 + σ P (t), . . . . . . while the coefficient β P (t) in front explicitly depends on the specific type of hadrons involved in the process.

Concluding remarks and prospects
=⇒ Certain apparently reasonable analyticity hypotheses of the line-line and loop-loop correlation functions imply both the Euclidean-to-Minkowskian analytic-continuation relations and (directly from this) also the crossing-symmetry relations.
=⇒ The reasonableness of the above-mentioned analyticity hypotheses comes essentially from perturbation theory.
=⇒ A real nonperturbative foundation of these properties is at the moment out of our reach.
=⇒ One is immediately faced with the following series of questions: • Are the analyticity hypotheses considered above maybe too strong and to what degree can they be relaxed?
• What about the Euclidean-to-Minkowskian analytic continuation if weaker analyticity hypotheses are kept in place of those discussed above?
• What about, finally, the crossing symmetry relation in the presence of a more complicated analytic structure, when the Euclidean-to-Minkowskian analytic continuation cannot be trusted, at least in the form presented above?
→ We have not the answers, at the moment . . .

=⇒
The Euclidean-to-Minkowskian analytic-continuation approach can, with the inclusion of some extra assumptions, easily reproduce a pomeron-like behaviour for the high-energy total cross sections, in apparent agreement with the present-day experimental observations.
→ The pomeron-like behaviour is, strictly speaking, forbidden (at least if considered as a true asymptotic behaviour) by the wellknown Froissart-Lukaszuk-Martin (FLM) theorem, according to which, for s → ∞: σ tot (s) ≤ π m 2 π log 2 s s 0 .
→ The pomeron-like behaviour can at most be regared as a sort of pre-asymptotic (but not really asymptotic !) behaviour of the high-energy total cross sections.
→ Immmediately the following question arises: why our approach, which was formulated so to give the really asymptotic large-s behaviour of scattering amplitudes and total cross sections, is also able to reproduce pre-asymptotic behaviours [violating the FLM bound] like the pomeron?