El problema: delimitación y directrices sobre la cuestión de la vida en SuZ

We have considered power corrections to hadronic final states in e⁺e⁻ annihilation and to DIS. Renormalon divergence also appears in the hard scattering coefficients in hadron-hadron collisions. The simplest hadron-hadron-hadron-hadron hard scattering process is Drell-Yan pro-duction of a lepton pair or a massive vector boson, A + B → {γ^∗, W, Z}(Q) + X, where X is any hadronic final state. At leading power dσ/dQ² = σ0W (τ, Q²), where σ0 is the Born cross section, τ = Q²/s, and

W (τ, Q²) = ^X

i,j

Z1

0

dxi

xi

dxj

xj

f_i/A(xi, Q²)f_j/B(xj, Q²) ωij(z, αs(Q)), (5.68)

with z = Q²/(x1x2s) and s is the centre-of-mass energy squared of A and B. In the following we are concerned with renormalon divergence and long-distance contributions to the hard scattering factor ωij(z, αs(Q)). It is convenient to work in moment space, in which

W (N, Q²) ≡

Z1

0

dτ τ^{N −1}W (τ, Q²) = f_q/A(N, Q²)f_q/B¯ (N, Q²) ωq ¯q(N, αs(Q)), (5.69) where the right hand side is expressed in terms of moments of the parton distributions (hard scattering factor) with respect to xi (z).

When Q is large, one can consider large moments 1 ≪ N ≪ Q/Λ. Conventional, fixed-order perturbation theory fails for high moments, because one encounters correc-tions αⁿ_sln^mN with m up to 2n. The physical origin of these corrections is that there exist three scales Q, Q/√

N and Q/N and the logarithms are ratios of these scales.

These scales appear, because for large N the moment integral is dominated by Q² ∼ s, which leaves little phase space for the hadronic system X. In a perturbative calculation the energy available for real emission is constrained to be of order Q/N and the IR cancellation between virtual and real correction becomes numerically ineffective.

The logarithmically enhanced contributions can be resummed systematically to all orders in perturbation theory (Sterman 1987; Catani & Trentadue 1989). The result has the exponentiated form⁵⁷

The function A is related to soft-collinear radiation and also referred to as ‘cusp’ or

‘eikonal’ anomalous dimension. The function B relates to the DIS process which enters in factorizing the parton densities. The function C, not needed for resummation of next-to-leading logarithms, relates to the Drell-Yan process (see (Sterman 1987) for details). The arguments of the coupling constants reflect the physical scale relevant to the respective subprocess.

Renormalon divergence is also related to soft gluons and one may ask what the precise relation to soft gluon resummation is. This question has guided the work on renormalons

57In the remainder of this section we restrict attention to the q ¯q annihilation subprocess.

in Drell-Yan production. Note that the integrals in (5.71) are formal, because they include integration over the Landau pole of the coupling. It was already noted in (Collins et al. 1989) that this implies sensitivity to the large-order behaviour in perturbation theory. (Contopanagos & Sterman 1994) performed the first quantitative analysis and found that the ambiguity due to the Landau poles in (5.71) in conventional leading or next-to-leading order resummations scales as Λ/Q. Leading order resummations of logarithms of N need only keep the first-order term in αs of A(αs) = a0αs+ . . .. B and C can be set to zero at this order. One then finds for the Borel transform (defined by (2.5) and using u = −β⁰t as usual) of the exponent

B[ELLA](N, u) ^u→1/2= 4(N − 1)

1 − 2u a0. (5.72)

The pole at u = 1/2 leads to the ambiguity of order Λ/Q in defining the exponent at leading-logarithmic accuracy which was noted by (Contopanagos & Sterman 1994). The question arises whether this ambiguity indicates a power correction of order Λ/Q to the hard scattering factor of the Drell-Yan cross section or whether the ambiguity appears as the consequence of a particular implementation of soft gluon resummation that was not designed to be accurate beyond leading power.

This question has been studied by (Beneke & Braun 1995b) at the level of one gluon virtual and real corrections with vacuum polarization insertions and accounting for gluon splitting into a q ¯q pair. Even in this approximation the functions A, B and C that enter the exponent become infinite series. The large-order terms in these series account for highly subleading logarithms in N and are not needed for the resummation of such logarithms to a given accuracy. On the other hand the Borel transform of the exponent becomes

B[E](N, u)^u→1/2= 4(N − 1) 1 − 2u



B[A](1/2) − 1

4B[C](1/2)



, (5.73)

and the residue of the pole at u = 1/2 involves the series expansion of A and C to all orders. (Beneke & Braun 1995b) found that when all orders are taken into account, the expression in square brackets is zero close to u = 1/2 and the pole is cancelled. After this cancellation the leading power correction to Drell-Yan production turns out to be of order N²Λ²/Q², at least in the approximation mentioned above. Note that the function B, related to the deep-inelastic scattering process, does not appear in (5.73). This is due to the argument of the coupling, which is larger, √

1 − zQ, in this case. In general, one finds that the terms introduced by performing collinear factorization in the DIS scheme are not relevant to the discussion of potential Λ/Q corrections. This is expected, because higher-twist corrections scale only as Λ²/Q² in DIS.

The physical origin of the cancellation becomes more transparent in terms of the sensitivity of the one-gluon emission amplitude to an IR cutoff. To this end we choose a cutoff µ and require the energy and transverse momentum of the emitted gluon to be

larger than µ. We are interested in terms of order µ in the cutoff. To this accuracy the one-gluon emission contribution in moment space can be written as

W_real^[1](N, µ) = 2 C_Fα_s The expansion at small µ of this integral starts with logarithms of µ. They would be cancelled by adding the virtual correction and collinear subtractions, both of which can be seen not to be able to introduce a linear dependence on µ. Expanding the square root in k_t/Q, one finds the following expression for the term of order µ/Q in the expansion at small µ: Hence there is in fact no linear sensitivity to an IR cutoff. One needs all terms in the expansion of the square root to obtain this cancellation. This means that to linear power accuracy the collinear approximation kt ≪ k⁰ ∼ Q(1 − z)/2, where k^t is the transverse momentum and k0 the energy of the emitted gluon, is not valid. It is essential to consider also large angle, soft gluon emission with kt ∼ k⁰. This conclusion (Beneke & Braun 1995b) is general and extends beyond the Drell-Yan process.

For the resummation of leading (next-to-leading etc.) logarithms of N an expansion in kt/k0 is justified. The leading logarithms are obtained by neglecting kt under the square root of (5.74). This leads to the first term only in the sum of (5.75) and a non-vanishing coefficient of µ/Q in agreement with the pole at u = 1/2 in (5.72) obtained in the same approximation.

The fact that the exact phase space for soft gluon emission is required to determine the coefficient of power corrections correctly relates to the fact that all terms in the expansion of the functions A and C in the exponent have to be kept for this purpose.

In particular the function C, not related to the eikonal anomalous dimension, is needed and this rules out the possibility discussed in (Akhoury & Zakharov 1995) that the universal parameter for 1/Q power corrections is given by the integral over the the eikonal anomalous dimension A(αs(kt)). Another implication is that the angular ordering prescription, according to which the emission angles of subsequent emissions in a parton cascade decrease, and which generates the correct matrix elements to next-to-logarithmic accuracy in N (see for example (Catani et al. 1991)), cannot be applied to power corrections. The intuitive argument that partons emitted at large angles can resolve only the total colour charge of the previous branching process does not hold true beyond leading power.

This argument also resolves a paradox raised by (Korchemsky & Sterman 1995b), who noted that 1/Q power corrections at large N and to 1 − T close to T = 1 should be

related, because the corresponding resummation formulae for logarithmically enhanced terms in perturbation theory are related. At present such a relation is known only to next-to-leading logarithmic accuracy (Catani et al. 1993). The fact that all orders in the exponent are needed for power corrections explains that it is consistent to expect Λ/Q power corrections to thrust but not to the Drell-Yan process.

Is it possible to organize the resummation of leading, next-to-leading, etc. logarithms in N without introducing undesired, because spurious, power corrections of order Λ/Q?

(Catani & Trentadue 1989) noted that one may substitute z^{N −1}− 1 → −Θ 1 − e^−γE

N − z

!

(5.76) in (5.73) to next-to-leading logarithmic accuracy. Then, for N ≪ Q/Λ which one must require for a short-distance treatment⁵⁸, the integration in (5.73) does not reach the Lan-dau pole and there are no power corrections to the exponent, unless the series expansions for A, B and C are themselves divergent.

(Beneke & Braun 1995b) addressed the question above in the fermion bubble ap-proximation, which provides a useful toy model, because the functions A, B and C are infinite series expansions in α_s. Ignoring complications from collinear subtractions, the partonic Drell-Yan cross section factorizes into ˆσDY(N, Q) = H(Q, µ) S(Q/N, µ) up to corrections that vanish as N → ∞, where H depends only on the ‘hard’ scale Q and S on the ‘soft’ scale Q/N. Following (Korchemsky & Marchesini 1993), the soft part is expressed as the Wilson line expectation value

S(Q/N, µ, αs) = and p1,2 denote the momenta of the annihilating quark and anti-quark. The ‘soft part’

S satisfies a renormalization group equation in µ that can be used to sum logarithms in N, because S depends only on the single dimensionless ratio Q/(Nµ). The solution to the RGE equation

58Recall that the expansion parameter for power corrections is N²Λ²/Q². For N ∼ Q/Λ the Drell-Yan process ceases to be a short-distance process and factorization breaks down.

reads

ˆ

σDY = H(αs(Q)) · S(α^s(Q/N)) · exp

Q²

Z

Q²/N²

dk_t² k²_t



Γeik(αs(kt)) lnk_t²N²

Q² + ΓDY(αs(kt))

!

, (5.80) where SDY(αs(Q/N)) denotes the initial condition for the evolution and in the end we have set µ = Q. From the analysis in the fermion loop approximation, one can draw the following, more general, conclusions.

The anomalous dimensions Γeik(αs) and ΓDY(αs) have convergent series expansions when defined in the MS scheme. Since the integrations in the exponent of (5.80) exclude the Landau pole for all moments N in the short-distance regime, it follows that the resummation, embodied by the exponent, can be carried out without ever encountering divergent series and power corrections implied by them. The conclusion is then that the renormalon problem is a problem separate from soft gluon resummation. Renormalons and power corrections enter in the hard part H and the initial condition S. Because S depends only on Q/N, the parameter for power corrections to S is NΛ/Q. One finds that all power corrections of order (NΛ/Q)^k to the Drell-Yan cross section are correctly reproduced in the soft part. In the approximation considered in (Beneke & Braun 1995b), terms with k = 1 do not exist. Note that if the exponentiated cross section is written in the ‘standard form’ (5.70, 5.71), the initial condition SDY(αs(Q/N)) is absorbed the exponent at the expense of a redefinition of C (ΓDY). With this redefinition the functions in the exponent are divergent series.

As always there is the question whether the absence of renormalon divergence that would correspond to a Λ/Q power correction is specific to the (essentially abelian) ap-proximation of (Beneke & Braun 1995b) and persists to more complicated diagrams.

The answer to this question is still open.

(Akhoury & Zakharov 1996; Akhoury et al. 1998; Akhoury et al. 1997) put the cancellation of 1/Q corrections to Drell-Yan production in the more general context of Kinoshita-Lee-Nauenberg (KLN) cancellations. Knowing that any potential 1/Q cor-rection would come from soft particles, but not collinear particles, one considers KLN transition amplitudes which include a sum over soft initial and final particles degenerate with the annihilating q ¯q pair. The KLN transition amplitudes have no 1/k0 (where k0

stands for the energies of the soft particles) contributions (collinear factorization is im-plicitly assumed). As a consequence, the amplitudes squared integrated unweighted over all phase space is proportional to dk0k0, which by power counting implies at most 1/Q² power corrections. To make connection with a physical process, one has to demonstrate that the sum over degenerate initial states can actually be dispensed of. The authors above use the Low theorem to show this for Drell-Yan production in an abelian theory.

For QCD this still remains an open problem.

(Korchemsky 1996) argued that non-abelian diagrams (involving the 3-gluon vertex) at 2-loop order would give a non-vanishing contribution to a certain Wilson line operator introduced in (Korchemsky & Sterman 1995a) to parametrize 1/Q corrections to Drell-Yan production. It would be very interesting to carry out the 2-loop calculation to see whether a non-zero linear infrared contribution is actually present in these diagrams.

(Qiu & Sterman 1991) extended collinear factorization for Drell-Yan production to 1/Q² corrections and showed that the same twist-4 multi-parton correlations enter as in deep-inelastic scattering. The factorization is carried out at tree-level and hence may not be conclusive on the issue of a 1/Q power correction, which would require a demonstration that soft gluon interactions cancel to all orders in perturbation theory to the level of 1/Q² accuracy. This is, at present, the missing element in a proof that there are no 1/Q long-distance sensitive regions in the Drell-Yan process to all orders in perturbation theory.

(Korchemsky & Sterman 1995a) have also considered power corrections to the trans-verse momentum (impact parameter) distributions in Drell-Yan production. In impact parameter space, they find that ambiguities in defining the perturbative contribution to the exponent require power-suppressed contributions of form

(bΛ)²(α ln Q + β) (5.81)

with b the impact parameter. The leading correction is quadratic in Λ and consistent with the parametrization of long-distance contributions suggested by (Collins & Soper 1981).