An Introduction to leading and next-to-leading BFKL Act. Phys. Pol. B30 (1999) 3679, G.P. Salam

1. IntroductionBFKL describes the high-energy behavior of the scattering of hadronic objects within the pQCD = (O(1))n the need of sum of whole series of Leading Logarithmic (LL) terms.result σ grows as a power of s 0.5 when 　 too largeSolution (?) next-to-leading (NL) correction But it needed about 10 years ! And not a satisfactory result !!!How one can go beyond the NL ?

1.1 problemCollisions of two perturbative hadronic objects:s >> Q2, Q2

0 >> Λ2 DIS at HERA, high-energy γγ scatt. at LEP, LHC etc.

2. Leading-logarithmic order2.1 DISAssume that one of the two hadronic objects is much smaller than the other: Mp

2 = Q20 << Q2 << s, Bj x x = Q2/s,

it is necessary to resum terms The cross section the quark distribution:∝

At small x, the splitting function is Pgg >> Pqq . So, let us examine the unintegrated gluon distribution: Purely gluonic DGLAP:

The first order (Pgg1/z):

The n-th order:

The double-logarithmic (DL) series. This resums ladders in which there is strong ordering of both the transverse and longitudinal momenta. 2.2 Summing the DL seriesModified Bessel function: ]exp[

)!()2/()(

02

2

0 znzzI z

n

n

If we set z = 2 ½, the result is

In a more general method, one has to solve the following equation:

Using the Mellin and inverse Mellin transformation, one finds

0lnln)( 22 AxgSaddle point method: with A= Q2/Q2

0.

assuming that lnA is large. Expanding the exponent around up to , and calculate the gaussian integral. The result is:

In the exponent:

This calculation is valid for the large 1/x andQ2/Q20.

2.3 BFKLHow shall we treat the case of Q2 ~ Q2

0 ?

lnlnlnexp Ax

2)(

)ln/1(ln)/1ln(2lnlnln AOAxAx ss

When Q2 ~ Q20, no longer the transverse momentum is ordered.

We have to reum all leading (single) logarithms (LL) of x BFKL eq.

with the collinear kernel ( collinear approximation )

It means that the scattering of a big object off a small one must be the same as that of the small one off the big one Symmetry between the collinear and anti-collinear reactions. Its Mellin tr. is then symmetric under χ is usually called as the characteristic function. 1/γ: collinear limit, 1/(1-γ): anti-collinear limit. These tell us how the BFKL characteristic function diverges.

Including the detailed structure of χ, the BFKL kernel is chosen as:

The last term describes the structure of the range for Q2 ~ k2. Then, the characteristic function is ψ(1) = -γ = -0.577 … saddle point

The inverse Mellin trans. gives

The exponent corresponds to in eq.(15), χ 1/γ. The saddle point is about γ = ½, which gives

The exponent is

vanishesχ(1/2) = 2(ψ(1) – ψ(1/2)) = 4ln2, if α = 0.2, the power is 0.5 !! too large !!Exp. L3, OPAL coll.: γ* + γ* collision gives the power is about 0.29.

...)5.0)(/1ln()5.0()()/ln(5.0)/1ln()5.0( 220

2 xtermQQx ss

3. Next-to-leading correctionsNLL corrections to the BFKL:

The determination of χ1 took about 10 years !!!Here we deduce the structure of the characteristic function in the NLL order.main, three corrections: Running coupling, Splitting functions, Energy scale

3.1 Running couplingFor the case of Q2 << k2, we takeand

This is again symmetric under the exchange of Q2 k2.The first term (MT)

The second term (MT & expand up to α2)

to get the common factor α in front of χ. anti-collinear part ( lack of symmetry)

3.2 Splitting functionAt NLL, we need to include the full splitting function. Its MT is , ( expand in terms of ω)

small-x branch nonsmall-x branch collinear anti-collinear

)/ln()/1ln()/ln( 00 QQxQQs

3/1

3.3 Energy scaleA more subtle source of NLL correction: energy-scale termsAt leading order: s0 is arbitrary. Changing s0 = introducing a whole set of higher order terms.Natural choice is s0 =Q0Q. Then, the L term changes

(note: x = Q2/s, Q2 >> Q0

2, )the second term: collinear log > the power of αFrom RGE, collinear logs power of α !! the 2nd term should be canceled by the NL correction.

(MT) in χ1. Now start from in the Mellin space

3/1

2/

and then

This implies that when s0 = Q2, it is just enough to make a shift: (note: replace )

and expand the 2nd term gives the term ln3Q2. Finally, we need

which includes not only the collinear but also anti-collinear terms.

3.4 Putting things togetherPutting together, we find in the collinear approximation.

The true NLL correction in the MS-bar scheme: A1 piece in Pgg

Energy-scale C.C.

terms free to double and triple polesThe collinear approximation works very well, within 7%.

3.5 Consequence of the NLL correctionsThe result is: the power is – 0.16 !!! no more in agreement with the data than the leading power !!!

saddle points

4. Beyond NLLLL took a year, NL took about ten years, NNLL will take 20-100 years.The only option left is to try and guess the higher-order terms. “a method based on the collinear approximation”For γ=1/2, one might NOT expect a collinear approximation to work too well, but at higher orders, it becomes better. So, a guess: collinearly-enhanced contributions give a significant part of the higher-order corrections even beyond the NLL: (1) 1/γ2 terms (single logs) from the splitting function and the running c.c. (2) 1/γ3 terms (double logs) from the energy-scale

4.1 Single collinear logsStraightforward to calculate the collinear NnLL corrections --- Ref.[13]Collinear poles:

Anti-collinear poles: expansion of α

4.2 Double collinear logsDIS for Q2 >> k2, anti-DIS for Q2 << k2: a change of energy-scale a shift of γLeading-order: Changing the scale to Q2:

4.3 The full resummed answerThe modified LL characteristic function is free from the unwanted double collinear logs, which must be subtracted from χ1.Also consider the shift due to the energy-scale change.

A point to note: it is no longer an expansion in α, but rather in ω the ω-expansion technique

4.4 ResultsLL, NLL BFKL exponents: 0.5, -0.16 at α=0.2.ωs = min. of 0.27 !ωc = position of the singularity of the gluon anomalous dimension. power growth of small-x splitting func.It reflects dif. between various processes.

Present uncertainty: about 15%.

),( s

5. Conclusions and outlook

We have seen how to deduce many of the properties of the BKFL pomeron in terms of the collinear and anti-collinear limit.

The resulting resummed power in the collinear approximation is much more compatible with the data than either the LL or NLL values.

For actual phenomenology, two more are required:(1)Understand the exponentiation of the χ function. (2)Need to know the virtual photon impact factors – the coupling of a virtual photon to the gluon-chain.

Despite initial fears, the large size of the NLL corrections is not an impediment to the use of BFKL resummention for predicting high-energy phenomena, but one needs to know the origin of the large corrections.