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

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

Introduction

BFKL 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 large

Solution (?) 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 problem

Collisions of two perturbative hadronic objects:

s >> Q2, Q20 >> Λ2

DIS at HERA, high-energy γγscatt. at LEP, LHC etc.

2. Leading-logarithmic order

2.1 DIS

Assume that one of the two hadronic objects is much smaller than the other: Mp2= 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 series

Modified Bessel function:

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

Saddle point method: with A= Q2/Q20.

- 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 BFKL

How shall we treat the case of Q2 ~ Q20?

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 thedetailed 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

vanishes

χ(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.

3. Next-to-leading corrections

NLL 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 coupling

For the case of Q2 << k2, we take

and

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 function

At 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

3.3 Energy scale

A more subtle source of NLL correction: energy-scale terms

At 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 >> Q02, )

the second term: collinear log > the power of α

From 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

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 together

Putting 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 poles

The collinear approximation

works very well, within 7%.

3.5 Consequence of the NLL corrections

The result is: the power is – 0.16 !!!

no more in agreement with the data than the leading power !!!

saddle points

4. Beyond NLL

LL 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 logs

Straightforward to calculate the collinear NnLL corrections --- Ref.[13]

Collinear poles:

Anti-collinear poles:

expansion of α

4.2 Double collinear logs

DIS for Q2 >> k2, anti-DIS for Q2 << k2:

a change of energy-scale a shift of γ

Leading-order:

Changing the scale to Q2:

4.3 The full resummed answer

The 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 Results

LL, 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%.

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:

- Understand the exponentiation of the χ function.
- 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.