An Introduction to leading and next-to-leading BFKL Act. 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
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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 ?
Collisions of two perturbative hadronic objects:
s >> Q2, Q20 >> Λ2
DIS at HERA, high-energy γγscatt. at LEP, LHC etc.
2. Leading-logarithmic order
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.
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.
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
Its Mellin tr. is then
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 …
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
χ(1/2) = 2(ψ(1) – ψ(1/2)) = 4ln2, if α = 0.2, the power is 0.5 !!
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.
Running coupling, Splitting functions, Energy scale
3.1 Running coupling
For the case of Q2 << k2, we take
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)
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
This implies that when s0 = Q2, it is just enough to make a shift:
(note: replace )
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.
A1 piece in Pgg
terms free to double and triple poles
The collinear approximation
works very well, within 7%.
The result is: the power is – 0.16 !!!
no more in agreement with the data than the leading power !!!
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
Straightforward to calculate the collinear NnLL corrections --- Ref.
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
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
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.
It reflects dif.
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:
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.