Two methods of solving QCD evolution equation

Aleksander Kusina,Magdalena Sławińska

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Multiple gluon emission from a parton participating in a hard scattering process.

The parton with hadron’s momentum fraction x0 emits gluons.After each emission its momentum decreases:

x0 >x1 > ... > xn-1 > xn

The evolution is described by momentum distribution function of partons D(x, t). t denotes a scale of a process. t = lnQ

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Evolution presented in a (t, x) diagram.

The change of momentum distribution function D(x, t) is presented on a diagram by lines incoming and outgoing from a (t, x) cell.

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Evolution equation for gluonsFrom many possible processes we consider only those involving one type of partons (gluons). The evolution equation is then one-dimentional:

where z denotes gluon fractional momenta kernel P(z, t) stands for branching probability density

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We use regularised kernel:

where Prepresents outflow of momentum and P – inflow of momentum. We discuss simplified case of stationary P.Proper normalisation of D, namely:

requires:

leading to:

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Monte Carlo Method

t0 t1 ... tn-1 tn

x0

x1

xn-1

xn

tmax

We generate values of momenta and ”time” according to proper probability distribution for each point in the diagram.(x0, t0 )->(x1, t1)->...->(xn-1, tn-1 )We obtain an evolution of a single gluon.

Each dot represents a single gluon emission.

Repeating the process many times we obtain a distribution of the momentum x.

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Monte Carlo MethodIterative solution

We introduce the following formfactor:

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By using substitution

we transform the evolution equation to the integral form:

and obtain the iterative solution:

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to obtain the markovian form of the iterative equation we define transition probability:

Which is properly normalized to unityApplying this probability to the iterative solution we obtain the markovian form:

Markovianisation of the equation

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Now we introduce the exact form of the kernel so that we can explicitly write the probability of markovian steps

The transmission probability factorizes into two parts

where

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Once more the final form of the evolution equation

The Monte Carlo algorithm:1.Generate pairs (ti, zi) from distributions p(t) and p(z)2.Calculate Ti = t1 + t2 + ... + ti, xi = z1z2 ... zi

3.In each step check if Ti > tmax (tmax – evolution time)4.If Ti > tmax , take the pair (Ti -1, xi -1) as a point of distribution

function D(x, tmax ) and EXIT5.Repeat the procedure: GO TO POINT 1

MC algorithm

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ResultsStarting with delta – distribution, now we demonstrate, how thegluon momenta distribution changes during evolution

t=2 t=5

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t=10

t=15

t=50

t=25

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From the histograms we see the character of the evolution – momenta of gluons are softening and the distribution resembles delta function at x=0.

Now we investigate how the evolution depends on coupling constant s:

15s=0.3s=1

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Semi- analytical MethodThe model

Problems:●How to interpret probability P(z) ?●Discrete calculations

Solutions:●Many particles in the system their distribution according to P(z)

distribution●Calculations performed on a grid

● evolution steps of size t● momenta fractions N bins of width x● kth bin represents momentum fraction (k + ½) x

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Since time steps and fractional momenta are descreet, so must be the equation

The interpretation of P(z) within this model:

In each evolution step particles move

- from k to k-1, k-2, ... , 0

- from N-1, N – 2, ... , k + 1 to k

where

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s=0.3

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This is to emphasise that both calculation methods and computational algorithms differ very much.

In MC the history of a single particle is generated according to probability distributions and its final momentum is remembered. These operations are repeated for 108 events (histories) so that a full momenta distribution is obtained. In semi- analytical approach, a momenta distribution function is calculated by considering all 104 emiter particles. At each scale a number of particles changing position from (t, i) to (t+1, k) is calculated. All particles are then redistributed and a new momenta distribution is obtained.

To compare the methods we divided corresponding histograms.

Comparison of the methods

20As we can see from division of final distribution

functions, both methods give the same distribution within 2%!

T = 4 T = 10

T = 18

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References:

[1] R. Ellis, W. Stirling and B. Webber, QCD and Collider Physics (Cambridge University Press, 1996)