two methods of solving qcd evolution equation
DESCRIPTION
Aleksander Kusina, Magdalena S ł awi ń ska. Two methods of solving QCD evolution equation. Multiple gluon emission from a parton participating in a hard scattering process. The parton with hadron’s momentum fraction x 0 emits gluons. After each emission its momentum decreases: - PowerPoint PPT PresentationTRANSCRIPT
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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-1tn
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 Method
Iterative 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 EXIT
5.Repeat the procedure: GO TO POINT 1
MC algorithm
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Results
Starting 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:
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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
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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)