Transcript Two methods of solving QCD evolution equation

Two methods of solving QCD
evolution equation
Aleksander Kusina,
Magdalena Sławińska
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 gluons
From 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 P represents 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
We generate values of momenta
and ”time” according to proper
probability distribution for each
x0
point in the diagram.
(x0, t0 )->(x1, t1)->...->(xn-1, tn-1 )
x1
We obtain an evolution of
a single gluon.
xn-1
Each dot represents a single
gluon emission.
Repeating the process many
times we obtain a distribution of
the momentum x.
xn
t0
t1
...
tn-1
tmax tn
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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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Markovianisation of the equation
to obtain the markovian form of the iterative equation we define
transition probability:
Which is properly normalized to unity
Applying this probability to the iterative solution we obtain the
markovian form:
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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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MC algorithm
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
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Results
Starting with delta – distribution, now we demonstrate, how the
gluon momenta distribution changes during evolution
t=2
t=5
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t=10
t=15
t=25
t=50
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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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s=1
s=0.3
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Semi- analytical Method
The model
Problems:
How to interpret probability P(z) ?
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Discrete calculations
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Solutions:
Many particles in the system  their distribution according to P(z)
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distribution
Calculations performed on a grid
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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
where
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
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s=0.3
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Comparison of the methods
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.
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T=4
T = 10
T = 18
As we can see from division of final distribution functions,20
both methods give the same distribution within 2%!
References:
[1] R. Ellis, W. Stirling and B. Webber,
QCD and Collider Physics (Cambridge
University Press, 1996)
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