Extracting the dipole cross section

from F2 data

Robin Marshall
aka
Joe the Plumber

01/22

Sunday, 7 December 2008 1

 The dipole picture of DIS:

F2 is given by an integral over the dipole-proton cross section

02/22

Sunday, 7 December 2008 2

 For completeness:

The kernel:

03/22

Sunday, 7 December 2008 3

 The task is to solve the inverse problem:

Measured Known To be
extracted

This is a Fredholm integral of the first kind.

Apart from e.g. Fourier types, most are difficult,
the rest are impossible.
04/22

Sunday, 7 December 2008 4

 The F2 <---> dipole problem

Is mathematically identical to the heat capacity problem:

I spent 3 years learning how to solve this for .

I can wheel in a set of tested and sharp tools.

05/22

Sunday, 7 December 2008 5

 Historical procedures, and even now still:

Try a model for the spectral function, integrate

forward and adjust the model parameters till a fit is
found.

e.g. Einstein and Debye models for heat capacity.

Einstein did not have a G5 with dual processor running

mathematica.

I wont spend much time on models here, but will

mention GBW, FS, IIM and Soyez in due course.

I will show you now how I back-transformed F2 to get

the dipole cross section, i.e. I map F2 back to
without using any model.
06/22

Sunday, 7 December 2008 6

 In general, there are 3 classes for the L.H.S. (F2)

A: F2 is a known function (not true for us)

B: F2 is a function that is known numerically.

C: F2 is a set of data with random errors

I will use B: solving numerically, an equation without random noise.

I will solve for , given the F2 behaviour.

07/22

Sunday, 7 December 2008 7

 For data, I used the neural net from Del Debbio et al
Penultimate layer of a NN is available as table of weights.
Fortran code to use it, which I “mathematicated”.
The NN was trained on all useable F2 data, not just H1 and ZEUS.
All the models I looked at, do not agree well with the NN because in
general, they used only one data set, often ZEUS, to fit to. Therefore, in
my opinion, based on this experience, all parameters in all models are
suspect.

My mathematica coding
of NN, compared to the
authors’ Fortran.
08/22

Sunday, 7 December 2008 8

 These are the data used by Del Debbio et al for the NN

09/22

Sunday, 7 December 2008 9

 The integral can be discretised into a matrix equation.
i.e.:

becomes

and this is where the trouble begins.

10/22

Sunday, 7 December 2008 10

 The problems in nut-shell

The matrix K is ill-conditioned.

There are a quasi infinite number of solutions,

nearly all unphysical, i.e. violent
oscillations: ±. 10^(enormous number)
Carry out an SVD of K, (powerful tool).

Compare Fourier coefficients with data (Picard

condition).

Truncate SVs or use something like Tikhonov

which is smoother, but essentially equivalent.

11/22

Sunday, 7 December 2008 11

 The solution to the problems in nut-shell

LS chi-squared doesn’t work

Naive regularisation

Add a priori knowledge

Extend to multi-dimensions
(My Weinkeller method)

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Sunday, 7 December 2008 12

 Possibilities for the a priori insertion

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Sunday, 7 December 2008 13

 Try rectangular and triangular a priori

The data demand a “dipole-ish” shape

The a priori shape does not “force” the outcome

Now choose a physically sensible a priori

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Sunday, 7 December 2008 14

 Possibilities for the a priori insertion

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Sunday, 7 December 2008 15

 Filters used:

is the identity matrix: it limits the norm size.

is the 2nd derivative matrix operator: it

controls smoothness.

and determine the balance between

residual norm, size norm and 2nd derivative norm.
I call these norms, r, s1 and s2.

and are determined rigorously by the

L-curve method.

15/22

Sunday, 7 December 2008 16

 The L-curve, the L-surface & the V-surface
The L-curve provides a rigorous
value for

optimal trade off point

Extend to multi-dimensions
(My Weinkeller method)
a new method obtained by
r
rotations in r, s1, s2 space.
16/22

Sunday, 7 December 2008 17

 Solution for x = 0.0004

This is the mapped solution, from F2 to . It is smooth,

stable and does not vary over a wide range of and 17/22

Sunday, 7 December 2008 18

 Integrating my solution to get F2:

This is a better solution than any of the models.

18/22

Sunday, 7 December 2008 19

 Look at small r region via log-log plot

Quasi-linear region indicates possibility of power law

But log-log hides a lot of detail . . . . . . . 19/22

Sunday, 7 December 2008 20

 Look at

Close to r2, but with possible deviations

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Sunday, 7 December 2008 21

 Compare with models GBW, FS04, IIM & Soyez

Within errors, agreement above 0.2 fm. Above 0.05 fm with Soyez.
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Sunday, 7 December 2008 22

 Next steps:
Repeat for a span of x, e.g. 0.00004. 0.004, 0.04 . . .
(or whatever is suggested by anyone with ideas).

Seek a way to “define” saturation from the data, e.g.

the point of inflexion on the rise edge of .
Define this to be and then

Draw conclusions, write up and publish.

Or else!

22/22

Sunday, 7 December 2008 23

 The End

Sunday, 7 December 2008 24