Update on fits for 25/3/08

AM Cooper-Sarkar

•Central fit: choice of parametrization

•Central fit: choice of error treatment

•Quality of fit to data

•PDFs plus experimental errors, compare to older fits

•Model uncertainties

•Model Variations

Possible forms of PDF parametrization: H1 style

Published H1 PDF2000

New H1-style parametrization optimised for combined data set

Optimization means starting without D,E,F parameters and adding parameters until there is no further χ2 advantage

PDFs: gluon, U=u+c, Ubar=ubar+cbar, D=d+s+b, Dbar=dbar+sbar+bbar

Sea flavour break-up at Q0: s = fs*D, c=fc*U AU=(1-fs)/(1-fc)AD

Possible forms of PDF parametrization: ZEUS style

ZEUS-JETS published

New ZEUS-JETS optimized for combined data set

E

PDFs: gluon, uv, dv, Sea= usea+ubar+dsea+dbar+s+sbar+c+cbar

Sea flavour break-up at Q0: sbar = (dbar+ubar)/4, charm dynamically generated,

dbar-ubar fixed to fit E866 data

Possible forms of PDF parametrization: ‘inbetween’

Original ‘inbetween’ parametrization

‘inbetween’ parametrization optimized to combined data set

PDFs: gluon, uv, dv, Ubar=ubar+cbar, Dbar=dbar+sbar+bbar

Sea flavour break-up at Q0: s = fs*D, c=fc*U AUbar=(1-fs)/(1-fc)ADbar

Can also dynamically generate charm, but need to keep AUbar to ADbar relationship as here

Lim x→0 u/d →1

Lim x→0 u/d →1

We have chosen the new ‘inbetween’ parametrization to be the form of the new h1/zeus combined PDF fit.

For each of the old parametrizations, which have non-zero D parameter for the gluon, there are two minima: ‘straight’ gluon and ‘humpy’ gluon solution.

Which look rather like the published ZEUS and H1 gluons respectively!

For the H1/ZEUS combined data set the χ2 of the straight solution is always lower by about 10 χ2 points. For the H1 data set alone the ‘humpy’ gluon had the lower χ2, whereas for the ZEUS data set alone the ‘straight’ gluon had the lower χ2This χ2 table compares humpy and straight gluon solutions for the old parametrizations and the straight solution for the new parametrizations: using a χ2 definition where all 47 systematic errors of the combined data set are added in quadrature.

433

Table from J.Feltesse excellent agreement on all results in this talk

Now compare methods of treating systematic errors: Quadratic/Hessian/Offset (using published ZEUS-JETS parametrization)

47 systematic errors added to statistical quadratically

47 systematic errors treated by Hessian method

43 original sources of systematic errors added to statistical quadratically and 4 procedural errors Offset

Central values very similar (not necessarily obvious for full Hessian treatment)

Errors generally largest for OFFSET procedural, but not much difference in treatments since systematic errors not so big now- chose this method.

Theoretical framework

NLO DGLAP evolution

Facttorisation and renormalisation scales Q2

Zero-mass variable flavour number heavy quark scheme

Model assumptions

Which will be varied to assess model uncertainty

Q02 = 4 GeV2 input scale

Q2min = 3.5 GeV2 minimum Q2 of input data

fs = 0.33D strange sea fraction, means s=0.5d

fc = 0.15U charm sea fraction, means c=0.176u

mc=1.4 mass of charm quark

mb=4.75 mass of beauty quark

αs(Mz) = 0.1176 PDG2006 value

Form of the parametrization: ‘in between’

PDF fit RESULTS

New H1/ZEUS combined PDF fit vs combined data- see comparison to ZEUS-JETS 2005 in EXTRAS

New H1/ZEUS combined PDFs with experimental uncertainty bands from statistical errors plus 43 sources of systematics in quadrature plus 4 procedural systematics OFFSET

First consider experimental uncertainties

Compare the experimental error bands of the new analysis using combined data to the experimental error bands of the ZEUS-JETS fit –obviously we are winning!

Compare to published ZEUS-JETS PDFs

Compare to published H1PDF2000 fit

New H1/ZEUS combined PDFs with total experimental uncertainty bands plus model uncertainty bands from 6 sources of model variation:

mc, mb, fs, fc, Q0, Q2min.

Now consider model uncertainties: Proposal for MAIN results

Variation with Q2

New H1/ZEUS combined PDFs with total experimental uncertainty bands plus model uncertainty bands from 6 sources of model variation:

AT THE STARTING SCALE Q20 = 4 GeV2

New H1/ZEUS combined PDFs with total experimental uncertainty bands plus model uncertainty bands from 6 sources of model variation:

AT Q20 = 100 GeV2 . Note how uncertainties are decreasing

New H1/ZEUS combined PDFs with total experimental uncertainty bands plus model uncertainty bands from 6 sources of model variation:

AT Q20 = 10000 GeV2 . At scales relevant to LHC physics uncertainties

are impressively small.

Examine model uncertainties

Include only variation of charm mass and beauty mass

mc:1.35→1.5 GeV and mb: 4.3 →5.0 GeV

This model dependence is invisible

Include only variation of strange fraction fs: 0.25 →0.40

Makes very little difference in the total Sea, but affects sea flavour break-up

Include only variation of charm fraction fc: 0.10 →0.20

A smaller effect than fs: makes very little difference in the total Sea, but affects sea flavour break-up

Include only variation of starting scale Q02: 2 → 6 GeV2

Has a small asymmetric effect on PDF ucertainties

Include only variation of minimum Q2 of fitted data Qmin2: 2.5 → 5 GeV2

Has an even smaller asymmetric effect on PDF ucertainties

Other possible model uncertainties

And we could have added in Include only variation of αs (Mz): 0.1156 → 0.1196 (PDG2006)

Looking at this uncertainty alone it affects only the gluon PDF

And if we add in the variation of αs (Mz) to the other 6 model uncertainties.

I chose not to add this in since MRST(MSTW) and CTEQ both use fixed alphas and I want the public to compare ‘like with like’.

There is another model uncertainty that we could add in- the variation if one uses different forms of parametrization: H1 style newly optimized paremetrization and ZEUS-JETs style newly optimized parametrization.

Looking at this uncertainty alone there is a small asymmetric effect, most visible in valence distributions.

Illustration of the difference between using 6 or 7 sources of model uncertainty: On the right the 7th uncertainty from form of parametrization: h1 vs zeus-jet is added in. There is excellent agreement with J Feltesse.

I chose not to add this in because its not the same kind of variation as the others- varying our new parametrization to two others is not an ‘up and down’ variation!

We could illustrate some of these other choices as

variations rather than adding them into model uncertainty.

Comparison of central fit plus total experimental errors to variations with αs(Mz)=0.1156 (left) and 0.1196 (right)

Confirms what we’ve already seen-variation is outside the gluon error bands even when other model dependence is accounted.

Comparison of central fit plus total experimental errors to parametrization variation using: New H1 optimised parametrization

Marginally outside normal error bands for valence even when other model dependence is accounted (but note this is at low x where valence isn’t very significant)

Comparison of central fit plus total experimental errors to parametrization variation using: New ZEUS-JETS optimised parametrization

Inside error bands if other model dependence is accounted

Comparison of central fit plus total experimental errors to parametrization variation using: the ‘humpy’solution for the h1/zeus parametrization.

At Q2=10 (left) and Q2=4 (right)GeV2 where the humpy structure is more visible.

Marginally outside normal error bands for valence more than for gluon even when other model dependence is accounted (but note this is at low x where valence isn’t very significant)

Finally we may be criticized for the use of a zero-mass scheme.

Comparison of central fit plus total experimental errors to variation of heavy quark scheme: using massive variable flavour number scheme of Thorne.

This cannot yet be checked by both H1 and ZEUS

Time to decide what we want to show the world

EXTRAS

Here’s how the new H1Z PDF fit fits the data e-CC (left) e+CC (right) but with ZJ2005 also superimposed H1Z fit is slightly lower.

Here’s how H1Z fits the data e-NC (left) e+NC (right) with ZJ2005 also superimposed – no I cant see it either!

ZEUS-Jets Parametrization in between H1 parametrization

xubar

xdbar

xcbar

xsbar

Now in terms of ubar, dbar, sbar, cbar

The similarity of these is perhaps even more remarkable given the different treatment of charm- clearly the fixed fraction fc=0.15 is about right compared to dynamical turn on at Q2=mc2

Model dependence: just fs and fc variations together

Model dependence: mc.mb,fs.fc.Q0

Illustration of the difference between using 6 or 7 sources of model uncertainty:

mc,mb, fs, fc, Q0, Q2min (and parametrization form h1 vs zeus-jets).

Excellent agreement with J Feltesse.

Comparison of central fit plus total experimental errors to variations with alphas=0.1156 (left) and 0.1196 (right)

Comparison of central fit plus total experimental errors to parametrization variation using: the ‘humpy’solution for the h1/zeus parametrization

FeltesseCooper-Sarkar

Chi2 old = 437.9

Chi2 new = 428.0

Look at parameters, agree that gluon distribution must be flatter at low-x

Bg= -0.03 ± 0.02

Cg= 7.8 ± 0.7

Buv= 0.69 ± 0.07

Cuv= 5.1 ± 0.2

Duv= 12.1 ± 2.3

Euv= -0.66 ±0.91

Cdv= 3.8 ± 0.4

BUbar = -0.206 ± 0.005

CUbar = 5.1 ± 0.9

ADbar = 0.159 ± 0.006

CDbar = 4.0 ± 1.1

Chi2 are for all 47 systematic sources added in quadrature