1 Emmanuelle Perez (CERN-PH) School on QCD, Low x physics, Saturation and Diffraction Calabria, Italy, July 1-14, 2007 Determination of parton distribution

$Page 1: 1 Emmanuelle Perez (CERN-PH) School on QCD, Low x physics, Saturation and Diffraction Calabria, Italy, July 1-14, 2007 Determination of parton distribution$
1

Emmanuelle Perez (CERN-PH)

School on QCD, Low x physics, Saturation and DiffractionCalabria, Italy, July 1-14, 2007

Determination of parton distribution functions,impact of HERA data & consequences for LHC

[email protected]

Plot shown by A. Martin yesterday.

In these lectures we see how we get there : - experimental data - QCD fit techniques

E. Perez Low x School, July 072

Lecture 1 : Deep-inelastic scattering and HERA

- Deep-inelastic scattering, formalism & basics- The HERA collider and experiments- DIS measurements at HERA

Lecture 2 : QCD fits and low x physics

- QCD fits : generalities fits using HERA data alone “global” fits- Back to the rise of F2 at low x

Lecture 3 : Looking at the future…

- Determination of uncertainties related to pdfs- Pdf uncertainties for LHC processes- A better knowledge of pdfs from LHC experiments ?- and beyond LHC…


• Universality of pdfs : the above formula holds for any process. i.e. measure pdfs from e.g. deep-inelastic scattering, apply them to predict cross-sections at the LHC.

Parton distribution functions• Parton distribution functions are the non-perturbative inputs necessary to calculate a process in a collision involving hadron(s).

• Ga/A (xa, F2) represents the probability to find a parton a in hadron A,

carrying a fraction xa of the hadron longitudinal momentum.

They cannot be calculated from first principles. Have to be measured.

• Beyond leading order, both the pdfs and the hard cross-section depend on

- the factorisation scale F (separates long & short distance parts of the

scattering process)

- the renormalisation scheme (usually MS , sometimes DIS ) and the

renormalisation scale R

F2) F

2) F2,

• is the cross-section of the partonic subprocess, calculable in perturbative

QCD at a given order in S.

^


Probing the nucleon…

nu

cle

us

nu

cle

on

qu

ark

NC

+C

C+

glu

on

Low

x,

diff

Scattering of a punctual (e.g. lepton) probe on a target : used for long tounderpin the target contents.

Deep inelastic scattering of alepton off a nucleon (nuclei)is the golden process to study theparton distribution functions.

Other processes also bringimportant constraints, as willbe seen later.

GE, GM(Q2) W1, W2() Fi(x,Q2)

Form factors, Bjorken scaling, structurefunctions.


Deep-inelastic scattering

q = k – k’, Q2 = -q2 squared momentum transfer

S = (p + k)2 square of center of mass energy

).(2

2

qPx

QBjorken

).(

).(

kP

qPy “inelasticity variable”

In nucleon rest-frame,y = (E – E’ ) / E

Q2 = x y S

W2 = squared mass of the hadronic system = ( P + q )2x

xQW 122

Partonic interpretation: xBjorken is the fraction of the nucleon longitudinal

momentum taken by the “struck” quark, in the frame of infinite momentum for

the nucleon (light cone variables : p+(quark) = x P+(N) )

V can be :- a or Z : Neutral Current DIS- a W : Charged Current DIS

eq

e

q

*

2

cos1 *y

PT2 = ( 1 – y ) Q2

PT

Cross-section depends on 2 variables, generally choose (x, Q2).


“Breit” frame :

P = (Ep, 0, 0, - Ep)

q = (E, 0, 0, qZ)

y

y

2cos

pin

Xein

eout

k

qP

PX

k ’

+z

Ep = qZ / 2x

A 2x P + q = 0

A02 = (2xP + q)2 = - Q2 + 4x(P.q) = Q2 (x = Q2 / 2(P.q) )

Hence Q2 = ( 2x Ep + E )2 = ( qZ + E )

2

Q2 = -q2 = qZ2 - E

2 = (qZ - E) (qZ + E)

q = (0, 0, 0, Q)

P = (Q, 0, 0, -Q) / (2x)

Lepton side : E(ein) = E(eout) since E = 0 same pT pZ(ein) = - pZ(eout) pZ(ein) = pZ(eout) + Q pZ(ein) = Q/2

(P.k) = (1/2x) (EeQ + Q2/2) = Sep/2 = Q2 / (2xy)

Ee = Q(2-y) / (2y)


JZ (boson) 0 +1 -1

Projection of

the spin of ein

on z axis

Projection of

the spin of eout

on z axis

2cos

2sin

2cos

2sin

2sin

2sin

2cos

2cos

ein

eout q

z

Example : amplitudes for right-handed leptons :

Projection of ein spin on z axis is

½ with d1/21/2, 1/2 = cos(/2)

and -½ with sin(/2).

• Parity violating term : (R + L) Jz=+1 – (R + L) Jz=-1 goes like :

• For transverse bosons :

• For longitudinal bosons :Contribution of

L vanishes

at y = 1 )1(2

2

2sin

2cos2 2

2

yy

x

)2(

)1(sincos 2

22

2

2

2

2 1cos1

41

22 yy

This gives the following terms in the differential cross-section :

)2(

cossincossin 2

444242 )2(

222cossin

22 yyy

avav


3

22 111211~ xFyFyFydL

P

T

[ (1-y)2 + 1 ] [ FT + FL ] - y2 FL FT + FL F2

Putting everything together : T : 1 + (1-y)2L : 2 (1-y) PV : y (2-y)

Contain the dependence on the nucleon structure

The sum of the first two terms can be rewritten as :

With Y = 1 (1-y)2 the Born-level cross-sections read :

L

NC FyxFYFYd

xQdxdQ2

324

2

2

2 2

WyxWYWYMQ

MGdxdQ

L

W

WFCC

x

d

2

32

2

22

22

2

2

4

0 FL F2

Effect of FL

large only atlarge y.


Partonic interpretation of Neutral Current DIS (I)

Cross-section for eq scattering : ( d / dy )q = 22/Q4 e2q [ 1 + (1-y)2 ]

qi(x) = probability to find a quark qi with momentum fraction x

d2 / dydx = (d / dy )qi [ qi(x) + qi(x) ]_

F2 = e2qi [ qi(x) + qi(x) ]

_

QCD improved parton model, qi(x) qi(x,Q2) : these two eq. hold at Leading Order.

At NLO : F2 and xF3 are given by convolutions of pdfs with coeff. functions.

- from Z interference :

5 coupling ~ ae aq

- from pure Z exchange : aeveaqvq

xF3 : given by the difference (e+ N) - (e- N) :

e+

q

-e-

q_

xF3 ~ ci [ qi(x) - qi(x) ] valence

_

In lepton-proton : ~ 4 (u + u) + (d + d)In lepton-neutron : ~ 4 (d + d) + (u + u )

_ __ _

un = dp ddn = up u In lepton-deuterium : ~ (u + d + u + d )

_ _

lp & ld allows to “separate” the flavors.


Partonic interpretation of Neutral Current DIS (II)

FL : back to the Breit frame :qin

qout

JZ,in = JZ,out

Quarks have spin 1/2Helicity conservation

must be transverse

FL = 0 at Leading Order

Non-zero FL at Next-to-Leading Order : when qg in the final state, or when the

incoming quark has a non-zero transverse momentum.

JZ0Happens when the interacting quark comes from a gluon splitting g qq. Hence :

- At NLO, FL is given by a convolution involving the “quark singlet”

(q + q) (contribution from q qg) and the gluon density (g qq).

- FL ~ gluon density (away from the “valence” region).

- FL sensitive to kT of quarks produced in g qq.

_


Partonic interpretation of Charged Current DIS (charged incident lepton, lp)

We+

R

dL

R

_

uL

uR

_dR

_

R

_

W

e+R

W couples to left-handed fermions

and right-handed antifermions.

e+R

e+R

qR

qR

e+R

e+R

qL

qL

J = 1

( d11,1 )

2 ~ ( 1 + cos *)2 ~ ( 1 – y)2

J = 0

( d00,0 )

2 ~ 1

Hence the e+p cross-section goes as (1 –y)2 xD + xU

_

the e-p cross-section goes as (1 –y)2 xD + xU

_

Comparing with eq. on slide 9 gives :W2+ = x ( U + D )

W2- = x ( U + D )

_

_ xW3+ = x ( D – U )

xW3- = x ( U – D )

__

with U = u + c, D = d + s ( + b)

CC DIS brings important information to separate up / down quarks.

CC(e-p) >> CC(e+p)


Neutrino Deep-inelastic scattering

dL, uR

_uL, dR

_

LW

-L

uL, dR

_dL, uR

_

RW

+R

_

p : (1-y)2 U + D_

p : (1-y)2 U + D

__

Weak cross-sections experiments use a heavy target, e.g. Pb (CCFR), Fe (NuTeV),which are nearly isoscalar (i.e. #n ~ #p)

On proton :

On neutron :

un = dp ddn = up u

n : (1-y)2 D + U_

n : (1-y)2 D + U

__

Hence : W2 = x ( U + D + U + D )_ _

xW3 = x ( U - U + D - D )

_ _

The data need to be corrected : - for nuclear effects ( q in heavy nuclei differs from q in proton) - to isoscalar target (e.g. 5.7% excess of n over p in Fe)Corrections are not trivial...

( + )_

( - )

_


xF3: 1< Q2 < 200 GeV2

FL

F2: 1< Q2 < 200 GeV2

approximateBjorkenscaling

scaling violations

DIS Fixed target results

Q2 > 5 GeV2 :data down to

10-2 only

Measurementsonly athigh x

Via N - N

_


What happens towards low x ??

What happens at high Q2 ??

Large spreadin the theoreticalpredictions.

Pre-HERA status…


The HERA electron-proton collider

H1

ZEUS

DESY, Hamburg

Two colliding experiments : H1 and ZEUS

HERA I (1992-2000) :

~ 130 pb-1

- mainly e+ p data

(only 20 pb-1 of e- p) - several analyses getting finalised on HERA I data

HERA II (2003 – 30/06/07) :

- ~ 500 pb-1 of data, ~ equally shared between

e+ and e-

- polarised leptons, P typically 30 -40 %

Equivalent to fixed target exp. with 50 TeV e±


Complete 4π detector

Tracking: - central jet chamber - z drift chambers - forward track. detector - Silicon μ-Vtx (operate in a B field of 1.2 T)

Calorimeters: - Liquid Argon cal. (em : 10%/E had: 50%/E) - Lead-Fiber cal. (SPACAL) (7% / E)

Muon chambers

Very forward detectors (e.g.“roman pots”) for diffractive physics.

p

e

Asymmetric detector : reflects thebeam energies asymmetry.

The H1 detector


Complete 4π detector

Tracking: - central tracking detector - Silicon μ-Vtx (operate in a B field of 1.43 T)

Calorimeters: - uranium-scintillator (CAL) σ(E)/E=0.18/√E [emc] σ(E)/E=0.35/√E [had] - instrumented-iron (BAC)

Muon chambers

The ZEUS detector


Up to very high

Q2 ~ 105 GeV2

Q2 = xyS with

S = (320)2 ~ 1.2 105 GeV2

Down to very low x, ~ 10-6 !

For Q2 above ~ 1 GeV2

(perturbative regime),

x down to ~ 10-5

Very low Q2 accessible,allow to study thepert – non-pert transitionregion.

Q2 from 0.1 to 105 GeV2

x from 10-6 to ~ 0.8

Huge extension ofthe kin. domaincompared tofixed targetexpts.

HERA kinematic domain


• Charged Current DIS : measure only the hadronic final state.

Ee

hZh

h

pEy

20

,

y

PQh

hT

h

1

2

,2 yQ

Q

hh

h

11

~2

2 Bad resolution

at high yh

• Neutral Current DIS : HERA experiments measure both the scattered lepton and the hadronic final state kinematics is over-constrained.

- From electron only : Q2e = 2 E0

e E’e ( 1+cos e) ye = 1 – (E’e/E0e) sin2(e/2)

Drawbacks : - resolution in y goes as 1 / ye i.e. bad at low ye

- in case of initial state radiation, E0e Ebeam- Combine e and had : replace 2E0

e in (1) by 2E0e = (E-pZ)h + (E-pZ)e and use

p2

T,e = Q2(1-y) to get Q2

Kinematic Reconstruction

- Kinematics can also be reconstructed only from the measured angles of e and of the had. final state (HFS).

Used for the calibration ofthe el. energy.

B. Heinemann, PhD Thesis

(1)


Kinematics (I)

electron Hadronic final state

“kinematic peak” :

Ee = Ee0,

x = Ee0 / Ep

H1 data,Q2 < 150 GeV2

Q2 below ~ 150 GeV2 : e “backward”, Ee limited

Higher Ee when e is “central”. Highest Q2 for < 90o.

backward

e pHFS goes morecentral as yincreases.

High x & y CCHigh x & y NC


Angular acceptance limits themeasurements down to

Q2 ~ 1.5 – 2 GeV2.

Going lower in Q2 requires specialtechniques or dedicated apparatus.

Kinematics (II) – focus on low x and low Q2

~ 180o : y = 1 – E’ / E i.e. high y low E’

Going towards highest y (for max. sensitivity to FL ) requires :

- dedicated triggers, with an energy threshold down to 2-3 GeV - a good understanding of experimental backgrounds (easy to “fake” a low E el.)


The first HERA results (1993)

Strong rise of F2 towards low x !

Q2 = 15 GeV2

HERA DIS measurements


Why this was somehow a surprise

Extrapolation from pre-HERA data indicated a “flattish” F2 at low x – that’s

also what came out from Regge-like arguments.

However, it was known (QCD) that the gluon should rise at low x, for Q2 high

enough. What was not known is “where” (in x and Q2) the rise should start.dg(x,Q2) / dlnQ2 ~ x dy/y Pgg(x/y) g(x/y) with Pgg(z) ~ 1/z

Approximate solution : g ~ exp ( -K(Q2) ln(1/x) ) with K(Q2) ~ (ln Q2)

Starting from a ~ flat gluon at the “starting” scale, a rising gluon is obtainedat higher scales.

The “starting” scale, i.e. the scale down to which pQCD was taken to hold, was

believed to be ~ a few GeV2.

The evolution between Q02 and Q2 ~ 15-20 GeV2 is not long enough to

generate a rising gluon from a flattish distribution.

Such a rise could be obtained : - from a steep input gluon which could be expected due to large ln(1/x) terms (resummed in the BFKL evolution equation)

- from DGLAP and a flattish starting gluon, but at a much lower Q02.

DGLAP equation :


Compared with where we are now…

stat. accuracy < 1 %systematics ~ 3 %

96-97 data

(not the final word …)

~ 20 pb-1

No “gap” between HERA & fixed target data.


What is measured is

actually not F2, but

rather

F2 – y2 FL / Y+

(cf slide 9)

The reduction of thecross-section asy 1 (lowest x),due to the disappearance of thecontribution oflongitudinal photons,is observed indeed.

FL

(more on FL later…)

Also note that there is no “gap”between HERA & fixed target data.


With increasing luminosity,important statistics over the fullkinematics domain.

- Good agreement between H1 and ZEUS- and with fixed target measurements

- Strong scaling violations observed at low x – sign of a large gluon density ( g qq )

- Negative scaling violations at high x ( q qg, a high x quark splits into a gluon and a lower x quark)

Overlaid curves are the results ofQCD fits based on the DGLAP equations (see later).

Excellent agreement with DGLAP, over

5 orders in magnitude in Q2 and 4 ordersof magnitude in x.

Within DGLAP : via F2/lnQ2,

access to the gluon density.


Very precise reconstruction of event’scharacteristics and kinematic variables:

Count nr. of events in appropriately defined (x,Q2) bins

Extract reduced cross section

NLOMC

bgdatameas σ

N

NNσ

- ~ =

Uncertainties are systematics dominatedfor Q2 < 800 GeV2

Slide from E. Tassi,CTEQ 2003

Phase-space region: 6x10-5 < x < 0.65 2.7 < Q2 < 30000 GeV2

How did we get there…(example of ZEUS analysis)


More on high Q2 NC measurements Recall that due to Z exchange,

NC(e-p) NC(e+p).

The difference gives access to xF3.

High Q2 measurements done both

in e-p and in e+p : - Z exchange contributes for Q2

above a few 103 GeV2

- Z interference is constructive

in e-p, destructive in e+pStatistics is limited !

Neglect pure Z contribution(small) and correct for propagator terms bring all data to

Q2 = 1500 GeV2

Test the x dependence of valence quarks.

~ 2uv + dv


Pushing towards highest x…

Quark pdfs are not well known at highest x : - highest measured points at x=0.75 (BCDMS) (data at higher x exist but are in the resonance region, can not easily be interpreted in terms of pdfs)

- HERA, standard techniques: xmax=0.65

increasing x

Method : if no jet is found at > 7o, don’ttry to reconstruct x, but put the event ina bin [ xedge; 1 ] and measure the integrated .

ZEUS Collab., EPJ C49 (2007) 523.

No jet found

(subset ot meas.)

Should bring constraints on high x quarks,especially with the full HERA statistics.


Measurements of CC DISRecall (slide 12) :

CC(e+p) goes as (1 –y)2 xD + xU

_

CC(e-p) goes as (1 –y)2 xD + xU

_

As an example, our e+p measurements are shownon the plot.

However, for x ~ 0.1, precision of about 15% is reached

(but at highest Q2 where thestat. is low)

HERA I measurements arestatistically limited.

Brings constraints on “flavorseparation”, which are

missing from F2p (4U+D) alone.


CC(e,SM) ~ (1 Pe)

e- p

21 pb-1

30 pb-1

69 pb-1

27 pb-1

e+ p

HERA I

Extrapolations to Pe = 1 consistent with no WR

• e+ p : first publication on HERA II data (PLB 634 (2006) 173)

• e- p : prelim. measurements with the full available 05 stat.(syst.) typically 4%, (stat.) from 2% to 8%


The charm and beauty contents of the proton

- Exclusive measurements :

D* D0 slow K slow

and b X, exploiting PT,rel() and impact parameter

- Semi-inclusive measurements : distributions of the significance of track impact parameters are used to fit simultaneously the light q, c and b contributions to F2. Use silicon vertex devices around the interaction point. H1 Central Silicon Tracker

2 cylindrical layers, at radiiof ~ 5 cm and ~ 10 cm.Impact parameter resolution:

9033 [ ]

T

mm GeV

P

As F2, F2bb,cc shows large scaling violations at low x.

Note the difference between the MRST andCTEQ predictions.Data now included in the most recent CTEQ analysis.


2 2 2

2 2 2

( , )/

( , )

HQ HQHQ d d x Q

fdxdQ dxdQ x Q

Charm fraction fcc

- roughly constant with Q2

- around 24% on average

Beauty fraction fbb - increases rapidly with Q2 from ~0.3% to ~3% (i.e. by a factor of 10)


Extending to lowest Q2 Q2 = 2 E0e E’e ( 1 + cos e)

To go lower in Q2, needs to : - access larger angles

- or lower the incoming energy E0e

• dedicated apparatus, e.g. ZEUS Beam Bipe Tracker : silicon strip tracking detector & EM calorimeter very close to the Beam Pipe.

Was present at HERA I. Allows to cover the range 0.045 < Q2 < 0.65 GeV2

• shift the interaction vertex in the forward direction. Two short runs runs such a setting, with zvtx = 70 cm.

Shiftedvertex

Larger angles :

• QED Compton events1

2

Final electron (2) in

the acceptance,

1 larger.

Lower the incoming energy :Exploit initial state radiation events.

Q2 = xyS, S reducedi.e. access higher xat a given Q2.

e

e

p


Example of measurements at lowest Q2

• Good agreement with fixed target exps.• F2 continues to rise at low x, even at the

lowest Q2…

• as Q2 0, F2 ~ Q2 as required by the

conservation of the EM current.


End of first lecture…

• F2 (ep) measures 4 ( U + U ) + D + D.

Precise measurements from HERA over a huge kinematic range, in particular

extending towards very low x, and towards very high Q2.

Strong rise of F2 towards low x. Unitarity ?

Strong scaling violations at low x.

• Withing DGLAP, the measurements of the scaling violations give access to the gluon density.

• The measurement of Charged Current DIS brings information on the flavor separation. The other source of information (F2

p vs F2d, N vs N) requires

non trivial corrections to the data.

• Measuring xF3 brings information on the valence quark distributions – but

statistics is quite limited.Next time :

• Extracting pdfs from these measurements• and from these measurements plus those from other experiments• Take a closer look at the low x behavior of HERA measurements.

_ _

_


QCD fits to DIS data

• Parameterize a set of pdfs at a “starting scale” Q02

e.g. xg(x) = A x (1-x) P(x)

and a set of quark pdfs, e.g. uval, dval, TotalSea = q, d - u

_ _ _

- quite some freedom in choosing what to parameterize - quite some freedom in choosing the form of the parameterization

• Choose the “order of the fit”, LO or NLO. Usually fits are performed at NLO. NNLO is coming – but so far not many processes calculated to NNLO.

• For NLO : choose the renormalization scheme, MS or DIS. When pdfs are used to predict e.g. a pp cross-section, scheme should match ! Most NLO calculations done in MS most often, fits in MS.

Choose the scales used in the calculation, e.g. R = F = Q2.

• Choose how to deal with heavy flavors.

• Choose the datasets, and kinematic cuts to be applied to the data points.

e.g. cut away very low x (where DGLAP may break down), very low W2

(higher twist effects), very high x (would need a resum. of ln(1-x) )


• DGLAP equations give f(x,Q2) at any Q2, once f(x, Q02) is known.

Allows to calculate theo (DIS, DY, jet data,…) and fit theory to data.

• Starting scale : not too high, to keep as much data as possible (mainly DIS) not too low, to be in the perturbative domain.

Typical value Q02 ~ a few GeV2.

• and do assumptions to supplement the lack of sensitivity of the fitted data. e.g. - s – s not well constrained yet, often assumed that s – s = 0. - If only lepton-p data are fitted, no information on d – u, set to zero or to something consistent with other data.

_ __ _

• Usually impose number sum rules :

1

02)]()([ dxxuxu

1

01)]()([ dxxdxd

1

00)]()([ dxxsxs

And momentum sum rule : 1

01]))()(()([ dxxqxqxgx

- Helps fix the gluon normalisation- “connects” the low x and high x behaviors of g(x)

Id. for c, b


First, need to define the set of pdf’s combinations that will be evolved(one does not fit the 11 q, q and g distributions, since the data do not contain enough information).

The gluon distribution is always one of those.

Common examples :

g, uVal, dVal, Total Sea, d – u

Possibly add strange combinations if some of the included data are sensitive.

_ _

H1 in “H1Pdf2000” fit : choice as close as possible to what is actually measured :g, U = u + c, D = d + s ( + b) , U = u + c , D = d + s ( + b )

_ _ _ _ _ _ _

Singlet & non-singlet evolve differently need at least two quark distributions. Need more if interested in more than just the gluon density.

“Flavor decomposition” : choice of input combinations

_

F2 can be written from a “singlet” and a “non-singlet” distribution.

E.g. at LO, below the charm threshold :

F2 = 4/9 ( u + u ) + 1/9 ( d + d + s + s ) = 4/18 + 1/6 ( ud – s+)

With = (q + q ) (singlet), ud = ( u + u ) – (d + d ) and s+ = s + s - / 3

_ _

_ _ _ _

singlet Non-singlet_


Early days : parameterize xg(x) and xq(x) as ~ x (1-x). Some “predictions” :

• Low x behavior : suggested by Regge phenomenology

Optical theorem relates F2p to tot(* p) at W2 = Q2 (1-x)/x

Regge : tot(* p) ~ A Pomeron ( W2 )P(0) - 1 + A Reggeon ( W2 ) R

(0) - 1

Pomeron has no contribution to the non-singlet part

xqval ~ ( 1 / x) R(0) - 1 with R(0) = 0.5 i.e. xqval ~ x 0.5

Singlet part, i.e. gluon and sea quarks :

xg and xq ~ ( 1 / x) P(0) - 1 with P(0) = 1 + 1.08 i.e. xq, xg ~ “flatish”

_ _

• High x behavior : simple “dimension arguments”, =2ns-1 where ns is the number of “spectator quarks” :

- Valence quarks : (qqq) ns = 2, = 3- Gluons : (qqqg) ns = 3, = 5- Antiquarks : (qqq qq ) ns = 4, = 7

_

Choice of the parametrisation


• Current parameterizations :

e.g. xf(x) = A x (1-x) ( 1 + p0 x + p1 x + p2 x2..)

May use some more or less ad-hoc “2 saturation” procedure to decide whether or not one adds a new term.

But : - at which Q2 should these naïve predictions hold ??

- current data show that x (1-x) is too simple !

• Param : needs to be flexible to allow for a good fit. But should still avoid unstable fits, secondary minima…

• The “best” parameterization depends on the starting scale.

E.g. one finds that, for a starting scale of a few GeV2, a param.

xg(x) = A x (1-x) ( Polynom ) works fine.

For a lower starting scale, ~ 1 GeV2, better fits can be obtained by giving more flexibility to the gluon density, e.g. (MRST global fits)

xg(x) = A x (1-x) ( Polynom ) - x (1-x)Big

d – u = A x (1-x) assumes that d – u 0 as x 0. Correct ?

• Don’t forget that the parameterization includes assumptions, e.g. :_ __ _


• Zero-mass Variable Flavor Number scheme (ZM-VFNS) :

c = 0 until Q2 ~ 4 m2c. Above, charm is generated by gluon splitting and is

treated as massless.

drawback: ignores mc near the threshold…

• Fixed Flavor Number scheme (FFNS) : No pdf for c, b. Only 3 active flavors.

For W2 above the threshold (4m2c)

a cc pair can be produced by photon-gluon fusion.

Drawback : at large Q2, large logs, ln(Q2/m2c).

• General-mass Variable Flavor Number scheme (GM-VFNS) : the “state-of-the-art”. somehow interpolates between the two above approaches. Not easy to implement esp. at NNLO.

Heavy flavor treatment

Several “schemes” exist :

g

c

c_

_


Higher twists and target mass corrections

“Higher twists” operators : generally related tocorrelations between partons in the nucleon.They contribute as additional terms in

f(x)/Q2, g(x)/Q4,… to the structure functions,with f(x), g(x) large at high x only. HT are important at low Q2 and high x A cut W2 ( + Q2 (1-x)/x) > typically 10 GeV2 allows to be on the safe side. Important esp. for the data from fixed target experiments. and/or one can correct the data for these HT effects, using some parameterized expressions for f(x) – but no general consensus.

Target mass effects : take into account the finite mass of the nucleon (d or heavier).

P

q ( P + q)2 = 0 = -Q2 + m2N + 2 (x.P)

Qmxx

222 /1

2

41

And the measured F2 is related to the “massless”

one by : QFQmx

Q mxxF

20

2222

2

2,

1

1),(

/4

“higher twist” like,

kinematic origin

Effects largest at low Q2

and large x.


x

x

x

x

x

x

W2 > 10 GeV2

MRST(HT) : fit with

F2 = F2LT ( 1 + D(x) / Q2)

Differences observed betweenMRST and MRST(HT) at low

W2.

A cut W2 > about 10 GeV2

allows to stay away from regionwhere HT are important.Note that this cuts out manyof the SLAC data points…

Illustration of HT :


Goal : determine the gluon density and S.

Datasets : H1 low Q2, BCDMS p data.

“Massive scheme”, assume symmetric sea : usea = u = d = dsea, s + s = ( u + d ) /2

Need only 3 input densities, the gluon & two quark combinations.

_ _ _ _ _

Choice :

AVF9

11

3

12

valval duV2

3

4

9 Pure valence sum rule ( V dx = 3 )

)2(4

1valval uduA closest to what is measured at low x

• Constraints on the gluon density only from scaling violations.• Correlation between g and S, can be

alleviated only with other data.• Only ep data : no binding corrections needed.• More data needed to disentangle the quarks !

H1 Collab., EPJ C21 (2001) 33

]1[)1()( xexdxxaxxf ffcb

fff

Example 1 : QCD fit to low Q2 ep DIS data


(slide from M. Klein)

Model uncertainty :

Scale uncertainties : > 5% !

1.4% 0.7%


Low Q2 ep DIS data alone (~ 4U + D) give no information about the flavor separation. Would need data on deuterium, see fits presented later.

Still with ep only : adding high Q2 CC, e+p and e-p, brings some information: - Charged Current in e+ p : mainly probes the d density - Charged Current in e- p : mainly probes the u density

- Neutral Current e-p vs e+p : brings xF3 i.e. combination of uval and dval

H1 Collab., EPJ C30 (2003) 1

Parameterize : xg(x), xU = xu + xc, xD = xd + xs ( + xb), xU , xD by :_ _

)(.)1()( xPolynomxxAxxf ff CBf

Assumptions :

DD

UU

AA

AA

Such that the valence distributions vanishwhen x 0.

DDUU BBBB Because no distinction of the riseat low x between xU and xD.

( “2 saturation” procedure )

xc constant fraction of xU, xs of xD, at starting scale Q02=4 GeV2

Relation between AU and AD such that d / u 1 as x 0_ _ _ _

Results in 10 free parameters.

Low xassumptionsneeded.

Example 2 : adding high Q2 DIS data


Chi2 / ndf = 0.88for 621 data points(H1 only)

Good determinationwith ep dataalone !

Best precision for

xU (1.5% at x=10-3,6.5% at x=0.4).

xD mainly from CC e+

(1.6% at x=10-3,27% at x=0.4)

Q20 = 4 GeV2


Example 3 : fit (ZEUS) to DIS data ZEUS Collab., PRD 67 (2003) 012007

Two fits performed :

- to ZEUS DIS data only (low Q2 and high Q2, NC and CC) - to ZEUS NC DIS data and DIS data from fixed target experiments :

- p and d data (BCDMS, NMC, E665)- NMC data for F2

d / F2p constraints d / u

- CCFR data on xF3 constraints the high-x valence

Starting scale Q02 = 7 GeV2, Q2

min = 2.5 GeV2

Flavor decomposition : g, uval, dval, total sea = (q + q ), d - u _ _ _

)1()1()(41

32 xpxxpxxf pp

Assumptions :

• p2 = 0.5 for uval and dval (little information on low x valence – cf Regge…)

• d - u : the less constrained pdf with only DIS data…

fix p2 = 0.5, fix the high-x parameter a la MRST, i.e. fit normalisation only.

( Hence d = u is imposed by the param. as x 0, cf H1 assumptions)• xs = 20% of Total Sea (as suggested by dimuon data from CCFR)

_ _

_ _

11 parameter fit


Reasonable agreement with e.g. the H1 and the CTEQ fit…

Differences however thatare not embedded in theerror bands, esp. forthe valence distributions.

Sensitivity to those has adifferent origin in theH1 and ZEUS fits :• H1 : uses W & Z to do the flavor separation• ZEUS : this comes mainly from p vs. d and xF3 measured in fixed target experiments.

ZEUS data + fixed target

E. Perez 51

Comparison of ZEUS alone / ZEUS with fixed target

ZEUS onlyZEUS + fixed target

ZEUS + fixed target

ZEUS only

xdval

xuval

xdval

• the gluon and sea densities are mainly determined by the ZEUS data ( for x below ~ 0.1)

• valence distribution : adding fixed target data reduces the uncertainties by a factor of ~ 2.

uval remains well determined from ZEUS alone.

For dval : deuterium data more constraining than

ZEUS high Q2 CC data.


Gluon and sea : backward extrapolation from Q02 to low Q2

For low Q2 : the sea continues to rise at low x, while the gluon density is suppressed !!

Gluon density even becomes negative

at Q2 = 1 GeV2.

Same features observed in the fitto ZEUS data alone.

This gluon density results in a

negative FL at lowest Q2.

Sign that the approximations donein the QCD calculations are notvalid in this regime.

Non-convergence could be due to important terms in S ln(1/x).

Cured by NNLO ? Or a full resummation of these ln(1/x) is needed ?


Example 4 : fit to HERA DIS and jet data

The precision on the gluon density obtained from the ZEUSdata alone can be improved when jet data from DIS and photoproduction are added in the fit.

HERA jet data do not bring strong constraints on the gluon at very high x, x > ~ 0.5,(for that better use Tevatron jet data). But useful for medium x, since HERA jet cross-sections have small systematic uncertainties (typically 5%, compared to ~ 60% for the Tevatron jet data).


comparison of gluon distribution from fits with and without jets :

no significant change in shape: no tension between jet and inclusive data (QCD factorisation)

HERA jet cross sections constrain gluon in range x = 0.01 – 0.4

reduction in gluon uncertainties by factor of ~2 in mid-x region

over the full range of Q2


Global pdf fits

Global fits performed mainly by the Durham and the CTEQ groups.

Non-inclusive DIS data that are usually included :• Tevatron jet cross-sections high x gluon• Drell-Yan measurements pN large x sea, d – u• Dimuon production in N and N ( s c X) s and s• asymmetry of W production at Tevatron d/u at medium x

_ __ _

Some data used to be included in global fits,as prompt photon production which inprinciple brings constraints on the gluondensity – but hampered by too largetheoretical uncertainties.

Recent fits also include HERA jet data and

F2b & F2

c measurements.

Typically this leads to ~ 2000 points in thefits, with a large number of systematicerror sources (see later…)

A. Martin


Follow-up by E866 (Fermilab) : fixed target, DY in pp and pd, Ebeam = 800 GeV.

)()(9

1)()(

9

42121xdxdxuxu

pp :

pn : )()(9

1)()(

9

42121xuxdxdxu

x1 = xBeam, x2 = xTarget

For x1 >> x2 one has :

)(

)(1

)()(

)()(

41

1

)()(

41

1

2

1~

22

2

2

2

1

1

1

1

xu

xd

xuxd

xuxd

xuxd

pp

pd

E866 measures this ratio down to < x2 > ~ 0.03.

Drell-Yan measurements constraints on d – u _ _

d = u was a “natural” assumption in global fits, until the NA51 experiment (CERN)reported that d > u at x = 0.18 (some hints before from NMC…)

_ _

u, d

u, d_ _

beam

Tx1 > x2

Note the spread of the predictions from pdfsbefore these data were included in the fits.

E866

_ _


What could explain a sea asymmetry ?

d – u > 0. _ _

E866

Several models… e.g. fluctuation of p in a meson + baryon pair :

(1)

(2) (1) is kinematically disfavored w.r.t. (2),hence the creation of uu pair is disfavored.

_

• Experimentally :

- from DY- NMC observed before a violation of the Gottfried sum rule : F2

p – F2n = 1/3 ( – 2/3 x( d – u ) )

_ _

• strange sea : s = 0.5 ( u + d ) ? _ _ _

Global fits tell this does not hold.- Mass effects- indications that s s. Could be explained by

_

carries most of the proton momentumhence s(x) > s (x) at “high” x.

_

Experimentally : NuTev, CCFR : W+ s c + X W- s c - X

_ _

Q2=56 GeV2


asymmetry in W production at the Tevatron

Tevatron : pp collider, s = 1.8 TeV u

d_

W+

l+

d

u_

l-

_W-x1,2 = (M2

W / S) exp ( W)

At central rapidity, x1 = x2 ~ 2 10-3

At ~ 2.5 : x1 = 2 10-2, x2 ~ 2 10-4

l+R uL

LdR

_

(1 – cos *)2 (1 + cos *)2

_

(W+) - (W-) ~

u(x1) d(x2) (1-cos)2 – d(x1)u(x2)(1+cos)2

At large : u(x1) > d(x2) hence W+ (W-)preferably emitted in the direction of theincoming proton (antiproton).

Asym. diluted when looking at (lepton) :

l-L

dLR

uR

_

Constraints on the d/u ratio.

p

p_

_


Jet production at the Tevatron

(slide from R. Thorne)


Comparison of global fits to data

But not allglobal fitsreproduce wellthe DIS dataat lowest x.

10-4 10-3 10-2

Q2 = 3.5 GeV2

F2

F2 - y2/Y+ FL

In general looks fine…

MRST fit

x

x

r

While HERAfits (to DISdata only) do.

e.g. MRST : Tevatronjet data require a quite high g(x) athigh x, resulting ina lower g(x) at low x,getting negative atQ2 ~ 2-3 GeV2.

Leads to FL < 0…


F2 rise towards low x well described by

the (ZEUS) QCD fit down to Q2 ~ 1 GeV2.And down to lowest x for such Q2.

i.e. the approximation :

dF2 / dlnQ2 ~ S(Q2) xg(x,Q2)

is not valid at all at NLO !

NB: cuts, and the treatment of system. errorsmay also explain part of the seen in various fits.

Note: the fit shown on the plot gives a gluon which goes negative at

Q2 ~ 1 GeV2. Also the case with theMRST gluon, or with the H1 fit when

extrapolated down to 1 GeV2.

The fact that • the lowest x region is described differently in “HERA fits” and in global fits • fits predict a negative FL at low x & Q2

might be a sign that the theo. formalism breaks down at lowest x.


Drastic approach: cut out the lowest x data in the fits…

Was tried by the Durham group (“MRST03 conservative pdfs”).

The cut x > xmin was made more and more severe, until fits are stable.

Stability was obtained for x > ~ 5. 10-3 !These fits do not describe the HERA data at lowest x.And give very different predictions from “standard” fits, for manyobservables at the Tevatron or the LHC.

Hence one needs to better understand the limitations of our calculationsat low x…

NLO DGLAP predictions at low x and Q2 could be wrong due to :

- Large terms in ln(1/x) look at NNLO, or at a resummation of these logs- unitarization (saturation) effects which tame the low x rise of F2, e.g. due to

gluon recombinations make the evolution equations non-linear.

To study this, one needs more observables than just F2. E.g :

- the longitudinal structure function FL

- the slopes of F2

- exclusive final states see lectures of C. Kiesling


The longitudinal structure function FL at low x

FL is more directly related to the gluon density than is F2.

Hence it is a good experimental observable to study the importance of theln(1/x) terms.

But no direct measure of FL at low x has been done yet !

Only indirect determinations so far : DGLAP fit to the data for y < ycut, i.e. cutting out the lowest x domain.

use the fit to predict F2 at lower x. The difference

between the measured cross-section and F2 is ~ FL.

This is more a “consistency check” of the overall framework.

… which fails e.g. for the MRST gluon…


A direct measurement requires measuring (x,Q2) ~ F2 – y2/Y+ FL for at least

two values of y = Q2/xS, i.e. at two different values of the center of mass energy.On March 21st, HERA started a “low energy run” with Ep = 460 GeV.

Was very successful ! More than 10 pb-1 collected in ~ 2 months.

On June 1st, moved to an intermediateenergy (575 GeV) for the last monthof data taking.

The plots show a simulation of what is

expected with 10 pb-1 at 460 GeV and

7 pb-1 at 575 GeV.


At NNLO, the gluon density becomeseven more negative.Compensated by positive terms inthe O(S

3) coefficient function

for FL FL at NNLO is positive.

FL data may tell us more about the

correct approach at low x :- NNLO enough ?- or need a full resummation of ln(1/x) terms ?

Which lead to in the gluon distribution from global fits :

R. Thorne, C. White, PRD 75 (2007) 034005


The slope of F2 and the low Q2 – high Q2 transition region

In the double asymptotic limit, DGLAP predicts that F2(x,Q2) is close to

x -(Q2) . A power-behavior is also predicted by the BFKL evolution, with ~ 0.3 – 0.5.

Extract

(x,Q2) ( F2 / lnx )Q2 .

A decrease of with decreasing x may sign a breakdown of the theory due to saturation effects.

No evidence for such adecrease in the data.


The previous plot shows that the data can be parameterized indeed by

F2(x,Q2) = c(Q2) x -(Q2)

For Q2 > 2-3 GeV2 : (Q2) depends logarithmically on Q2 and c ~ constant – as ~ expected from the DGLAP equations.

For Q2 < ~ 1 GeV2 : (Q2) deviates from a ln(Q2) behavior and tends to a value

close to Pom(0)-1 ~ 0.08

Observation of a “confinementtransition” between“partonic degrees of freedom”to “hadronic degrees of freedom”at a scale of about 0.3 fm.


The low Q2 – high Q2 transition in dipole models

Dipole models provide a nice description of this transition :

At low x, * qq and the long-liveddipole scatters from the proton

Original model was improvedby relating (x,r) to1/g(x,Q2).

Describes the slopes both

at low and high Q2.

Golec-Biernat, Wustoff

See lectures by F. Gelis.


GBW model predicts a “geometric scaling” property :

Which is borne out by the low x data indeed. Transition between

(*p) ~ 0 ( small) to (*p) ~ 0 / ( large) observed for ~ 1.

Not a proof of saturation… but shows that the low x HERA data have many of theattributes of a saturated system.

Slide from M. Cooper


End of second lecture…

We saw how QCD fits are performed. - That at low-medium x, the gluon density is largely driven by the HERA data. - That fixed target data provide a better sensitivity to the flavor separation than do the HERA CC data. - that fits make assumptions…

We saw that global fits generally describe well the data. But not so well thelow x & low Q2 HERA data.Many fits predict a negative gluon and, more problematic, a negative FL atLow x and Q2. This may be a sign that the DGLAP NLO calculations arenot sufficient, which is ~ expected because of : - large ln(1/x) - possible saturation effects

FL is a nice observable to study the effects of the ln(1/x) terms. Expect adirect measurement from HERA soon !

F2 slopes show no indication of a “tame” of the rise of F2 at low x. But thelow x data exhibit the “geometric scaling” behavior as predicted by color dipole models with saturation. Possible hint that the saturation has been observed at HERA – but low Q2…


Uncertainties on parton densities

Old days : take many pdf fits available on the market and compare them…

Not correct :

• no statistical interpretation of the spread (what is then the “1” error band on xg(x) or on a related quantity ?)

• the envelope is even not “representative” of the true uncertainty, since all fits share the same data – with their exp. errors.

A lot of work done over thepast ~ 5 years to assess rigorously the pdf uncertainties.

Note : the plot also shows that one shouldnot extrapolate a fit beyond the region wherethere are data… Before HERA, there was noinformation on xg(x) at low x !


The parameters of the QCD fit (i.e. which describe a given set of pdfs at a

given starting scale) are usually obtained by minimizing a 2 function.

Sum runs over data points di

i = error on di

ti(p) = theoretical prediction for parameters (p)2 / pk = 0 gives the parameters pk

Simplest : if statistical errors >> systematic errors :

Hessian matrix :

For a quantity F (a density, or a cross-section) which vary approximately linearly with p around the minimum, the standard formula for errorpropagation gives :

1 error on F corresponds to

2 = 1.


Experiments give a “normalization” uncertainty.

Comes usually from the uncertainty on the luminosity.When several datasets are fitted together, theory + dataset A (B) can constrain the normalization of dataset B (A) :

22 12

2)(

k

N

k

N

datasetski

fptdf

i

iiN

In addition to the theory parameters (p), the “scale factors” fNk are

also fitted.

Nk is the normalization uncertainty of dataset k.

ti = xi + b

dataset A

dataset B

fNA < 1 fN

B > 1


With the stat. uncertainty of the experimental measurements getting smallerand smaller, one needs to consider carefully all systematic errors.

“Generalization” of the previous formula :

With 2i = 2

i,stat + 2 i,uncorrelated syst.

The sources of correlated systematic uncertainties are labeled by k. They move“coherently” the data points.

ki = the amount of change of di, when source k (e.g. energy scale) is changed

by 1 (i.e. when sk = 1)

sk = new parameters, the “systematic shifts”.

They should be Gaussian distributed, G(0,1), if the errors are gaussian and well estimated.

Moving datapoints by ki sk causes a “penalty term” sk2 in the 2 function.

(1)


This definition of the previous 2 is equivalent to the “standard” one :

with

Proof : start from (1), minimize w.r.t. sk :

Inserting sk into the 2 expression (1) on previous page gives :

gives with

(2)

(1 prime)

CTEQ,Refs.

With one can write :(Npoints x Npoints)

(Nsyst x Nsyst)


i.e. Vij = 2i ( 1 + M )ij

A kk’ = ( 1 + N ) kk’

Putting this expression of ( V -1 )ij into (2), one gets indeed (1 prime).

-


CTEQ, MRST, H1 and ZEUS all adopt this definition of the chi2.

With 2i = 2

i,stat + 2 i,uncorrelated syst.

But use it differently :

- The “offset” method : set sk = 0 in the central fit, i.e. the central fit is

performed without the correlated systematic errors.

Then, for each source of system. error, set sk = 1, redo the fit, get e.g.

the gluon density, and add in quadrature all differences to the “central gluon” to get the “1 error band”.- The “Hessian” method : sk are not fixed but are parameters of the fit.

Technically, they are obtained analytically (cf previous slides : 2 is

quadratic in sk, hence 2 / sk leads to a simple expression for sk).

This means that the central fit is not a fit to the “raw data”, but to the data shifted by the optimal setting for the systematic shifts.

Error band obtained from 2 = T2, with T = 1 (or larger…)


The “offset” method

• Gives fitted theoretical predictions which are as close as possible to the “raw” data points.

• Does not assume that the errors are Gaussian.

• Numerically, it is equivalent to define two Hessian matrices :

Cov = Vstat + Vsyst

And the error on a quantity F is obtained with :

C. Pascaud & F. Zomer, LAL-95-05

• The method does not use the full statistical power of the fit to correct the data for the best estimate of the systematic shifts, since it chooses to distrust that syst. uncertainties are Gaussian distributed.


NB: to get the terms convenient to perform a rotation in the (pk) space such that the Hessian is diagonal (Pumplin et al).

The “Hessian” method

Minimize

We saw that this is equivalent to minimizing

• A (Nsys x Nsys) easier to invert than V (Npoints x Npoints)

• if i << ki, V can be close to a singular matrix (X . tX) Expression ( 1 ) avoids numerical problems.

( 1 )

Advantages of using (1) w.r.t. (2) :

• Get for free “better” estimates of the systematic uncertainties. E.g. the fit may tell you that normalisation of dataset A is off by + 2%. This means that we let the theory “calibrate” the measurements.

( 2 )

The error 2F is then obtained from formula on slide 72, i.e. assumes that the

syst. errors are Gaussian distributed.


Getting pdf uncertainties in practice

With Cov ij = H-1ij for the Hessian method

and is given on slide 78 for the offset method.

A rotation of the parameter space (pi) (z) allows to diagonalize Cov.

This leads to

Modern fits as “stored” in LHAPDF (http://hepforge.cedar.ac.uk/lhapdf)consist of : - the central fit

- 2Nsys fits which are the fits obtained when one eigenvector

z is shifted by its error : pdf “sets” S+ and S-

i.e.

Hence: x 2

2

x 2


Comparison of the “offset” and “Hessian” methods

• Offset method is conservative. Results in larger uncertainties than the

“Hessian” method when 2 = 1 is used to get the error band.

• With Hessian method : model uncertainties

(e.g. vary S, Q02,…) are usually larger than

the fit uncertainty. Because each “model choice” can result in different values of the syst. shifts, i.e. when changing the “model”, one does not fit the same data points.

As long as 2 < N + 2N, the fit is acceptable. The change in the parameters can be much larger than the central fit uncertainty when the latter

is estimated with 2 = 1.• Comparing H1 error (Hessian method) to the ZEUS error (offset method) on the gluon density : the full errors are somehow of similar size.


Hessian method : the “tolerance” parameter

Especially when fitting data from various experiments, it can happen that somedatasets are “marginally” compatible with the others.

Example : CTEQ6MNMC p data are globally described, but

“outliers” which result in a bad 2 for thisdataset.

2/ndf ~ 1.5 for 200 points

CTEQ method : Move from the global minimum in the direction of one eigenvector.Determine the distance from the global minimunthat is allowed by each experiment individually,as defined by the criterium :

One does not wish to drop the dataset (wouldloose constraints !).But the level of inconsistency between datasetsmust be reflected in the quoted uncertainties.


Take the most restrictivebound on each side.

In this example, thisgives distance ~ 10.

The average of this distanceover the eigenvectors isthen taken as the “tolerance”.

i.e. give errors corresponding to 2 = T2 with T = 10.

This differs from the well-known strict statistical criterium 2 = 1.But keep in mind that the conditions for the application of this criterium,namely Gaussian errors, are also not met.

CTEQ gives error corresponding to 2 = 100, MRST to 2 = 50.

Not a rigorous definition, but based on how far the parameters can be variedwhile still giving an acceptable description of all the datasets.


Pdf uncertainties : bottomline

Keep in mind that the uncertainties given by the pdf sets do not include those due to :

- Dataset choice and cuts - parameterization choice- theory (treatment of HQ, target mass / nuclear corrections, higher twists, low x effects…)

medium x


Very different shapes for the errorband on the gluon density in differentglobal fits.

Related to the parameterization choice.

Example of changing the parameterization choice

Q2 = 5 GeV2

- MRST parameterize at Q02 = 1 GeV2

and allows the gluon to become negative.

- CTEQ param. at Q02 = mc

2 = 1.7 GeV2.

Input gluon is valence-like and very small at low x, i.e. very small absolute error.

At higher Q2, all uncertainty is due to the evolution driven by the higher x gluon, which is well-determined.


Example of “changing the theory” – e.g. treatment of heavy quarks

Latest CTEQ fit (CTEQ6.5) : use the general VFNS with quark mass effectsaccounted for.

Main effect : rescaling of the momentum fraction carried by the incoming

quark. E.g. in c c, x c = x ( 1 + 4 M2c / Q

2)

Largest effect when xf(x,Q2) varies quickly,

i.e. at low x and Q2. New formalism suppressesthe HQ contributions relative to the zero-mass case.

x c

Fit without mass effects

Fit with mass effects

Causes u and d to increase, with differences

persisting at higher Q2.Impact on LHC predictions as will be seen later.

xc(x)


The near future

• Analysis of the full HERA data

high Q2 measurements benefit from the large increase of luminosity with

the inclusion of HERA II data ( 100 pb-1 700 pb-1)

In particular e-p luminosity : 20 pb-1 (HERA I) 250 pb-1 (HERA II)

i.e. constraints from xF3, CC DIS, high Et jets will be much stronger.A “projection fit”to HERA dataalone shows asignificantimprovement inthe determination ofthe valence quarkdistributions, and ofthe gluon densityat high x.

(C. Gwenlan, A. Cooper-Sarkar, C. Targett-Adams)


• Combined H1 + ZEUS dataset : averaging of the measurements in a model-independent way.

“cross-calibration” of syst. uncertainties leads to an improvement which is better than 2 in regions where the measurements are dominated by systematics.

Final analysis of the low Q2 HERA I data : Larger statistics ( up to x2) compared to what is currently included in the fits, better understanding of systematics and Monte-Carlo, smooth 2000 data taking. Expect a precision of 1 – 1.5 %.

Direct FL measurement with a precision of ~ 20%.

• The LHC comes in operation. pp collider, s = 14 TeV. First collisions at 14 TeV expected in July 08.

“Low luminosity” : 1032 cm-2 s-1 by the end 08 (i.e. > 1 fb-1 / year !) See lectures by A. De Roeck.

S. Glazov, HERA-LHCWorkshop

E. Perez 89

• SM processes : to understand the detector at the beginning, and for precision measurements.

• huge discovery potential for physics Beyond the SM.

X-section (pb) Tevatron LHC Ratio

W± e (80 GeV)

2600 20000 10

tt (2 x 172 GeV) 7 800 100

g g -> H (120 GeV)

1 40 40

q q (2 x 400 GeV)

0.05 60 1000

g g (2 x 400 GeV)

0.005 100 20000

Z’ (1 TeV) 0.1 30 300

Basic processes at the LHC

~

~~

~


W and Z production at the LHC

Thought of as “standard candles”. Do we know them so well ?

(J. Stirling)

W, Z

But W & Z at the LHC : <x> ~ 7.10-3 at central rapidity. In the measurable range,

x between ~ 5.10-4 and ~ 5.10-2

i.e. not in the valence region.i.e. not in the region where quarks are bestknown…

Small theoretical uncertainty :


Uncertainty obtained when using all pdf setswithin a given group : usually ~ 5%.

Note by how much the HERA data have allowed thisuncertainty to be reduced. This is due to themuch improved precision on the gluon densityat x ~ 10-3.

But the central values from different groups differby more than 5%, typically 8%.

And (W) has moved by ~ 8% when going fromCTEQ6.1 to CTEQ6.5, mainly due to the newtheo. treatment of heavy quarks !

i.e. notpreciseenough yetto be usedas a lumi.monitor…


Pdf uncertainties and their impact on the W mass measurement

Need to predictprecisely the spectrum ofPt(lepton) !

Hope to “derive” thisspectrum from thatof Pt(ll) in Z productionand thus, to reduce thispdf uncertainty to ~ 1 MeV.

OK provided that we trustour predictions for Z/Wvs. .

CDF II

But d – u at lowx is not wellknown. Smallunc. are artificial,due to the fitassumptions…

_ _


Remark : The interpretation of discoveries in AA at Alice may require directmeasurements of pdfs in A… – not covered.

Focus on ATLAS & CMS discovery potential : - Higgs - new particles or deviation from SM at high mass. Large M large x… and still limited knowledge esp. on d(x) and g(x).

Current pdf uncertainties : implications for LHC discoveries ?

d(x) g(x)

Gq + qG

G-Gq-qbar gluon takes part to the production of dijets

up to highest masses. This means that the

current predictions for Mjj at high mass are

quite uncertain…


How do pdf uncertainties affect the Higgs discovery potential ?

Not too bad… Cross-sections are known to within ~ 10%. Same for backgrounds.


This eigenvector is dominated by thehigh-x gluon parameter.

Limited knowledge of proton structure might “fake” a discovery…

Recall the excess of high Et jets reported byCDF in 1995…

was initially interpreted as new physics (quarksubstructure ?) until it was realized that a higher gluon density at high x could accommodatethese data, while remaining in agreement withother data.

Now we have pdf uncertainties, i.e.better handle even if there might stillbe some “uncertainty” on theseuncertainty bands…

The Tevatron jet data are in reasonableagreement with global fits today, takinginto account the large unc. due to theunc. on the high-x gluon.


Some NP models predict deviations in dijet mass spectrum at high mass.Example : qqqq contact interactions, some extra-dimension models.

Due to pdf uncertainties, sensitivity to compactification scales reducedfrom 6 TeV to 2 TeV in this example.This is due again to the large uncertainty on the high-x gluon.

Mjj (GeV)

Mc = 4 TeV

S. Ferrag, hep-ph/0407303

Mc = 2 TeV

Mjj (GeV)

Limited knowledge of proton structure might limit the discovery potential


d

u

• _ W ’

s

c

• ’

_ ~

RpV SUSY : reach would depend on the strength of the coupling ’ .

With sea quarks involved, uncertainties large already well below the kinematical limit.

(reach for a W’ withSM like couplings)

CMS Physics TDR Vol II

40% uncertaintyon part. lum. Fora 6 TeV W ’. g(W ’) ?

Would make the measurement of the coupling difficult.

Limited knowledge of proton structure might limit the precision on BSM _

New Physics in qq processes : Example: new W ’, resonant slepton production in RpV SUSY


Example : new Z’ boson, KK gravitonsin Randall-Sundrum models etc.. Signal = a mass peak.

But close to the discovery limit, couplings of a Z’ boson may not be measured accurately.

CMS Physics TDR Vol II

(shown uncertainties: from CTEQ 61 sets)

Partonic luminosities can be “normalised” to the side-bands data if enough stat.

4 TeV 8 TeV

New physics in qq processes : DY mass spectrum_


All pdf fits usually assume that charm is “radiatively” generated, i.e. originate

only from QCD evolution, starting from a null distribution at mc.

But an “intrinsic” charm component of non-perturbative origin could exist.E.g. models of Brodsky et al, p > = uud > + uuduu > + uudcc > + …

_ _

In a frame where the proton is moving, the uudcc state can exist only if all partons

travel at the same rapidity, yi = ln (ki+ / mT,i) ~ ln (xi / mi)

i.e. the intrinsic charm quarks should be at high x.The existing data have no sensitivity to such a component because it is at toolarge x. They allow that IC carries a few % of the proton momentum.

(Pumplin et al., hep-ph/070122)

no IC

IC

Could enhance drastically e.g. the production of H+ via c s H+ _

Limited knowledge of proton structure might bring “good surprises”

_


Improvement of our pdf knowledge from LHC data : EW processes

• If systematics can be controlled at the level of ~ 5%, measuring d(W)/d at the LHC should improve our knowledge of pdfs. Main improvement expected for the low x gluon parameter.

• Other observables, as W charge asymmetry, are less sensitive to systematics and could bring interesting constraints :

Visible differences between the MRSTAnd CTEQ predictions. Due to diff. in the valence distributions.

qdu

du

dudu

duduA

valval

valval

W 2

Assuming (low x) u = d q :_ _ _

WW


Quantified by M. Cooper-Sarkar :

generate pseudo-data (the “lepton”asymmetry, i.e. what will beactually measured) with an error of 4%,and include them in the ZEUS fit.

lepton

The large reduction of the error bandshow that this will put constraints,for the first time, on valence distributions at low x.

(M. Klein, B. Reisert, EP)

Relaxing the constraint that d – u 0 as x 0 (usually imposed…) : theuncertainty on the W charge asym. getsmuch larger, ~ 20% (for the plot, the fit includesH1 & BCDMS DIS data, and E866 DY data).i.e. measurement of the W charge asymmetry should bring new constraints on d – u at low x.

_ _

_ _


Quarks at high and low x using Z at LHCb

coverage : between 2 and 5

Look at Z : quark from

beam has x ~ 10-1, quark from

target has x ~ 10-4

Possibility to bring constraints on quarks

at high Q2 and high / low x

T. Lastovicka, M. Ferro-Luzzi,HERA-LHC Workshop


Improvement of our pdf knowledge from LHC data : jet production

So far our most stringent constraints on the gluon density at high x comefrom inclusive jet data at the Tevatron.What about inclusive jet at the LHC ?

At 1 TeV (2 TeV) , in the range 1 < eta < 2, the pdf uncertainty on the inclusivejet cross-section is about 15% ( 25%). This means that one must control thejet energy scale very well to bring significant pdf constraints !


(fits from C. Gwenlan)

Uncertainties on the gluon densities including LHC jet pseudo-data :


The future besides / beyond LHC…

• Theory side : two examples :

- a better understanding of prompt photons may allow to have them included in global fits ( q g q dominates)

- the data on diffractive J/ production at HERA should be able to constrain the low x gluon :

~ 2-gluon system in colorless stateAt lowest order,cross-section proportionalto the square of the gluon density.

But beyond leading ln(1/x), needto introduce “skewed” pdfs.

“NLO” predictions do exist(Martin, Ryskin, Teubner), but not yetas “solid” as the predictions in e.g. theDGLAP formalism.


Elastic J/in photoproduction

These data should be ableto constrain the low x gluon,once predictions areformulated which are notmodel-dependent.

More in the lecturesof C. Kiesling.

x ( Q2 + M2 ) / W2

x 10-3x 10-4


• Experimental side : several projects – not approved yet

- Upgrade of the accelerator complex at Jefferson Laboratory :

Fixed-target DIS with a 12 GeV beam,ep, en, eA, polarisation.Large luminosity ( 1038 cm-2 s-1)allows to access x values as large as 0.8.

Would allow to constrainmuch better esp.the ratio d/u athigh x.

- LHeC : ep collisions at s = 1.4 TeV

- eRHIC (Brookhaven) and ELIC (JLAB) eRHIC : polarised e (3-20 GeV) on polarised p (30-250 GeV) or A. ELIC : e (3-9 GeV) on p (30-225 GeV) or A. Rich program on low x, high parton density, hard diffraction, polarized pdfs.


Consider the feasibility of pursuingthe DIS programme using the 7 TeV LHC proton (A) beam and bringing itin collision with a 70 GeV electronbeam in the LHC tunnel: LHeC.

s = 1.4 TeV i.e. Q2 up to 2. 106 GeV2

Lumi ~ 1033 cm-2 s-1, i.e. integrated luminosities of about10 fb-1 per year can be considered.Polarised e beam.

J.B. Dainton, M. Klein, P. Newman,F. Willeke, EP

See talks of M. Klein and P. Newmanat DIS’07.

LHeC : a future DIS experiment at the LHC ?

JINST 1:P10001,2006


LHeC : kinematic plane

For Q2 = 1 GeV2,

x down to 5.10-7 !

i.e. very low x accessible atQ2 values where “partonic”language is applicable.


PDFs with LHeC…


Examples of low x measurements at LHeC

Clearly establishes saturation atnot-too-low Q2.

Precision allows to disentangle betweenexisting saturation models.

(P. Newman)


The end …

We have seen :

- the experimental data which allow to constrain parton distribution functions - how these fits are done technically - that they generally describe well the datasets, with a few exceptions, e.g. the low x HERA data not well described by NLO DGLAP fits.

- how the current pdf uncertainties translate into uncertainties for LHC measurements : pdf uncertainties do not jeopardize discoveries in most cases, but may affect “BSM precision”, e.g. the measurement of a very massive Z’ couplings. - LHC and future facilities may still improve our knowledge of the proton structure.

Thank you !


Backups…




Comparison global fit vs. fit to DIS data only

Pdfs should be constrained by all “reliable” data.

HERA-LHC Workshop : R. Thorne compared the MRST global fit with a fitto DIS data alone (“benchmark” fit).

Valence d, Q2 = 20 GeV2 Gluon, Q2 = 20 GeV2


Documents

1 Emmanuelle Perez (CERN-PH) School on QCD, Low x physics, Saturation and Diffraction Calabria, Italy, July 1-14, 2007 Determination of parton distribution