# Presentation: 786544

## Determination of parton distribution functions,

**School on QCD, Low x physics, Saturation and Diffraction**
**Calabria, Italy, July 1-14, 2007**

- impact of HERA data & consequences for LHC

- Emmanuelle.Perez@cern.ch

- Plot shown by A. Martin yesterday.
- In these lectures we see how we
- get there :
-   - experimental data
-   - QCD fit techniques

## 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...

## Parton distribution functions

-   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.

- F2)

- F2)

- F2,

## Probing the nucleon...

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

- Deep inelastic scattering of a
- lepton off a nucleon (nuclei)
- is the golden process to study the
- parton distribution functions.

- Other processes also bring
- important constraints, as will
- be seen later.

- GE, GM(Q2)  W1, W2()  Fi(x,Q2)

- Form factors, Bjorken scaling, structure
- functions.

## Deep-inelastic scattering

- q = k – k’,  Q2 = -q2 squared momentum transfer
- S = (p + k)2  square of center of mass energy

- “inelasticity variable”

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

- Q2 = x y S

- 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

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

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

- “Breit” frame :

- P = (Ep, 0, 0, - Ep)
- q = (E, 0, 0, qZ)

- Ep = qZ / 2x

- 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)

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

## ein

| JZ (boson) | 0 | +1 | -1 |
| --- | --- | --- | --- |
| Projection of the spin of ein
on z axis |             

 |  |  |
| Projection of the spin of eout on z axis |               |  |  |

- ein

- eout

- q

- 

- 

- z

- Example : amplitudes for right-handed leptons :

- Projection of ein spin on z axis is
- 1⁄2 with d1/21/2, 1/2 = cos(/2)
- and -1⁄2 with sin(/2).

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

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

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

- FT + FL  F2

- Putting everything together :

- T :  1 + (1-y)2

- L :  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 :

- 0  FL  F2

- Effect of FL
- large only at
- large y.

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

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

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

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

- _

- _

- _

- _

- In lepton-deuterium :  ~ (u + d + u + d )

- _

- _

- lp & ld allows to “separate” the flavors.

## FL : back to the Breit frame :

- FL : back to the Breit frame :

- JZ,in = JZ,out

- Quarks have spin 1/2
- Helicity 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.

- Happens when the interacting quark comes
- from a gluon splitting g  qq.
- Hence :

## ( d11,1 )2  ~ ( 1 + cos *)2 ~ ( 1 – y)2

- W

- W couples to
- left-handed fermions
- and right-handed
- antifermions.

- J = 1

- ( d11,1 )2  ~ ( 1 + cos *)2 ~ ( 1 – y)2

- J = 0

- ( d00,0 )2  ~ 1

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

- CC(e-p) >> CC(e+p)

## Neutrino Deep-inelastic scattering

- 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 :

- Hence :

- 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

- xF3:     1< Q2 < 200 GeV2

- FL

- F2:     1< Q2 < 200 GeV2

- approximate
- Bjorken
- scaling

- scaling violations

- DIS Fixed target results

- Q2 > 5 GeV2 :
- data down to
- 10-2 only

- Measurements
- only at
- high x

##  What happens at high Q2 ??

-  What happens at high Q2 ??

- Pre-HERA status...

## The HERA electron-proton collider

- 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±

## p

**Complete 4****π**** 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.**

- e

- Asymmetric detector : reflects the
- beam energies asymmetry.

- The H1 detector

## The ZEUS detector

**Complete 4****π**** 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**

## Q2 ~ 105 GeV2

- 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 the
- pert – non-pert transition
- region.

- Q2 from 0.1 to 105 GeV2
- x from 10-6 to ~ 0.8

- Huge extension of
- the kin. domain
- compared to
- fixed target
- expts.

- HERA kinematic domain

## at high yh

- Bad resolution
- at high yh

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

-  Combine e and had : replace 2E0e in (1) by 2E0e = (E-pZ)h + (E-pZ)e and use
-    p2T,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 of
- the el. energy.

- B. Heinemann,
- PhD Thesis

- (1)

## Ee = Ee0,

- 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

- p

****

- HFS goes more
- central as y
- increases.

- High x & y CC

- High x & y NC

## Q2 ~ 1.5 – 2 GeV2.

- Angular acceptance limits the
- measurements down to
- Q2 ~ 1.5 – 2 GeV2.

- Going lower in Q2 requires special
- techniques 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.)

## HERA DIS measurements

**HERA DIS measurements**

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

- 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 obtained
- at 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...

- No “gap” between HERA & fixed target data.

## actually not F2, but

- What is measured is
- actually not F2, but
- rather
- F2 – y2 FL / Y+
- (cf slide 9)

- The reduction of the
- cross-section as
- y  1 (lowest x),
- due to the
- disappearance of the
- contribution of
- longitudinal photons,
- is observed indeed.

- FL

- (more on FL later...)

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

## 5 orders in magnitude in Q2 and 4 orders

- With increasing luminosity,
- important statistics over the full
- kinematics 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 of
- QCD fits based on the DGLAP
- equations (see later).

- Excellent agreement with DGLAP, over
- 5 orders in magnitude in Q2 and 4 orders
- of magnitude in x.

- Within DGLAP : via F2/lnQ2,
- access to the gluon density.

## Count nr. of events in appropriately 

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

- Extract reduced cross section

- Uncertainties are systematics dominated
- for 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)

## NC(e-p)  NC(e+p). 

- 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+p

- Statistics 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

##   - HERA, standard techniques: xmax=0.65  

- 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’t
- try to reconstruct x, but put the event in
- a bin [ xedge; 1 ] and measure the integrated .

- ZEUS Collab., EPJ C49 (2007) 523.

- No jet found

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

## CC(e+p) goes as  (1 –y)2  xD + xU

- Measurements of CC DIS

- Recall (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 shown
- on the plot.

- However, for x ~ 0.1,
- precision of about 15% is
- reached
- (but at highest Q2 where the
- stat. is low)

- HERA I measurements are
- statistically limited.

- Brings constraints on “flavor
- separation”, which are
- missing from F2p (4U+D) alone.

## CC(e,SM) ~ (1  Pe)

- 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%

##      D* D0 slow  K  slow

-  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 radii
- of ~ 5 cm and ~ 10 cm.
- Impact parameter resolution:

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

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

##   Charm fraction fcc 

-   Charm fraction **f****cc**** **
-  **    ****- roughly constant with Q****2**** **
**     ****- around 24% on average**
-   Beauty fraction **f****bb**
**     ****- increases rapidly with Q****2**** **
**       ****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.

- Shifted
- vertex

- Larger angles :

-  QED Compton events

- 1

- 2

- Final electron (2) in
- the acceptance,
- 1 larger.

- Lower the incoming energy :

- Exploit initial state radiation events.

- Q2 = xyS, S reduced
- i.e. access higher x
- at a given Q2.

- 

- e

- e

- p

## Example of measurements at lowest Q2

- 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.

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

-  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 (F2p 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

-  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.

-  Usually impose number sum rules :

- And momentum sum rule :

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

- Id. for c, b

## g, uVal, dVal, Total Sea, d – u 

- (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.

- _

- _

- 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

- _

## 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

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

- Choice of the parametrisation

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

-     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

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

-   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

- _

- _

## f(x)/Q2, g(x)/Q4,... to the structure functions,

- “Higher twists” operators : generally related to
- correlations 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)

- And the measured F2 is related to the “massless”
- one by :

- “higher twist” like,
- kinematic origin

- Effects largest at low Q2
- and large x.

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

- MRST(HT) : fit with
- F2 = F2LT ( 1 + D(x) / Q2)

- Differences observed between
- MRST and MRST(HT) at low
- W2.

- A cut W2 > about 10 GeV2
- allows to stay away from region
- where HT are important.
- Note that this cuts out many
- of the SLAC data points...

- Illustration of HT :

##  Correlation between g and S, can be

- Choice :

- Pure valence  sum rule  (  V dx  = 3 )

- 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

- Example 1 : QCD fit to low Q2 ep DIS data

## Model uncertainty :

- (slide from M. Klein)

- 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

- Assumptions :

- Such that the valence distributions vanish
- when x  0.

- Because no distinction of the rise
- at low x between xU and xD.

- ( “2 saturation” procedure )

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

- Results in 10 free parameters.

- Low x
- assumptions
- needed.

- Example 2 : adding high Q2 DIS data

## xU (1.5% at x=10-3,

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

- Good determination
- with ep data
- alone !

- 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

##    - to ZEUS DIS data only  (low Q2 and high Q2, NC and CC)

- 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 F2d / F2p		 constraints d / u
- 	- CCFR data on xF3		 constraints the high-x valence

- Starting scale Q02 = 7 GeV2, Q2min = 2.5 GeV2

-  11 parameter fit

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

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

- Differences however that
- are not embedded in the
- error bands, esp. for
- the valence distributions.

- Sensitivity to those has a
- different origin in the
- H1 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

## xdval

- Comparison of ZEUS alone / ZEUS with 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

- 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 fit
- to ZEUS data alone.

- This gluon density results in a
- negative FL at lowest Q2.

- Sign that the approximations done
- in the QCD calculations are not
- valid 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 ZEUS
- data 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).

##   over the full range of Q2

- 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.

- Some data used to be included in global fits,
- as prompt photon production which in
- principle brings constraints on the gluon
- density – but hampered by too large
- theoretical uncertainties.

- Recent fits also include HERA jet data and
- F2b & F2c measurements.

- Typically this leads to ~ 2000 points in the
- fits, with a large number of systematic
- error sources (see later...)

- A. Martin

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

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

- x1 = xBeam, x2 = xTarget

- For x1 >> x2 one has :

- 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

- _

- _

- 

- 

- 

- x1 > x2

- Note the spread of the predictions from pdfs
- before these data were included in the fits.

- E866

- _

- _

## What could explain a sea asymmetry ?

- 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 :

-  strange sea :

- Global fits tell this does not hold.

- Q2=56 GeV2

## l-

- Tevatron : pp collider, s = 1.8 TeV

- d

- u

- _

- l-

- 

- _

- W-

- x1,2 = (M2W / S) exp ( W)

- At central rapidity, x1 = x2 ~ 2 10-3
- At  ~ 2.5 : x1 = 2 10-2, x2 ~ 2 10-4

- (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 the
- incoming proton (antiproton).

- Asym. diluted when looking at (lepton) :

- l-L

- 

- 

- 

- 

- dL

- R

- uR

- _

- Constraints on the d/u ratio.

- p

- p

- _

- 

- 

- _

## Jet production at the Tevatron

- (slide from R. Thorne)

## r

- But not all
- global fits
- reproduce well
- the DIS data
- at lowest x.

- In general looks fine...

- MRST fit

- x

- x

- r

- While HERA
- fits (to DIS
- data only) do.

- e.g. MRST : Tevatron
- jet data require a
- quite high g(x) at
- high x, resulting in
- a lower g(x) at low x,
- getting negative at
- Q2 ~ 2-3 GeV2.
-         Leads to FL < 0...

## F2 rise towards low x well described by

- 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. errors
- may 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 the
- MRST 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.

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

- 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 many
- observables at the Tevatron or the LHC.

- Hence one needs to better understand the limitations of our calculations
- at 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 the
- ln(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 

- 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 intermediate
- energy (575 GeV) for the last month
- of 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.

## for FL  FL at NNLO is positive.

- At NNLO, the gluon density becomes
- even more negative.

- Compensated by positive terms in
- the O(S3) 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 a
- decrease in the data.

## F2(x,Q2) = c(Q2)  x -(Q2)

- 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 “confinement
- transition” 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

- 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-lived
- dipole scatters
- from the proton

- Original model was improved
- by relating (x,r) to
- 1/g(x,Q2).
-  Describes the  slopes both
-     at low and high Q2.

- Golec-Biernat, Wustoff

- See lectures by F. Gelis.

## (*p) ~ 0 ( small) to (*p) ~ 0 /  ( large) observed for  ~ 1.

- 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 the
- attributes 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 the
- low x & low Q2 HERA data.
- Many fits predict a negative gluon and, more problematic, a negative FL at
- Low x and Q2. This may be a sign that the DGLAP NLO calculations are
- not 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 a
- direct measurement from HERA soon !
- F2 slopes show no indication of a “tame” of the rise of F2 at low x. But the
- low 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 the
- past ~ 5 years to assess
- rigorously the pdf uncertainties.

- Note : the plot also shows that one should
- not extrapolate a fit beyond the region where
- there are data... Before HERA, there was no
- information on xg(x) at low x !

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

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

- 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 error
- propagation 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) :

- In addition to the theory parameters (p), the “scale factors” fNk are
- also fitted.
- Nk is the normalization uncertainty of dataset k.

## With   2i = 2i,stat + 2 i,uncorrelated syst.

- and smaller, one needs to consider carefully all systematic errors.

- “Generalization” of the previous formula :

- With   2i = 2i,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)

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

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

## With   2i = 2i,stat + 2 i,uncorrelated syst.

- With   2i = 2i,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...)

## Cov = Vstat + Vsyst

-  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.

##  if i << ki, V can be close to a singular matrix  (X . tX)

-                                               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.

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

- 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

- Hence:

- x  2

- 2

- x 2

##   “Hessian” method when 2 = 1 is used to

- 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.

## “outliers” which result in a bad 2 for this

- Hessian method : the “tolerance” parameter

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

- Example : CTEQ6M
- NMC p data are globally described, but
- “outliers” which result in a bad 2 for this
- dataset.

- 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 minimun
- that is allowed by each experiment individually,
- as defined by the criterium :

- One does not wish to drop the dataset (would
- loose constraints !).
- But the level of inconsistency between datasets
- must be reflected in the quoted uncertainties.

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

- Take the most restrictive
- bound on each side.

- In this example, this
- gives distance ~ 10.

- The average of this distance
- over the eigenvectors is
- then 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 varied
- while 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...)

## Q2 = 5 GeV2

- Very different shapes for the error
- band on the gluon density in different
- global 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 = mc2 = 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.

## quark. E.g. in c  c,  x  c = x ( 1 + 4 M2c / Q2)

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

- Main effect : rescaling of the momentum fraction carried by the incoming
- quark. E.g. in c  c,  x  c = x ( 1 + 4 M2c / Q2)

- Largest effect when xf(x,Q2) varies quickly,
- i.e. at low x and Q2. New formalism suppresses
- the HQ contributions relative to the zero-mass case.

- 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 data
- alone shows a
- significant
- improvement in
- the determination of
- the valence quark
- distributions, and of
- the gluon density
- at high x.

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

##  Final analysis of the low Q2 HERA I data :

-   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.

##  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

- ~

- ~

- ~

- ~

## But W & Z at the LHC : <x> ~ 7.10-3 at 

- 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 best
- known...

- Small theoretical uncertainty :

## at x ~ 10-3.

- Uncertainty obtained when using all pdf sets
- within a given group : usually ~ 5%.

- Note by how much the HERA data have allowed this
- uncertainty to be reduced. This is due to the
- much improved precision on the gluon density
- at x ~ 10-3.

- But the central values from different groups differ
- by more than 5%, typically 8%.

- And (W) has moved by ~ 8% when going from
- CTEQ6.1 to CTEQ6.5, mainly due to the new
- theo. treatment of heavy quarks !

- i.e. not
- precise
- enough yet
- to be used
- as a lumi.
- monitor...

## Pdf uncertainties and their impact on the W mass measurement

- Need to predict
- precisely the spectrum of
- Pt(lepton) !

- Hope to “derive” this
- spectrum from that
- of Pt(ll) in Z production
- and thus, to reduce this
- pdf uncertainty to ~ 1 MeV.
- OK provided that we trust
- our predictions for Z/W
- vs. .

- CDF II

## current predictions for Mjj at high mass are

- measurements 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 ?

- 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.

## Limited knowledge of proton structure might “fake” a discovery...

- Recall the excess of high Et jets reported by
- CDF in 1995...
- was initially interpreted as new physics (quark
- substructure ?) until it was realized that a
- higher gluon density at high x could accommodate
- these data, while remaining in agreement with
- other data.

- Now we have pdf uncertainties, i.e.
- better handle even if there might still
- be some “uncertainty” on these
- uncertainty bands...

- The Tevatron jet data are in reasonable
- agreement with global fits today, taking
- into account the large unc. due to the
- unc. 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 reduced
- from 6 TeV to 2 TeV in this example.
- This is due again to the large uncertainty on the high-x gluon.

- Limited knowledge of proton structure might limit the discovery potential

## RpV SUSY : reach would depend on the 

- 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’ with
- SM like couplings)

- CMS
- Physics
- TDR
- Vol II

- 40% uncertainty
- on part. lum. For
- a 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 gravitons

- in 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

- _

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

- 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.

- 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 too
- large x. They allow that IC carries a few % of the proton momentum.

- Limited knowledge of proton structure might bring “good surprises”

- _

## W

-  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 MRST
- And CTEQ predictions.
- Due to diff. in the valence distributions.

- W

- W

## Quantified by M. Cooper-Sarkar : 

- Quantified by M. Cooper-Sarkar :
- generate pseudo-data (the “lepton”
- asymmetry, i.e. what will be
- actually measured) with
- an error of 4%,
- and include them in the
- ZEUS fit.

- lepton

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

- (M. Klein, B. Reisert, EP)

## beam has x ~ 10-1, quark from

-  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 come
- from 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 inclusive
- jet cross-section is about 15% ( 25%). This means that one must control the
- jet 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 state

- At lowest order,
- cross-section proportional
- to the square of the gluon density.

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

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

## x  ( Q2 + M2 ) / W2

- Elastic J/
- in photoproduction

- These data should be able
- to constrain the low x gluon,
- once predictions are
- formulated which are not
- model-dependent.

- More in the lectures
- of C. Kiesling.

- x  ( Q2 + M2 ) / W2

- x  10-3

- x  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
- constrain
- much better esp.
- the ratio d/u at
- high x.

- - LHeC : ep collisions at s = 1.4 TeV

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

- the DIS programme using the 7 TeV
- LHC proton (A) beam and bringing it
- in collision with a 70 GeV electron
- beam 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 about
- 10 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. Newman
- at DIS’07.

- LHeC : a future DIS experiment at the LHC ?

- JINST 1:P10001,2006

## For Q2 = 1 GeV2, 

- LHeC : kinematic plane

- For Q2 = 1 GeV2,
- x down to 5.10-7 !

- i.e. very low x accessible at
- Q2 values where “partonic”
- language is applicable.

## PDFs with LHeC...

## Examples of low x measurements at LHeC

- Clearly establishes saturation at
- not-too-low Q2.
- Precision allows to disentangle between
- existing 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...

## Slide 114

## Slide 115

## Valence d, Q2 = 20 GeV2

-  Pdfs should be constrained by all “reliable” data.

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

- Valence d, Q2 = 20 GeV2

- Gluon, Q2 = 20 GeV2

## Slide 117