Transcript transparencies

Diffractive PDF fits to
the ZEUS Mx data
Michael Groys, TAU
HERA-LHC, 22.03.2005
Outline
•
•
•
•
•
•
Very brief overview
Experimental Data
Regge Factorization tests
Fits of Data
Interpretation of the fit results
Conclusion
Regge Theory and the Pomeron
• It is assumed that the diffractive
interactions are due to the Pomeron
exchange, where the Pomeron is a Regge
trajectory and can be parameterized as,
 (t )   (0)    t
IP
IP
IP
• Then total hadron-hadron cross section is,
 (ab) ~ s
 IP ( 0 ) 1
tot
• Results from hadron-hadron interactions by
Donnachie and Landshoff
 IP (0)  1.08
• In current study we set,
• αIP (0) – free parameter,
• α’IP = 0.25
   0.25 GeV
IP
2
Regge Factorization
• Using previous assumption we can write
D( 4)
F2

12 IP ( 0 ) IP
N
2
2
|  p IP ( t ) | x IP
F2 ( x IP , t ,  , Q )
16
• Here βIP(t) represents the Pomeron-proton coupling.
It may be obtained from fits to elastic hadronhadron cross section at small t ,
 (t )  4.6mb e
1, 9 GeU 2 t
1/ 2
p IP
• Regge factorization states that
F ( x , t ,  , Q )  F ( , Q )
• And so F2IP can be treated as the Pomeron structure
function. To simplify expressions we can introduce
the Pomeron flux factor,
IP
f
2
2
IP
IP / P
( x , t, ) 
IP
IP
2
2
N
[  (t )] x
16
2
p IP
1 2  IP ( t )
IP
• Then diffractive structure functions become,
F
D(4)
F
D(4)
F
D(4)
2
( x , t,  , Q )  f
IP / p
( x , t )F (  , Q )
( x , t,  , Q )  f
IP / p
(x ,t)
2
IP
1
IP
2
IP
IP
IP
2
2
1
F ( , Q )
x
IP
2
1
IP
L
( x , t,  , Q )  f
2
IP
( x , t ) F ( , Q )
IP
IP / p
IP
L
2
Regge Factorization cont.
• F2D(3) and FLD(3) have the same xIP dependence
• In the experiment, reduced crossection is
measured.
xIP
D ( 3)
r
 xIP F
D ( 3)
2
y2

xIP FLD (3)
2
2(1  y  y / 2)
• Its longitudinal part contains kinematic
factor that is xIP dependent.
y
Q2
xIP  s
• This factor is small for small y, thus
longitudinal part is usually neglected for
y<0.45. In the current study, FLD(3) was
included and no cut on y was done.
Pomeron parton distribution functions
•
•
Within the Ingelman and Schlein model the Pomeron
structure functions are defined in exactly the same way
as the structure functions of the proton.
Some constraints must be applied in order to get object
with vacuum quantum numbers.
• Self-charge-conjugation implies that
f ( x)  f ( x)
q / IP
q / IP
• Isoscalar implies that
f ( x)  f
u / IP
•
•
d / IP
( x)  f ( x)  f
u / IP
d / IP
( x)  f ( x)
q / IP
Evolution equations allows to obtain PDFs at any scale by
providing PDF at some initial scale Qini .
In the massless scheme, below mass threshold the PDF
of corresponding quark is 0.
f q / IP ( x, Q2 )  0 if Q2  4 mq2
•
For strange quark we make following assumption:
f ( x)  sf ( x)
s / IP
u / IP
• where 0 ≤ s ≤ 1
Experimental Data
• HERA collider
• ZEUS experiment
• FPC – forward plug calorimeter
− Mx Method
• LPS – leading proton spectrometer
− Direct proton measurements
• H1 experiment (partial sample)
− Large Rapidity Gap
• Values xIPσrD(3) at different β, Q2 and xIP .
Kinematical ranges
Kinematical ranges
Regge Factorization Test
• Check only xIP dependence.
• Fit with following function:
x IP F2D (3) ( x IP ,  , Q 2 )  N (  , Q 2 ) f ( x IP )
•
where N(β, Q2) factor will incorporate
f(xIP) represents xIP dependence.
β and Q2 dependence and
• Two types of f(xIP) were checked
f ( x IP ) 
1
A
x IP
f ( x IP )  f IP ( x IP ;  IP )
• Values of N(β, Q2) were determined from the fit to
each (β, Q2) bin independently.
• The values of parameters A, αIP are global.
• Two sets of fits were done:
• Fits in different Q2 bins independently with aim to find Q2
dependence.
• Fits in different
dependence.
β bins independently with aim to find β
Q2 dependency test of ZEUS FPC data
Q2 dependency test of ZEUS LPS data
Q2 dependency test of H1 data
β dependency test of ZEUS FPC data
Fits of Data
Parameterization of Pomeron PDFs.
• Guess Pomeron parton distribution functions at
initial scale.
• Following parameterization was chosen:

q

g
xq( x)  Aq x q (1  x)
xg( x)  Ag x g (1  x)
xs( x)  0
• The constraints are,
Aq , Ag  0
 q ,  g ,  q ,  g  1
• Evolve using CTEQ package with massless scheme
Data Selection
• MX > 2 GeV
• Higher twist effects
• Q2 > Q2ini = 3 GeV2
• xIP < 0.01
• Single Pomeron exchange
Fit results for ZEUS FPC data
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
Fit results for ZEUS LPS data
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
Fit results for H1 data
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
Parton Distribution functions
ZEUS FPC
ZEUS LPS
H1
Two solutions of ZEUS LPS data
gluons >> quarks
gluons ≈ quarks
ZEUS LPS fit with charm data points
Alexander Proskuryakov
Comparison:
ZEUS FPC data vs. H1 fit
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
Comparison:
H1 data vs. ZEUS FPC fit
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
Interpretation of the fit results
Fraction of Pomeron
momentum carried by
quarks/gluons
H1
1
Pq (Q )    dx xqi ( x, Q 2 )
2
i
0
1
Pg (Q )   dx xg( x, Q 2 )
2
0
ZEUS FPC
Pg
Pg  Pq
ZEUS LPS
Pq
Pg  Pq
Probability of diffraction
• The probability of diffraction for the action of the
hard probe which couples to quarks or gluons is:
  dx d  ( x  x  ) f ( x ) q ( , Q )
P ( x, Q ) 
 q ( x, Q )
IP
D
q
2
IP
IP
IP
i
2
IP
IP
i
2
i
i
D
g
P ( x, Q
2
 dx
)
IP
d  ( x  xIP  ) f IP ( xIP ) g iIP (  , Q 2 )
g iIP ( x, Q 2 )
• If the interaction in the gluon sector at small x
reaches a strength close to the unitarity limit then
Pg is expected to be close to ½ and be larger than
Pq.
• L.Frankfurt and M.Strikman,
"Future Small x physics with ep and eA Colliders”
Probability of Diffraction
ZEUS FPC fit
H1 fit
Conclusions
• Regge Factorization tests succeeded for xIP < 0.01
• Simple parameterization of Pomeron parton
distribution function allows to describe well
existing data in selected kinematical range.
• We didn’t succeed to fit ZEUS FPC and H1 data
using the same parameterization even with
introducing some overall normalization factor.
• The fraction of the Pomeron momentum carried by
gluons was found to be 70-90% for ZEUS LPS/H1 data
and 55-65% for ZEUS FPC data.
• Although the probability of diffraction can be
calculated at any value of x, the results below 10-4
are unphysical. Possible reason is gluon saturation.
The End
Thank you
Computation of Strong coupling
constant.
DIS Formalism
•
Cross section can be expressed as,
d 2 4 2 s  y 2
2
2 


2
xF
(
x
,
Q
)

(
1

y
)
F
(
x
,
Q
) 
1
2
dxdy
Q 4  2

•
•
The structure functions F1 and F2 are process dependent
At high Q2 it can be represented as incoherent
sum of lepton quark interactions
d 2 ep
d 2 eq

dxdy
dxdy
q
•
In the leading order the structure functions are,
F1 ( x) 
•
1
 qi2 f i ( x)
2 i
F2 ( x)  2 xqi2 f i ( x)
i
f(x) is a probability density of finding parton with momentum x
• Then the Callan-Gross relation should hold,
2 xF1  F2
Evolution Equations
•
•
In QCD partons interacts one with another through the
exchange of gluons and so the parton distribution functions
become Q2 dependent. The following processes must be
considered:
iii
i
ii
i.
Gluon Bremsstrahlung,
ii. Quark pair production by gluon,
iii. ggg coupling.
The DGLAP evolution equations describe the evolution of
parton distribution functions with Q2
dq i ( x, Q 2 )

 s
2
2
d log Q
dg i ( x, Q )

 s
2
2
d log Q
2
•
1

x
1

x
x
 x 
dy 
2
2
q i ( y, Q ) Pqq    g ( y , Q ) Pqg  
y 
 y
 y 
x
 x 
dy 
2
2
 q i ( y , Q ) Pgq    g ( y, Q ) Pgg  
y  i
 y
 y 
The emission of non-collinear gluons by quarks will induce an
appearance of non-vanishing σL. This leads to the violation of
Callan-Gross relation which can be quantified by longitudinal
structure function,
FL  F2  2xF1
Diffractive DIS Formalism
•
In an analog to the DIS, the diffractive cross section
can be expressed as,

d 2 D 4 2 s  y 2
 2 xF1D ( x, Q 2 )  (1  y) F2D ( x, Q 2 ) 

4
dxdy
Q  2

•
Then the four-fold differential X sec. can be written as,
 
d 4 DD
44
yy2 2  D (D4() 4 )
y 2y 2 D ( 4D) ( 4 ) 2 2
22
2 2

1

y

F
(
x
,
Q
,
x
,
t
)

FLFL ( x,(Q
t ),t ) 

1

y

F
(
x
,
Q
,
x
,
t
)

x, Q, x IP, x, IP
IP IP
 2 2
22
2 2 

22 
22
dxIPIP dtd
dtdxdQ
xQ
dx
dQ
Q 
 
•
Let us introduce reduced X sec.,
d 4 D
4 2 s 
y 2  D ( 4)

 r ( x IP , t ,  , Q 2 )

1

y

2
4

2 
dx IP dtdxdQ
Q 
•
F
D ( 4)
2
y2

FLD ( 4)
2
2(1  y  y / 2)
Two quantitative conclusions can be made.
i.
•

D ( 4)
r
FL affects σR at high y.
ii.
If FL = 0 then σR = F2.
Three-fold variables are defined in the following way:
X D (3)   dt X D ( 4 )
Diffractive parton distributions
•
QCD Factorization holds for diffractive processes,
d    d f
(b, x / , t ;  ) dˆ
( D)
i
i
i
• where,
• the index i is the flavor of the struck parton,
• the hard-scattering coefficients dσ are perturbatively calculable
and are the same as for the corresponding fully inclusive cross
section,
• the renormalization/factorization scale should be of the order of
Q,
• the diffractive parton distribution function f(D) is the density of
partons conditional on the observation of a diffractive proton in
the final state.
•
Then F2D can be represented in the following way:
d 2 f i D (  , x IP , t ;  ) ˆ
d 2 F2D ( x, Q 2 , x IP , t )
   d
F2,i (  , Q 2 ;  )
dx IP dt
dx IP dt
i
•
The probability of diffraction looks like,
Pi ( x, Q 2

)
f i ( D ) (  , x IP , t ;  )  ( x  x IP  ) dx IP d dt
f i ( x, Q 2 )
Regge Theory
• Hadrons with different
spins but same other
quantum numbers lie on
Regge Trajectories
l   (t )  (mi2 )  J i
 (t )   0    t
• In hadron-hadron
interactions it is useful to
consider the exchange of
the whole trajectory and
not of a single particle.
• Then the scattering
amplitude is given by,
A( s, t ) ~ s  (t )
ω
ρ
φ
π
Cut on xIP and Regge
Factorization Test.
e P scattering
• Elastic Scattering
• Deep Inelastic Scattering (DIS)
• Diffractive DIS
• Inclusive diffraction
• Vector Meson production
What is Pomeron?
• Regge Trajectory
• Object with vacuum
quantum numbers
• Colorless
• Self-charge-conjugate
• Isoscalar
• Two gluon exchange
• Ladder structure
develops
• Quark constituent may
also exist.
≡ IP
Theoretical
Framework
Deep Inelastic Scattering
e(k) + P(p)
γ*
e(k’) + H(p+q)
Q 2  ( k  k ' ) 2
W 2  ( p  q) 2
Q2
x
2pq
p  q Q2
y

p  k sx
Diffraction and the Pomeron
e(k) + P(p)
γ*(q) + P(p)
e(k’) + P(p’) + H
IP
P(p’) + H
•
•
•
•
Ingelman and Schlein model of
diffractive scattering
If p’ > 0.99 p - Pomeron exchange
Pomeron has vacuum quantum
numbers
Large Rapidity Gap
t  ( p  p' ) 2
x IP
M X2  Q 2  t
q( p  p' )


q p
W 2  Q 2  m P2
M X2  Q 2

W 2  Q2
Q2
Q2
 
 2
2q  ( p  p ' )
Q  M X2
x  x IP 
Elastic Scattering
e
γ*, Z
p
Deep Inelastic Scattering
e
γ*, Z
p
X
Diffractive Scattering
Inclusive Diffraction
e
γ*
X
p
Diffractive Scattering
Vector Meson production
e
γ*
VM
p
HERA accelerator at DESY
•
•
length – 6.5 km
since – 1992
EP  920 GeV
Ee  30 GeV
s  330 GeV
H1
ZEUS
Detectors at HERA
ZEUS
H1
Experimental signatures of
diffraction
•
•
Outgoing proton with high momentum. Unfortunately,
very often proton goes down the beam pipe and thus can
not be observed. Only ZEUS LPS detector allows to
detect diffractive protons directly.
Large Rapidity Gap.
1 E p 1 W
y  ln
 ln
2 E p 2 m
P
L
P
L
P
2
2
P
1 E p 1 W
y  ln
 ln
2 E p 2 M
X
L
X
L
X
2
2
X
y  y  y  ln
P
X
2
W
mM
P
X
• Problem: if proton is not seen then a selection on absolute
value of rapidity must be done.
Proton dissociation
•
There are also processes where proton doesn’t remain
intact and also dissociates into some system Y. In this case
the LRG can be still observed. This process is called double
diffractive dissociation.
e
γ*
γ* + P
IP
X
X+Y
p
•
•
If the proton breaks up one can not be sure that the
exchanged object had vacuum quantum numbers. In
addition the Pomeron proton coupling can change.
In order to get pure sample it is necessary to exclude
proton dissociation events. Unfortunately we often do not
know whether dissociation occurred or not.
Y
Differences between data sets
• Background evaluation
• The LRG can be also observed for non-diffractive events. It is
exponentially suppressed for small values of Mx, but
becomes very important for high masses.
• H1 group uses Monte-Carlo simulations in order to evaluate
background, while ZEUS FPC group developed their own
Mass Decomposition Method.
•
Proton dissociation
• It is often impossible to determine whether the proton
indeed remained intact and hadn’t dissociated. Detector
structure and kinematics provide only an upper limit for
mass of the object that went down the beam pipe. For H1
detector this value is 2GeV while for ZEUS detector it is 4GeV.
To overcome this problem appropriate corrections must be
done.
•
LPS part of the detector doesn’t have such problems
because it measures outgoing proton directly.
Unfortunately it has low acceptance which leads to small
statistics.
Fit results for H1 data
Q 2  3GeV 2
  0.04
  0 .1
  0.2
  0.4
  0.65

0
0..9
9
Q 2  3GeV 2
Q 2 (GeV 2 )
6.5
8.5
Q 2  15GeV 2
Q 2  15GeV 2
12
15
20
Q 2  50GeV 2
Q 2  50GeV 2
Q 2  100GeV 2
Q 2  100GeV 2
25
35
45
60
90
Fit results for ZEUS FPC data
Q 2  3GeV 2
Q 2  3GeV 2
Q 2  15GeV 2
Q 2  15GeV 2
Q 2 (GeV 2 )
4
6
8
Q 2  50GeV 2
Q 2  50GeV 2
Q 2  100GeV 2
Q 2  100GeV 2
14
27
55
Fit results for ZEUS LPS data
  0.03
  0.13
  0.48
Q 2  3GeV 2
Q 2  3GeV 2
Q 2  15GeV 2
Q 2  15GeV 2
Q 2  50GeV 2
Q 2  50GeV 2
Q 2  100GeV 2
Q 2  100GeV 2
Q 2 (GeV 2 )
3.7
6.9
13.5
39
Q2 Dependence of H1 data
  0.005
  0.025
  0 .1
  0.3
  0.65
  0.9
xIP
.00015
.0003
.0006
.0012
.0025
.005
.01
.02
Data with
Mx<2 GeV
Q2 Dependence of ZEUS FPC data
  0.005
  0.025
  0 .1
  0.3
  0.65
  0.9
xIP
.00015
.0003
.0006
.0012
.0025
.005
.01
.02
Data with
Mx<2 GeV
ZEUS FPC fit vs. H1 data
  0.04
  0 .1
  0.2
  0.4
  0.65
  0.9
Q 2 (GeV 2 )
6.5
8.5
12
15
20
25
35
45
60
90
H1 fit vs. ZEUS FPC data
Q 2 (GeV 2 )
4
6
8
14
27
55
Overview
• Obtain experimental diffractive data
• Test validity and limitations of the
Regge Factorization
• Fit the experimental data assuming
the validity of the Regge factorization
in the selected kinematic range
• Calculate some physical quantities
using the fit results
Probability of Diffraction
H1 data
PqD ( x, Q 2 ) 
  dx
i
IP
d  ( x  xIP  ) f IP ( xIP ) qiIP (  , Q 2 )
q
IP
i
( x, Q 2 )
i
PgD ( x, Q 2 ) 
 dx
IP
d  ( x  xIP  ) f IP ( xIP ) g iIP (  , Q 2 )
g iIP ( x, Q 2 )