Small-x and Diffraction in DIS at HERA
I
Henri Kowalski
DESY
12th CTEQ Summer School
Madison - Wisconsin
June 2004
H1 detector
ZEUS detector
Ep = 920 GeV, Ee = 27.5 GeV,
# bunches = 189
Ip = 110 mA, Ie = 40 mA
Linst= 2 x 1031 cm-2 s-1
ZEUS detector
Q2 ~ 2 –100 GeV2
Q2 ~ 0.05-0.6 GeV2
Q2 - virtuality of the incoming photon
W - CMS energy of the incoming
photon-proton system
x - Fraction of the proton momentum
carried by struck quark x ~ Q2/W2
d 2 σ eP
2
2 π αem
2
2
2
2


[Y
F
(x,
Q
)

Y
xF
(x,
Q
)

y
F
(x,
Q
)]

2

3
L
dxdQ 2
xQ 4
Y  1  (1  y)2
Infinite momentum frame
Proton looks like a cloud of
noninteracting quarks and gluons
F2 measures parton density
in proton at scale Q2
F2 = f e2f x q(x,Q2)
y – inelasticity
Q2 = sxy
there is a change of slope at small-x, near Q2 = 1 GeV2
Gluon density
Gluon density dominates F2 for x < 0.01
Gluon density known with good precision at
larger Q2.
For Q2 ~1 GeV2 gluons tends to go negative.
NLO, so not impossible
BUT – cross sections such as L also
negative !
MX - invariant mass of all particles seen in the central detector
t - momentum transfer to the diffractively scattered proton
DY ~ log(W2 / M 2X)
Diffractive Signature
diff
Non-Diffraction
Nondiff
Diffraction
- Rapidity
uniform, uncorrelated particle emission along the
rapidity axis =>
probability to see a gap DY is ~ exp(-<n>DY)
<n> - average multiplicity per unit of rapidity
dN/ dM 2X ~ 1/ M 2X =>
dN/dlog M 2X ~ const
Slow Proton Frame
incoming virtual photon fluctuates into a quark-antiquark
pair which in turn emits a cascade-like cloud of gluons
 qq 
1
1

 10  1000 fm
DE m p x
Transverse size of the quark-antiquark cloud
is determined by r ~ 1/Q ~ 2 10-14cm/ Q (GeV)
σ
γp
tot
1
 2 ImAel (W 2 , t  0)
W
σ
γ* P
tot
4 π 2 αem
(W, Q ) 
 F2 (x, Q 2 )
2
Q
2
Diffraction is similar to the elastic scattering:
replace the outgoing photon by the diffractive final state
r , J/Y or X = two quarks
Rise of gptot with W is a measure of radiation intensity
Radiation process
3 s n
dk12t
dkn21t
  C  ( )  dy1  2     dyn1  2  dyn ( yn , Q02 )

k1t
kn1t
n
gp
tot
emission of gluons is ordered in rapidities
0  yn  yn 1      y2  y1  ln( W 2 / Q 2 )  ln( 1 / x)
QCD Toy Model:
integrals over transverse momenta are independent
of each other
2
2
(Q 2 ) d k
3 s kmax
it

2
2

 Q0
kit
  C 
gp
tot
n
ln(1/ x )
n

0
n
dy1  dy2      dyn  C  (ln( 1 / x)) n
n n!
0
0
y1
yn1
gp
 tot
 C exp(  ln( 1 / x))  C (1 / x)  ~ (W 2 ) 
Rise of gptot with W is a measure of radiation intensity
Dipole description of DIS
 tot
g *p
1



  d rˆ  dzY(Q 2 , z, r )* qq ( x, r 2 )Y(Q 2 , z, r )
2
0
g p



d VM
1
*
|t 0 
|  d 2 r  dzYVM
(Q 2 , z, r )  qq ( x, r 2 )Y(Q 2 , z, r ) |2
dt
16
0
*
g p
d diff
1
*
dt
1
|t  0 
16
1



*
2
2
2
2
d
r
dz
Y
(
Q
,
z
,
r
)

(
x
,
r
)
Y
(
Q
,
z
,
r
)
q
q


2
0

Y ( z, r )
g  qq QCD wave function
 qq ( x, r 2 )
cross section for scattering of
qq pair on proton (dipol cross section )

1
2


 T , L ( x, Q )   d r  dz  YTf, L (r , z, Q 2 )  qq ( x, r )
g *P
2
2
0
0
f
2

3
YTf (r , z, Q 2 )  em2 eq2 {[ z 2  (1  z ) 2 ] 2 K12 (r )  mq2 K 02 (r )}
2
2

3
YLf (r , z, Q 2 )  em2 eq2 {4Q 2 z 2 (1  z ) 2 K 02 (r )}
 2  z (1  z )Q 2  mq2
2
K1 (r )  1/ r
for r  1
Q2~1/r2
K1 (r )   / 2 x exp( r )
for r  1
exp(-mq r)
GBW Model
K. Golec-Biernat, M. Wuesthoff
r2
 qq ( x, r )   0 [1  exp( 
)]
4 R02
1
R ( x) 
GeV 2
2
0
 x 

x 

 0

Scaling in
r 2 / R02

Geometrical Scaling
A. Stasto & Golec-Biernat
J. Kwiecinski
Parameters fitted to DIS F2 data:
0 = 23 mb  = 0.29 x0 = 0.0003
Parameters fitted to HERA DIS data: c2 /N ~ 1
0 = 23 mb  = 0.29 x0 = 0.0003
Saturation Model
Predictions for Diffraction
Geometrical Scaling
A. Stasto & Golec-Biernat
J. Kwiecinski
  Q 2 R02 ( x )
1  x
 
R02 ( x) 
2 
GeV  x0 

GBW model, in spite of its compelling success has some obvious
shortcomings:
The treatment of QCD evolution is only rudimentary
remedy => incorporate DGLAP into dipole cross-section
J. Bartels, K. Golec-Biernat, H. Kowalski
 qq ( x, r )   0 (1  exp( 
2
r
));
2
R0
1  x
2
 
R0 
2 
GeV  x0 
GBW
2 2
 qq ( x, r )   0 (1  exp( 
r  s xg( x,  2  C / r 2  02 )))
3 0
Recently: BFKL motivated Ansatz proposed by
Iancu, Itakura, Munier
The dipole cross section is integrated over the transverse coordinate
although the gluon density is expected to be a strongly varying function
of the impact parameter.
Impact Parameter Dipole Saturation Model
H. Kowalski
D. Teaney
hep-ph/0304189
b – impact parameter
Proton
well motivated:
Glauber
Mueller
Levin
Capella
Kaidalov
d qq ( x, r )
d 2b


2 2
 2  1  exp( 
r  s (  2 ) xg( x,  2 )T (b)) 
23



T(b) -
proton shape
 T (b)d
0
2
b 1
Derivation of the GM dipole cross section
P(b)  1 
d
2
2
NC
r 2 s (  2 ) xg( x,  2 ) r (b, z )dz
bdzr (b, z )  1
S (b)
2
 2 2

 exp  
r  s (  2 ) xg( x,  2 )T (b) 
 NC

T (b)   dz r (b, z )
d qq
2
d b

d qq
 2(1  Re S (b))
<= Landau-Lifschitz


2 2
2
2

2

1

exp(

r

(

)
xg
(
x
,

)
T
(
b
))


s
d 2b
2

3


probability that a dipole at b
does not suffer an inelastic
interaction passing through
one slice of a proton
S2 -probability that a dipole
does not suffer an inelastic
interaction passing through
the entire proton
t-dependence of the diffractive cross sections determines the b distribution
2
t D

D - transv. momentum (2 - d)

b - impact parameter (2 - d)
g p
 1

 

d VM
1
2 2
2
2
ib D
*
2

|  d r  d be  dzYVM (Q , z, r ) 1  exp( 
r  s xg( x,  2 )T (b)) Y(Q 2 , z, r ) |2
dt
16
23


0
*
d diff
~ exp( B  t )
dt


T (b) ~ exp( b 2 / 2 B )
2
TG (b)  exp( b / 2wG )
wG  4.25 GeV -2
  2
TGY (b)   d b exp( (b  b ) / 2wG ) K 0 (b / wE )
2

g *p
1

 2
 

2 2
*
2
   d r  d b dzY f (Q , z, r ) 1  exp( 
r  s xg( x,  2 )T (b))Y f (Q 2 , z, r )
23
f


0
2
2 
C
  02
2
r
g
1
xg( x,  02 )  Ag   (1  x)5.6
 x
Q2 > 0.25 GeV2
mu = 0.05 GeV
mc = 1.30 GeV
Fit parameters
g = -0.12 C= 4.0
Q02 = 0.8 GeV2
c2/N = 0.8
x < 10-2
GBW Model
r2
 qq ( x, r )   0 [1  exp( 
)]
4 R02
IP Saturation Model
d qq ( x, r )
d 2b


2 2
 2  1  exp( 
r  s xg( x, C / r 2  Q02 )T (b)) 
23


 g p ~ (W 2 )
*
tot ( Q
2
)
~ (1 / x)tot (Q
2
)
xg( x,  ) ~ (1 / x)
2
eff (  2 ( r ))
----- universal rate of rise of all
hadronic cross-sections
Smaller dipoles  steeper rise
Large spread of eff characteristic for
Impact Parameter Dipole Models
Saturation region
-------------------------------------------------------------------------------------------------------
d qq ( x, r )
d 2b


2 2
 2  1  exp( 
r  s xg( x,  2 )T (b)) 
23


All quarks
1
2

2r  dz  Y (r , z, Q 2 )  qq ( x, r )
f
T ,L
0
f
mu ,d , s  100 MeV
Charmed quark
1

2r  dz qq ( x, r )YTc, L (r , z, Q 2 ) 2
0
mc  1.3 GeV
mc  1.3 GeV
Gluon density
Charm structure function
Photo-production of Vector Mesons
Absolute values of cross sections are strongly dependent on mc
Absorptive correction to F2
from AGK rules
Example in Dipole Model
d
2

2
(
1

exp(


/
2
))




/ 4  ....
2
d b
F2
~
-
Single inclusive
pure DGLAP
Diffraction
2 2

r  s (  2 ) xg( x,  2 )T (b)
NC
A. Martin
M. Ryskin
G. Watt
Fit to diffractive data using MRST Structure Functions
A. Martin M. Ryskin G. Watt
A. Martin M. Ryskin
G. Watt
Density profile
2 2
D
 s (  2 (r )) xg( x,  2 (r ))T (b)
3
grows with diminishing x and r
approaches a constant value
Saturated State
- Color Glass Condensate

r2 
S  exp   D  

2 

2
S2 -probability that a dipole
does not suffer an inelastic
interaction passing through
the entire proton
Saturated state =
= high interaction
probability
S2 => 0
multiple scattering
S 2  e 1
rS - dipole size for
which proton consists
of one int. length
QS2  Saturation scale
= Density profile at the saturation radius rS
S = 0.25
S = 0.15
QS2 
2
rS2
Saturation in the un-integrated gluon distribution
kT factorisation formula
dipole formula
GBW - - - - - - - - - - - - - - - - - - - - x = 10-6
BGBK
___________________________________
x = 10-2
GBW - - - - - - - - - - - - - - - - - - - - x = 10-4
BGBK
___________________________________
-
numerical evaluation
x = 10-2
_
Diffractive production of a qq pair
Inclusive Diffraction
LPS - Method
END of Part I