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QCD issues in photon-photon
total cross-section : why we
need a photon collider
Giulia Pancheri - INFN-Frascati
Giulia Pancheri - INFN-Frascati
December 12, 2007Hiroshima University,
Higashi-Hiroshima, Japan
AdA : Where e+e- collisions were born
LEI 2007, Hiroshima, December 2007 1 12/28/07
Points for discussion
• The data
• Why the total cross-section is interesting
• A QCD model for protons to account for
– The rise
– The saturation, i.e. Froissart bound
• Application to γγ
LEI 2007, Hiroshima, December 2007 2 12/28/07
The data
LEI 2007, Hiroshima, December 2007 3 12/28/07
Start with
present
γγ σtot data
Large errors, scattered
values:
•Difficult to establish proper
normalization
•Need to check data
normalization from LEP
(there is an issue 2007
LEI 2007, Hiroshima, December here) 4 12/28/07
The region 2-10 GeV
• For a good model low
energy data are crucial
for normalization
• A photon collider at low
energy could give very
useful information
LEI 2007, Hiroshima, December 2007 5 12/28/07
The region 10-100 GeV
• In this region one sees
the beginning of the rise
• Errors are large at the
high end, where QCD
models could be
distinguished from one
another
• Need to have a photon
collider with good beam
energy resolution to fix
the rate of increase
LEI 2007, Hiroshima, December 2007 6 12/28/07
Beyond 100 GeV
• Comparison with
proton σtot can really
only be done if data
are available
for Wγγ ~ 500 GeV
LEI 2007, Hiroshima, December 2007 7 12/28/07
Why the total cross-section
is interesting
LEI 2007, Hiroshima, December 2007 8 12/28/07
The strong limit of QCD
• The most important yet unresolved
issue in hadron physics and QCD is the
infrared behaviour of αs
• One way to get information on αs is
from total cross-section studies
As I shall try to show
with a toy model
LEI 2007, Hiroshima, December 2007 9 12/28/07
Do all total cross-section look alike?
LEI 2007, Hiroshima, December 2007 10 12/28/07
Do all total cross-section
look alike?
• Yes
– They all start falling and s-η sε
then rise with energy
and
• No
– They fall with different slopes
at low energy
– They may be rising with different slopes
at high energy
LEI 2007, Hiroshima, December 2007 11 12/28/07
Difference at low energy is
understood from unitarity and
analiticity
• Quantum numbers in the s-channel give rise to
different resonances in the very low region
• Quantum numbers in the t-channel bring in different
Regge pole exchanges and a decrease with energy:
through Finite Energy Sum Rules related to s-
channel resonances and different quantum numbers
in s and t channel
LEI 2007, Hiroshima, December 2007 12 12/28/07
Difference at high energy?
• Not well understood yet
• Pomeron exchange was supposed to give
universal behaviour but one now
distinguishes
– Soft Pomeron : same for everybody
– Hard Pomeron :could differ
• Here is where QCD comes in
LEI 2007, Hiroshima, December 2007 13 12/28/07
Models for total cross-section
• Is factorization all
there is to go from pp
to γγ?
• Can one disentangle
the QCD role ?
• How much can
ILC/LPC help ?
A.de Roeck, R. Godbole, A. Grau, G.Pancheri,
JHEP 2003
LEI 2007, Hiroshima, December 2007 14 12/28/07
sε : Should ε be the same for all hadronic
cross-sections?
• Yes if the model
– is based on Regge poles
and a universal Pomeron
pole exchange
σ=Bs-η + Asε
• Not necessarily if
– The model has some
connection with QCD and
parton densities play a
role
LEI 2007, Hiroshima, December 2007 15 12/28/07
A fit to LEP data shows that ε is not the
same for proton and photon cross-
sections
LEI 2007, Hiroshima, December 2007 16 12/28/07
σ=Bs-η + Asε+ Csε1
• Fit3
C≠ 0 ε=0.093
ε1=0.418
• Fit 1
C=0 ε=0.250
• Fit2
C=0 ε=0.093 as in pp A.de Roeck, R. Godbole, A. Grau, G.Pancheri, JHEP 2003
LEI 2007, Hiroshima, December 2007 17 12/28/07
A QCD model to account for
The rise
The saturation, i.e. Froissart
bound
LEI 2007, Hiroshima, December 2007 18 12/28/07
The Bloch-Nordsiek Eikonal Minijet model
includes kt resummation
R.Godbole, A. Grau, G.Pancheri, Y.Srivastava PRD 2005
A. Corsetti, A. Grau, G.Pancheri, Y. Srivastava PLB 1996
A. Achilli, R. Godbole, R. Hedge. A. Grau, G.P., Y. Srivastava, TbP in PLB
1. Multiple parton interactions : optical theorem and
eikonal representation for Tel(s,t)
2. Hard scattering to drive the rise due to 1/x
3. Soft gluons down to zero momentum to tame the
rise
LEI 2007, Hiroshima, December 2007 19 12/28/07
1. Hard component of scattering responsible
for the rise of the total cross-section
DGLAP
Parton densities
x1
x2
LEI 2007, Hiroshima, December 2007 20 12/28/07
Jet cross-sections at LO
Using current
DGLAP evoluted
PDF’s :
GRV, MRST, CTEQ
LEI 2007, Hiroshima, December 2007 21 12/28/07
1.
Different ε
for different
Hard PDFs
component of
scattering
responsible sε
type
for the rise of behaviour
the total
cross-section
LEI 2007, Hiroshima, December 2007 22 12/28/07
In mini-jet models densities make all the difference between
photon and proton processes
Proton-proton and γ−proton and γγ
proton-antiproton
Most commonly used Most commonly used
densities densities
• GRV • GRV
• GRS
• CTEQ
• CJKL
• MRST
LEI 2007, Hiroshima, December 2007 23 12/28/07
Choose ptmin = 1÷2 GeV for mini-jets
and parton densities
For photons, LEP data
suggest
ptmi ~ 1.3 ÷ 1.8 GeV
n
• Gluck Reya Vogt
• Gluck Reya Shielbein
• Cornet Jankowski Lorca
Krawczyk
LEI 2007, Hiroshima, December 2007 24 12/28/07
3. eikonal
transformation
implies multiple
scattering and
requires impact
parameter
distribution in
hadrons
LEI 2007, Hiroshima, December 2007 25 12/28/07
2. soft gluon emission from scattering
particles which softens
the rise and gives b-distribution
Soft gluon emission factor
LEI 2007, Hiroshima, December 2007 26 12/28/07
Soft gluon emission factor
~
qmax is the maximum transverse momentum
allowed by kinematics to single soft gluon emission
in a given hard collision, averaged over the parton
densities.
M. Greco and P. Chiappetta
LEI 2007, Hiroshima, December 2007 27 12/28/07
qmax for ptmin=1.15 geV
CTEQ GRV98
GRV
CTEQ
LEI 2007, Hiroshima, December 2007 28 12/28/07
What about the kt 0 limit for αs ?
Modeling the infrared behaviour
• frozen
•Our choice : singular but integrable, phenomenological choice
LEI 2007, Hiroshima, December 2007 29 12/28/07
Zero momentum quanta
• Soft gluons need to be Models for infrared behaviour
resummed if they are
indeed soft ≈1/k
• Resummation implies
integration over dkt
• What matters will be
∫αs(kt )dkt f(kt)
and not αs(0)
LEI 2007, Hiroshima, December 2007 30 12/28/07
Ansatz
The saturation mechanism
which reinstates the
Froissart bound comes from
zero momentum gluons which
are emitted in an indefinite
number
LEI 2007, Hiroshima, December 2007 31 12/28/07
Model results for the
proton-proton case and
application to γγ
LEI 2007, Hiroshima, December 2007 32 12/28/07
How the model works
• Choose ptmin = 1÷2 GeV for mini-jets
• Choose parton densities
• Calculate minijet x-section
• Calculate qmax for soft gluons
• Calculate A(b,s) for given qmax
• Calculate nhard (b,s)=A(b,s) σjet(ptmin,s)
• Parametrize nsoft
• Evaluate n(b,s)= nsoft + nhard
• Eikonalize σtot≈2∫d2b [1-e-n(b,s)/2]
LEI 2007, Hiroshima, December 2007 33 12/28/07
Comparison
with proton
data
R.Godbole,
A. Grau
R. Hedge
G. Pancheri
Y. Srivastava
Les Houches 2005
Pramana 67 (2006)
GGPS PRD 2005
LEI 2007, Hiroshima, December 2007 34 12/28/07
For all pdf’s
• For different PDF , with soft gluon emission to give
an energy dependent size and QCD hard gluon
minijets to drive the rise
• All the Bloch-Nordsieck type curves
LEI 2007, Hiroshima, December 2007 35 12/28/07
Now apply the model to γγ
LEI 2007, Hiroshima, December 2007 36 12/28/07
Applying the mini-jet+IR gluons to γγ
• Factorization from pp
Or
• Using photon densities
in minijets and soft
gluon emission factors
LEI 2007, Hiroshima, December 2007 37 12/28/07
Factorization
from pp
• Multiply band of our
model predictions for pp
by appropriate factor
1/330 which reproduces
γp
• Multiply pp by (factor)2
LEI 2007, Hiroshima, December 2007 38 12/28/07
Using photon densities in minijets and
soft gluon emission factors
• Choose densities
• Calculate minijets
• Calculate qmax for chosen densities and b-
distribution from soft gluons
• Calculate nhard
• Eikonalize with
– nsoft from pp
– Normalization for nhard from Vector Meson Dominance
and Quark Parton model
LEI 2007, Hiroshima, December 2007 39 12/28/07
Eikonalize σtot≈2Phad∫d2b [1-e-n(b,s)/2]
and compare with data
Phad is a
Phenomenological
input describing
the hadronic
content of the
photon in eikonal
models
R.Fletcher, T.Gaisser. F.Halzen, 1993
LEI 2007, Hiroshima, December 2007 40 12/28/07
Conclusions (I)
• We have built a model for the total
cross-section which
– Incorporates hard and gluon effects
– Satisfies the limits from the Froissart
bound
– Can be used to study other minimum bias
effects
– Easily extended to γ p and γ γ
LEI 2007, Hiroshima, December 2007 41 12/28/07
Conclusions (II)
• Predictions at ILC vary according to which
densities better describe the behaviour at low
x
• Total cross-sections measurements in
Collider mode would allow clean information
on γγ cross-sections, reducing the errors due
to modelling of diffractive components
• Even in regular mode, difference in the
model predictions are measurable and can
give insights into the soft or non perturbative
region of QCD.
LEI 2007, Hiroshima, December 2007 42 12/28/07
Back up slides
LEI 2007, Hiroshima, December 2007 43 12/28/07
The saturation mechanism
which reinstates the
Froissart bound
LEI 2007, Hiroshima, December 2007 44 12/28/07
At very large energies :
LEI 2007, Hiroshima, December 2007 45 12/28/07
soft gluon resummation effects
Virtual Energy-momentum
gluons conservation factor
for real soft gluons
Model
LEI 2007, Hiroshima, December 2007 46 12/28/07
From power law to log behaviour
Main
result
LEI 2007, Hiroshima, December 2007 47 12/28/07
The hard cross-section
• Mini-jet cross-section
Σ∫ densities ∫dpt dσ/dpt
LEI 2007, Hiroshima, December 2007 48 12/28/07
σjet for ptmin=1.15 GeV
GRV
GRV98
green band MRST CTEQ
LEI 2007, Hiroshima, December 2007 49 12/28/07
Soft resummation
Probablity of total KT from infinite # of soft
gluons
∫ d2b eiK b exp{-∫d3n(k)[1-e-ik b]}
T t
depends upon single gluon energy
maximum : use Kinematics
minimum : 0 from Bloch-Nordsieck theorem
LEI 2007, Hiroshima, December 2007 50 12/28/07
Role of resummation
An infinite number of soft quanta
• down to zero momentum but how?
next slides
• Up to an energy dependent limit qmax
– Higher hadron energy possibility
of more small x partons with “high energy”
(≈1-2 GeV) higher qmax
LEI 2007, Hiroshima, December 2007 51 12/28/07
Resummation of soft gluons
down to kt=0
• Gluon emission in kt changes the collinearity of initial partons
• And for same energy and ptmin, acollinearity of initial partons will
bring loss of luminosity of the parton beams and parton-parton
cross-sections will decrease
• As the energy available for soft gluon emission increases, so
does the acollinearity of the parton-parton collision
• The rate of rise of total cross-sections due to rising minijet
cross-section is reduced (softened by) by soft gluon emissions.
• Softening effect more important the more singular αs
LEI 2007, Hiroshima, December 2007 52 12/28/07
Which parameters in soft
emission?
ptmin and p regulate how large is the
maximum energy , but PDF’s also play a
role
• Ptmin hard scattering lowest scale
• p infrared (integrable) behaviour
LEI 2007, Hiroshima, December 2007 53 12/28/07
An aside on the Froisart
bound and minijet cross-
sections
LEI 2007, Hiroshima, December 2007 54 12/28/07
About the Froissart bound
and QCD minijets
LEI 2007, Hiroshima, December 2007 55 12/28/07
Interpolating from the
infrared to asymptotic freedom
• Singular but integrable
• Singularity regulated by p < 1
LEI 2007, Hiroshima, December 2007 56 12/28/07
QCD Mini-jets violate the
Froissart bound
• Consequence of infinite range of QCD
• One needs to introduce a finite distance
of the interaction
• The eikonal does it through the hadron
finite size
LEI 2007, Hiroshima, December 2007 57 12/28/07
How about nsoft?
•Parametrized with a constant σ0
•With ptmin dependence through A(b,s)
LEI 2007, Hiroshima, December 2007 58 12/28/07
Minijets : how to reduce
their growth?
• Embedding the minijets into the eikonal
but
• It depends on how A(b,s) 0
as b infinity
LEI 2007, Hiroshima, December 2007 59 12/28/07
Finite size of hadrons
• The finite size can be introduced through
the Form Factor
A(b)~ e -b constant as b ~ very large :
not enough to tame the rise because the growth of
σjetPDF is too strong!!
or
energy dependent soft gluon emission where satisfaction of the
energy dependent soft gluon emission where satisfaction of the
Froissart bound would imply that the the finite range of the interaction is
Froissart bound would imply that the the finite range of the interaction is
restored through soft gluon emission down into the infrared
restored through soft gluon emission down into the infrared
LEI 2007, Hiroshima, December 2007 60 12/28/07
For all pdf’s
• For different PDF , with soft gluon emission to give
an energy dependent size and QCD hard gluon
minijets to drive the rise
• All the Bloch-Nordsieck type curves
LEI 2007, Hiroshima, December 2007 61 12/28/07
Photons and QCD
• Photons probe the QCD vacuum
• How a photon becomes a hadron
• Two photons have a unique signature in
producing scalar or pseudoscalar resonances
• If new strong WW interactions would be
detected at LHC possible new scalar states
could be probed to extract threshold
dynamics information not unlike what
happens in γγ π0π0 around the σ-meson
• One can study γγ γγ if enough luminosity
– Light-by-light and hadronic contributions
– Insight into the trace anomaly
LEI 2007, Hiroshima, December 2007 62 12/28/07
Total cross-sections are a testing
ground of our understanding of
QCD beyond perturbative regime
• A popular models for
total cross-section
from L3 fits
LEI 2007, Hiroshima, December 2007 63 12/28/07
σtot=102 mb
at LHC GRV
LEI 2007, Hiroshima, December 2007 64 12/28/07
Comparison with proton data(I)
R.Godbole,
A. Grau
R. Hedge
G. Pancheri
Y. Srivastava
Les Houches 2005
Pramana 67 (2006)
GGPS PRD 2005
LEI 2007, Hiroshima, December 2007 65 12/28/07
Maximum soft gluon energy
• q1 and q2 : any
two partons
• X : the 2-jet final
state
• Q2≥4 p2tmin
• qmax depends on
x1,x2
• We average over
densities
LEI 2007, Hiroshima, December 2007 66 12/28/07
Soft gluons give b-distributions
In eikonal representation
σtot≈2∫d2b [1-e-n(b,s)/2]
• n(b,s)=average # of collisions at distance b,
at energy √s
• b-distribution is needed
Our ansatz:
b-distribution =
Fourier transform of soft gluon Kt distribution
LEI 2007, Hiroshima, December 2007 67 12/28/07
