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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

				
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