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Recombination and Scaling

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					 Hadronization at high pT:
Recombination and Scaling


     Rudolph C. Hwa
    University of Oregon

      Heavy-Ion Collisions
     McGill, June 24-28, 2003
Outline
     High pT in AA collisions: p/p anomaly
     Fragmentation versus Recombination
     Scaling in centrality and energy

     Hadron production at high pT
     Fractional energy loss
     Conclusion




                                             2
Hadronization in heavy-ion collision
                                                 rare




                             dominant                     dominant
                                                 rare
       before
                                              after

       p,n                              p,n
A                                                     A


                q, g         q, g

         hard collision calculable in pQCD                      3
Quark fragmentation                            q

 In vacuum                e+                               e-

                                               q
                                               

          q                            p           phenomenological
                                                   fragmentation function
                                                       D p /q(z)
                            not calculable         momentum fraction z<1
 In medium


      q                                    p
                                    fragmentation function
          energy loss
                                         modified

               XNWang, XFGuo, NPA696, 788 (2001)
                                                                        4
If hadronization in nuclear medium is by quark fragmentation,
then the p/p ratio
             R p/p = ( q     p ) / ( q   p ) in AA

should be about the same as R p/p in pp, since
• nuclear effects on quarks cancel in the ratio
• fragmentation into p vs p has to do with the masses and the
quark contents of the hadrons, so
• one would expect R p/p to be small in both cases.

But that is not the case:

                R   p/p   ~ 1    at pT ~ 3-4 GeV/c
                                in Au-Au at RHIC

                                                                5
             PHENIX PRL 88, 24301(2002)




 central
peripheral




                                          6
The black box of fragmentation


                   q
               1                                      p
                                                z


  A QCD process from quark to
  pion, not calculable in pQCD   Momentum fraction z < 1


                       Dp/q

   Phenomenological
   fragmentation
   function

                                         z
                                     1
                                                           7
Let’s look inside the black box of fragmentation.


                     q
                 1                                           p
                                                      z


                                                    fragmentation
         gluon radiation

                            quark pair creation


     Although not calculable in pQCD (especially when Q2 gets low),
     gluon radiation and quark-pair creation and subsequent
     hadronization nevertheless take place to form pions and
     other hadrons.


                                                                 8
Drawing a smaller circle
Let’s look inside the black box of fragmentation


                         q                   x1
                                             x2                p
                     1                                    z


                                          recombination
                                     z = x1 +x2 , x1,2 < z <
   In AA collision
                                     1
                         q                   x1
                                             x2                    p
                                                               p
            X        X       X
                                                               p

One can adopt the view that all hadrons are formed by recombination.
                                                                9
Coalescence
       Dover,Heinz,Schnedermann,Zimanyi, PRC44,1636(1991)
  Nuclear physics at intermediate energy or higher (AGS)
       p + n  d at low relative momentum
       n + , S + S, L + L  H dibaryons
  Input: thermal distribution
  Output: yields, no momentum distributions

Recombination
       Das,Hwa, PL68B,459(1977); Hwa, PRD22,1593(1980).
  Particle physics at high energy in p + p  p + anything
       u + d  p+ , d + s  K-, …
  Input: parton distributions in proton
                         
  Output:  meson distributions at large pL (or large xF)

Now, both models generalized to particle production
      at high pT in AA collisions at RHIC energies.

                                                            10
In recombination/coalescence approaches
        Fries,Meuller,Nonaka,Bass PRL 90,202303 (2003)
        Greco, Ko, Levai, PRL 90, 202302 (2003)
   Inputs: soft + hard components
        Soft: F(pT )  exp( - b pT )
        Hard: F(pT )  ( 1 + pT /p0 ) -n
  It is claimed that h+h recombination is not important
        h+h: p1T -n  p2T -n ~ (pT/2)-n  (pT/2)-n ~ pT-2n « pT-n
        s+s: exp[- b (p1T+p2T )] ~ exp( - b pT )
    Quark fragmentation is still important at very high pT
        q(p1T )  h(z p1T) : (p1T )-n  (z p1T)-n ~ p1T -n
         Frag > Recom at high pT
  Assumption: piT ~ pT/2
                                           p1T
Pion recombination function allows                          pT
                                           p2T
         p1T » p2T
                                      p1T
   then the result may be viewed as quark fragmentation
                                                    pT
                                      p2T
                                                                    11
Crucial issue in recombination model: Input Quark Distributions

  Soft:    thermal distr.
           (hydrodynam)
                              Quark Distr.         Hadron Distr.
   Hard:     pQCD
           (energy loss)
                                       hadronization

            evolution

Our approach    (tests Recomb only, w/o free parameters)

                                                       p
                                    q, q
                                       
                                             qqq
                                                       p
                            Hwa, CBYang, PRC67,034902(2003)   12
                              [HY-I]
It is easy to get p/p ~ 1 in the recombination model



Fq                             Fq




     x2          x1                 x2   x3   x1

          pion                           proton

                      quark momenta add
                                                       13
The results (2): Central p0 SPS  RHIC 130  RHIC 200

• Slow increase of power- law slopes with sqrt(s)




                                                        14
        The results (3): pp vs AuAu (peripheral & central)
                           p0 from PHENIX pp @ 200 GeV
       • Peripheral:               (Hisa Torii's talk) •   Central:



         70-80% PERIPHERAL                          0-10% CENTRAL
            Ncoll =12.3 ±4.0                          Ncoll =975±94




•    Consistent with
                                        Large suppression (increasing with pT)
    Ncoll scaling                         above pT~1 GeV/c compared15  to
                                          scaled pp.
                  The results (1): High-pT p0 spectra

                                            •    Spectra up to pT~10 GeV/c
• 30 Million events analyzed.
    “ min. bias” trigger, |z_vtx| < 30 cm


                                                                         




                                                                           16
                                                Min. bias, most central & periph.
     Scaling          (in centrality and energy)
             Npart         N                    pT
                                          z             scale change
                                             K (s, N)
                          1            dNp
             (z)  A(N)
                         2p       pT     dpT            A(N) normalization
                                               d
                                K(s, N) K(s,N)                change


   dN                                    (z)
pT dpT d


                            peripheral


            central


                                 pT                             z   17
Scaling distribution for all centralities and all energies




        Hwa, CBYang, PRL 90, 212301 (2003)
              [HY-II]                                        18
K(s,N) is normalized to 1GeV
at s=200GeV, N=350.
                               Ncoll = 0.44N1.33
                                                   19
KNO scaling of multiplicity distribution

                     variable rescaling

                   normalization rescaling




                                             20
        KNO scaling of nch in e+e- annihilation
5 GeV




                34 GeV




                                                  21
     Lepton-proton scattering


p           np                 np
                                




                                     22
Actually, (z) is only analogous to KNO scaling function.
      The KNO-type scaling function is Y(u).




                                z    pT
                            u    
                               z  pT 




                                        HY-II




                                                            23
To have a universal curve raises a question:
   QGP is supposed to give a different nuclear suppression effect
      from that of the normal nucleus.
   Peripheral (small Npart)     cold, uncompressed
   Central    (large Npart)     hot, dense (perhaps QGP)

A universal scaling curve (z) suggests that
       there does not exist a drastic change of suppression effect
       ---- at energies up to s = 200 GeV

Implications:
   either (1): no QGP yet
    or    (2): no drastic change of suppression effect
               i.e., large pT is not a good region to look for the
               signature of QGP.



                                                                     24
Is there a universality in fragments
  in intermediate energy nuclear collisions?




                                               25
Outline
     High pT in AA collisions: p/p anomaly
     Fragmentation versus Recombination
     Scaling in centrality and energy

      Hadron production at high pT
      Fractional energy loss
      Conclusion




                                              26
Outline
     High pT in AA collisions: p/p anomaly -- goal
     Fragmentation versus Recombination -- vehicle
     Scaling in centrality and energy    --   avenue

      Hadron production at high pT
      Fractional energy loss
      Conclusion




                                                      27
Outline
     High pT in AA collisions: p/p anomaly -- goal
     Fragmentation versus Recombination -- vehicle
     Scaling in centrality and energy     --   avenue

      Hadron production at high pT    --    DRIVE ON
      Fractional energy loss
      Conclusion




                                                      28
Recombination model

 d 3 Np  3    3
        d p1 d p2
E 3             F ( p1 , p2 )Rp (p1, p2 , p)          Hwa(II) (1980)
  d p    E1 E2

 Apply now to high transverse momentum
  Integrate out y (at midrapidity) and f (collinear)
         and change to scaling variable z

                                                          HY-I(03)
        dN p
               dz1 dz2 z1 z 2 F(z1 , z 2 )Rp (z1, z2 , z)
         zdz
        Parton distributions                      Recombination
                                                     function
 Should be for partons just before            Should depend on the
 recombination, thus after hydro-             wave function of the
 expansion and Q2 evolution.                  pion in terms of the
                                                constituent quarks

                                                                29
  Recombination model

    d 3 Np  3    3
           d p1 d p2
   E 3             F ( p1 , p2 )Rp (p1, p2 , p)          Hwa(II) (1980)
     d p    E1 E2

    Apply now to high transverse momentum
     Integrate out y (at midrapidity) and f (collinear)
            and change to scaling variable z

                                                             HY-I(03)
           dN p
                  dz1 dz2 z1 z 2 F(z1 , z 2 )Rp (z1, z2 , z)
            zdz

Since q and q are copious in AA collisions,          Recombination
assume                                               function
           F(z , z )  F (z )F (z )
               1   2      q   1    q   2

 Further, since  / p ~ 0.7const, assume
                p                                   1
                                                       (z1 + z 2  z)
                                                    z
       Fq (z)  (0.7)1/ 3 Fq (z)                                    30
Recombination function Rp(z1, z2,z)
 The wave function of hadrons in terms of momentum fractions
can best be determined in the Valon Model.
                                             Hwa(I) PRD22,759(80)
 Proton: use parton distributions from
                                             HY, PRC66,025204(02)
       deep inelastic scattering
                                   CTEQ4LQ, http://zebu.uoregon.edu~parton



     p                    u
                  U               u
                      U
                 D
  Pion:   use Drell-Yan production data in pp collision
                              Sutton,Martin,Roberts,Stirling,PRD45,2349(1992)
                      U
                      
 p                D                        GD/p(x)
                                      e+
             U
                                      e-
 p                                                                         x
                      U                              0           1    31
              D
Valon distributions are independent of the virtuality Q
                       HY (02)




                                                          32
Recombination function        (time reversal of valon distribution)
                       1                 1
  Rp (z1 , z 2 , z )  2 Gp (x1 , x2 )  2  (x1 + x2  1)
                      z                 z
                                     1
      xi  z i / z                  (z1 + z 2  z)
                                     z

  It is as likely for x1 >> x2 as for x1 ~ x2 .

 Recall
           dN p
                  dz1 dz2 z1 z 2 Fq (z1 )Fq (z 2 )Rp (z1 , z 2 , z)
            zdz
 From the observed pion distribution, Fq(z) can be
    determined without assumptions about their origin.

           Fq (z)  15(z 2 + 0.47z + 0.72) 4.25
                                                                        33
Quark distribution (just before hadronization)


        ~ exponential


                                HY-III(03)




                            Power-law: ~z-8.5




                                                 34
This quark distribution Fq(z) can then be used to calculate
        qqq recombination to form proton


dN p
       dz1dz 2 dz 3z1 z 2 z3 F(z1, z2 , z 3 )Rp (z1, z2 , z 3 , z)
zdz
         F(z1, z 2, z3 )  Fu (z1 )Fu (z2 )Fd (z 3 )
Proton recombination function            valon distribution of proton
                                                        HY(02)

      Rp (z1, z2 ,z3 ,z)  R0 z 2 Gp (x1 ,x 2 , x3 )

      Gp (x1 ,x 2 ,x3 )  g(x1x2 ) x3 b (x1 + x2 + x3  1)

                       1.755, b  1.05
                                                                       35
In the invariant form for dN p there is no dependence on
                             zdz
the proton mass: the formalism is valid only for pT values
where the mass effect is insignificant.
       We take it to correspond to z  2



To extend the calculation to z<2, one would have to consider
recombination in the rest frame by Lorentz boost, assuming
some form of the wave function of the proton.

Using the relativistic formalism we now can calculate
the p/p ratio.
                           dN p / zdz
                Rp / p 
                             (z)

                                                               36
                HY-I(03)




PHENIX (preliminary)




          pT in GeV/c      37
String fragmentation cannot explain the large p/p ratio.
       Baryon junction was suggested.
       (There are basic questions about strings for AA
       collisions.)


In the recombination picture,
       no exotic mechanism is needed.
       Just add the quark momenta to form proton.




                                                           38
                             Color strings
e+e- annihilation
                        e+                   e-


pp collision
                    p                             p




 AA collision


                A                                     A


                                                      39
Energy loss

      Centrality dependence of quark distribution Fq
                  Fq(x,N)            Fq(z)   scaling



 Fq                                       Suppression F(x,N) < F(x,2)
                                          Shift        F(x-X,N) = F(x,2)

                                 N=2
                            N
            x-X x

· In pQCD, calculate quench factor: medium effect vs vacuum (N=2)
         X  x             Baier,Dokshitzer,Mueller,Schiff,
                                  JHEP 0109 (2001) 033
                                                                     40
Pion distribution in AA collision
        dN p
              (z)  Y(u)             x  pT / p0
        xdx                                            10 GeV/c
Normalized probability P(x,N)

       P(x,N)  (dNp / xdx)/  dxxdNp / xdx
Shift X:       P(x-X,N) = P(x,2)
   Using the scaling distribution (z), one can solve for X.
       Get:              X(x,N) = x0(N) + f(N) x
                                                           HY-IV(03)
Fractional energy loss
                                         small
       X/x ~ f(N) ~ f N,        f5.3104,      (at N=350, X/x ~ 0.17)
Quark distribution behaves similarly, due to scaling.

       Data disagree with pQCD’s prediction: f ~ 1/pT
                                                                       41
Origin of Scaling
For DN = , there is a corresponding Dx = .
                                                          x
      Dx    DN                                    x-           N
If           ,
       x     N                                                  N+

then F(x,N)  F(z) scaling, with z = x / K(N), requires

               N  
       K(N) ~  
              N                   except for N very small
               0

 We re-examined scaling and found

                                                      HY (very recently)

 Implication
                               
                        13.4
                   x            N
                                                                      42
    new scaling
    K(N)  N 0.07 46




-




                        43
CONCLUSION
 Hadron production at high pT in AA collisions
 • Recombination is the mechanism     (can’t ignore the low-pT
                                         soft partons)
 • No anomaly in the p/p ratio
 • Scaling in centrality and energy
 • Fractional energy loss is independent of pT and increases
         linearly with Npart (or fractional increase of N)



 pQCD not reliable at pT < 10 GeV for RHIC experiments.
 Centrality scaling : where are the effects of QGP?



                                                                 44

				
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