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					      Proposing a new constraint for
       predictions of         tot.c.s.
           and      ratio at LHC
                    Muneyuki ISHIDA
                   Meisei Univ., Japan
                New Trends in High Energy Physics,
                     Yalta, Sept. 16-23, 2006
              In collabotation with Keiji IGI
K.Igi and M.Ishida, Phys.Lett. B 622 (2005) 286;
Prog.Theor.Phys. 115 (2006) 601; hep-ph/0605337      1
           Contents of the talk
• We propose to use rich inf. on the
  tot.cross section      in low energy regions
  through duality to predict the         and
                       ratio at very high energies.
• Using FESR as constraints for high energy
  parameters, we search for simultaneous best
  fit to     and       ratio up to some energy to
  determine high-energy parameters.
• We then predict          and       in the LHC
  energy regions.
                                                2
  Last Year : Present. by K. Igi
• We analyzed crossing-even forward scatt.
  ampl.
• Using one FESR as a Constraint,
        n=1 mt. FESR called Constraint(2),
      and       up to Tevatron energy
 (Ecm=1.8TeV) are fitted simultaneously.
• Two conflicting measurements at 1.8TeV.
• We analyzed two cases:
  E710/E811 and CDF, respectively.           3
      CDF
             E710
             E811




E710/E811 data input : green lines
CDF datum input :         blue lines;
       K. Igi presented in ’05.         4
Table 3
                       
  The predictions of  tot  and     at LHC energy , s  14TeV
  Last Year presented by Prof. Keiji Igi.

                     
                   tot       s  14TeV             s  14TeV   
     E710
     E811

     CDF

Large systematic difference between E710/E811
input and CDF input.
Prediction
at LHC :                                                                   5
      This Year : New topics
• We propose a new Constraint(1),corres. to
  n= -1 mt. FESR, free from unphys. region.
• We have used two constraints, (1) and (2),
  and search for the best fit of ,     up to
  SPS energy(0.9TeV).
   prediction at Tevatron(1.8TeV) to invest.
  E710/E811 and CDF discrepancy.
• We have also analyzed crossing-odd ampl.
  and obtain pred. at LHC(14TeV).
                                            6
               Such a kind of attempt
               K.Igi, PRL 9(1962), 76
• The following sum rule has to hold under the
  assumption : there is no sing.with vac.q.n.
  except for Pomeron(P).
           f2                                        N
                      dk  tot  k    tot     
                                          
                        N
 1       a                                      
  M            M     0                                   
   0.0015        -0.012           2.22
• Evid.that this sum rule not hold
    prediction of the P’ traj. with  P  0.5 .       '


  f meson         was discovered on the P’ .
                                                                 7
      Brief survey of our prev.work on
       sat.of Froissart -Martin bound:
                    K.Igi and M.Ishida,PRD66(2002)034023
• As is well known,
               tot.c.s.
 increase above 70GeV.
• It had not been known,
however, if this increase
behaved like

 consistent with
 Froissart-Martin
 bound before 2002.

                                                     8
• We proposed to use rich
  inf.of      in low energy
  reg. to investigate high
  energy behaviours.
• FESR(1),(2) as constraints
• Searched for best fit to
       above 70 GeV.
• We can conclude that our
  analysis strongly favours
              behaviours.

                               9
    Main Topic :Derivation of new
           constraint(1)
• Crossing-even forward scatt.amp.
                                    : proton Lab. Energy
                                    :   Lab. momentum



• We assume (log ν)2 behaviour


 at high energies:
• Difference between real ampl. and its asymp. form
                                                      10
     We can write down FESR in the spirit of P’ sum rule.




        Unphysical region    physical reg.


If               this eq. reduces to so-called P’ FESR.
• This FESR suffers from unphys. region com. from
  boson poles below      thres.                  11
• FESR, not at ν=M but at some
  intermediate energy ν=ν1 .



• Difficult to estimate       exactly enough.

• Taking difference between N=N1 and N=N2.




Constraint(1) : relation between                 .
                                                12
 n= -1 mt. FESR, free from unphysical region.
                     Constraint(2)
 The second FESR corres. to n=1 mt. FESR:


 Unphysical region    Physical region




• n=1 mt. FESR.
• relation between
• Contributions from unphys. region are negligible,
   due to the suppression factor ν.
                                                      13
    Contribution to Constraint(2)
 from unphysical region: negligible
• Why? Ave. of im. part from boson poles
  below p p threshold
                                 2 2
= smooth extrapolation of t-ch q q exch.
  contribution from high energy to   M
  due to FESR duality.




                                           14
  • Since Im Fq q    Im F   ,
                                                         
                                     2 2




1st term                             
                                                    d  0  Im F  d
                                                                          
                    M                                       M
of LHS of
FESR(2)
                
                0
                         Im Fq          2
                                             q2


                                                    M2
                                     d                          k  o
            M
                  Im f      pp
                                                        Im f   pp
                                                                                3.2GeV
            0   2                                    4

             1                                                                  2nd term
                            k  tot  k  dk  3403  20GeV
                                   
                        N
                    
                             2
                                                                                of LHS of
            4          0
                                                                                FESR(2)

    So, resonance contributions from unphysical regions
    is less than 0.1% of resonance contributions from phys. reg.
       Therefore : negligible
                                                                                            15
                        
               The     ratio
         
• The      ratio = the ratio of the real to
  imaginary part of F  
                       




                              : subtraction const.   16
            General approach
                                       
• Constraints(1),(2) , formula of  tot and
        
  the  ratio : our starting point.
• In Constraints(1),(2), the integrals of
                are expressed in terms of
  high-energy parameters: c0 , c1 , c2 ,  P' .
• The integrals are evaluated with error
  (less than 1%).
   Constraints(1),(2) 
                                                  17
• 3 Parameters :
• We search for simultaneous best fit to
  the data of  tot  k  and   k  to determine
                             

  values of params. giving the least  .
                                             2


• fitted energy region:             and below
  SPS energy:
• We can predict      and
  in Tevatron energy and in LHC
  energy, using these params.
                                                18
           Data:
 We use rich data
  (PDG 2004)

to evaluate the
relevant integrals
of cross sections
appearing in
Constraints(1),(2).

                      19
        Evaluation of integrals:
                        in Constraint(1).
                  in Constraint(2).
Phenomenological fits to           ,    in



   c0     β       d     f       χ2/NF
   6.34   11.08 5.28   -0.15 36.6/(70-4)      7pt. below 10GeV
                                              are removed by stat.
   6.32   4.25   -12.6 24.4    71.8/(103-4)   method(Sieve algor)
                              : fixed
                                                            20
Based on these phenom. fits,
For Constraint(1): (N1,N2,ν1)=(10,70,40) GeV

 Integrals for


 Averaging



   Error is less than 1%, negligible.
                                           21
 For Constraint(2): N=10GeV.
  Integral region is devided into two parts:


 polygonal line graph↑     ↑phenom. fit
 Area:
 Error:

 Integrals for




Consistent with prev. estim.:

       Error is less than 1%, negligible.      22
                Constraints
• Constraint(1) : (N1,N2,ν1)=(10,70,40)GeV
   3.316βP’+31.98c0+141.1c1+610.9c2=230.81
 (normalized : 0.104βP’+c0+4.41c1+19.1c2=7.22)
• Constraint(2) : N = 10 GeV
   140.7βP’+383.6c0+781.6c1+1635.c2=3396.3
 (normalized : 0.367βP’+c0+2.04c1+4.26c2=8.85)



                                                 23
                           data
• When        and     data points
  are listed at same value of k,
  we make               data
  points by averaging them.
  27(11) points for          up to
  ISR,k=2094GeV(               ).
• 5(1) points in SPS energy region:

• In summary, we have
  32(12) points of
  in                           .      24
                   Analysis:
         
• Both  tot and                   
                      (actually Re F ) data in
                     (SPS) are fitted simult. .
• Constraints(1),(2) are used.


• The fitting is done by three parameters:



                                             25
Result




LHC                     LHC
      input E. region




                         26
  Table 1
   The values of parameters and χ2 in the best fit
                  
   N F and N N  are the degree of freedom and
                    
   the number of  tot  
                         
                                data points in the fitted energy region.


      c2          c1           c0      P    '


   0.0473 - 0.173 6.32 7.32                      10.01


             NF
             2
                          N
                           2
                                       N
                                         2


         15.9/41 10.5/32 5.4/12

The reduced χ2 and respectiveχ2 devided by
# of data points for      and    are less
than unity : Fit is successful .          27
   Predictions for    and
     at Tevatron and at LHC

By using these
values of params.,
we can predict
    and       at     CDF .
Tevatron and at
                         . E710
                             E811
LHC.                                LHC
                                          28
          Prediction at Tevatron
• We have investigated a possibility to resolve
  the discrepancy among the
  E710, E811 and CDF at 1.8TeV.
          E710
          E811
          CDF

• Our pred.

  E710 is consistent with the prediction in one standard deviation.
  Both CDF and E811 are in two standard deviation.
• E710 preferable,but CDF,E811 not excluded.
                                          29
     Prediction at LHC(14TeV)
Based on the result of analysis of crossing-even
amplitude, we obtain the prediction at LHC,




                                                   30
Analysis of Crossing-odd Amplitude

• We can also analyze the crossing-
  odd amplitude :

• It is simply described by exchange of
  vector-meson trajectories (ρ,ω).
• Its Asymptotic form:


                                          31
              Its benefit

• Analyses of both even and odd
  amplitudes make it possible to use all
  the        data, not selected for
  even-amplitude.
• Number of fitted data is about 5 times
  compared with the analysis of even-
  amplitude.

                                       32
          Formula
•               Using the crossing-odd property,




•   amplitude




                                           33
          General approach
• The αV is determined by the fit to the
  difference between        and      .
• Fixing the value of αV, then, 4 Data sets of
  are fitted simult..
• Constraints(1),(2) are used for even-
  amplitude.
• Number of parameters is 4:                ,
  determined by the fit.
    Prediction of         and      at LHC. 34
                    Value of αV
• Using two constrs,
both       and
are fitted
simultaneously in
20GeV < k < SPS.




: fixed in the following
                                  35
analysis.
  /GeV
                                   TeV


         Result of the analysis of
         both even and odd
         amplitudes using two
         constraints. Data in
         10GeV < k < SPS are
         fitted simultaneously.
LHC
         Dashed lines : 1-standard deviation   36
   Table 2 Result of Analysis of both even and odd
   amplitudes: ND , Nσ and Nρ are the number of data points
  in the fitted energy region: 10GeV < k < SPS.



      c2         c1        c0      P   '


   0.0486 -0.201 6.44             7.14      11.43 3.64




266.3/(250-4) 24.4/58 9.8/12 112.3/112 119.8/69

The reduced χ2 for total data is almost unity,
although the fit to ρ(pp) is not successful.
                                                              37
 Table 3 Result of the fit to the data set removing ρ(pp).
           fitted energy region: 10GeV < k < SPS.


      c2        c1        c0      P   '


   0.0482 -0.203 6.45           7.11       6.36 3.66




142.7/(181-4) 24.7/58 7.3/12 110.7/112                ------

Fit is successful. The result is almost the
same as that of the fit including ρ(pp) data.
                                  reliable !                   38
            Prediction at LHC
• By using these values of parameters we can
  predict the total cross section and ρ ratio at
  LHC(14TeV).


  • Consistent with the result of analysis of
   crossing-even amplitude.

# of data becomes 5 times, but error does not reduce
so largely, since the added data are below 2100GeV.
                                                  39
 Comparison with other groups
• Predicted central values of 
                                                   pp
                                          :       tot

  in good agreement with Block and Halzen
       tot  107.4  1.2mb,
         pp                   pp  0.132  0.001
   They adopt a statistical method of data selection, Sieve algorithm,
   and obtain smaller errors without systematic errors.
   They also used duality but different method with ours.

• In contrary to our results, however, their
  values are not affected so much about
  CDF, E710/E811 discrepancy.
   In our case, the measurement at LHC energy will discriminate
   which solution is better at Tevatron.                             40
• Our prediction has also to be compared
  with Cudell et al.(PRL 89(2002) 201801)
                                + 4.1
        pp
        tot    111.5 1.2syst  2.1stat mb,
                                    + 0.0058
      pp  0.1361  0.0015syst  0.0025stat
  who’s fitting techniques favour the CDF
  point at s  1.8TeV .
• Predictions by Bourrely et al. based on
  the impact-picture phenomenology.

                                               41
          Concluding Remarks
• We analyze all the available      data in
  10GeV < k .
• Using FESR as two constraints we obtain the
  prediction at LHC(14TeV):

• We emphasize that precise measurements of
  both    tot and  pp in coming LHC exp. will
            pp


  resolve Fermilab discrepancy of        at 1.8TeV.
• It is also important to investigate  pp
  in extremely high-energy regions    5 1019 eV . 42
43
           Fit by Cudell et al.
• They use the same formula of (log ν)2 .


Fitting all the processes,
by using common values of parameters.



while Z, Y1, Y2 are process-dependent.



                                            44
• Their                            is
  smaller than ours αV=0.517.
    fitted energy region:
     Ecm > 5GeV (k > 12.3GeV):smaller
• Their conjecture : value of B(=0.308+-
  0.010 mb) is process-independent.
   corresponding to c2 = 0.0554+-0.0018 .
• Larger value of c2 predicts larger σ at
  LHC .                            +4.1
             tot  111.5 1.2syst  2.1stat mb,
               pp

• It should be checked if their values of
  parameters satisfy duality constraints or not.
                                             45
• Our c2 in πp process: c2 = 0.00173
    B(πp) = 4π c2 / mπ2 = 0.4345 mb
• This B(πp) in our prev. anal. is larger than
  B(pp)( = 0.270 +- 0.018 mb) in our present
  analysis.
• Conjecture: Universality of B (coeff. of (log
  ν)2 ) should be checked through the re-
  analysis of πp in our case.


                                              46
• Since amplitude is crossing-even,we have
                                        e  i               
                2c0  c2 2  c1  log                 log     
           i                             M                  M 
 R                                                           ,
          2M 2             e  i                            
                c2  log 2           log 2                    
                             M               M                 
                      e  i M  P'   M  P'
                P'                                   
  FP'                                             
               M                sin  P'             
                                                      
 and so obtain
                                 
  Re R            c1  2c2 log  ,
                2M 2              M
                      P   0.5
   Re FP'    
                        '

                          ,                                          47
                     M M 
         Determination of αV
• It is necessary to pay special attention to
  determine αV , since the prediction of
  tot.c.s. at LHC is sensitive to αv.
• We determine αV directly by fitting
  and          .
• Fitted energy regions are changed:
    10,20,30,50,70GeV < k (below SPS).

                                                48
                         Result
Fitted
E.range
10GeV<k                    102.3/(169-3) 28.6/58 73.7/111
20GeV<k                    82.5/(145-3)   21.1/51 61.4/94
30GeV<k                    77.2/(126-3)   19.6/45 57.6/81
 50GeV<k                66.3/(101-3) 15.3/29 51.0/72
 70GeV<k                46.4/(88-3)   13.4/26 33.0/62
Results are mutually consistent for 20,30,50GeV < k .
10GeV seems to be too small as a lower limit.

We use αV = 0.517 :fixed to predict the σ and ρ at
                                                        49
LHC.
   Three cosmic-ray
     data sample
  1. N.N.Nikolaev
  2. T.K.Gaisser et al.(GSY)
  3. M.M.Block et al.(BHS)
                         
  Nikolaev claims the  tot is
      approximately 30mb
      larger than BHS’s results.
                    NGSY BHS
  fit 2             22.9/7 1.3/7
  fit 3             18.0/7 0.9/7
                        
Our pred. values of  tot are in good agreement with BHS.
                                                     50
• There are also very interesting prob. above
  the GZK cutoff energies 5 1019 eV .
• Because of the reaction p   CMB    n,
    is supposed to drop off above this
     pp

  energy.
• According to AGASA,     pp is still increasing,
  but soon Pierre Auger Observatory in
  Argentina may give new results.
• LHC experiments may also be important for
  searching for this energy regions.

                                                51

				
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