Proposing a new constraint for
predictions of tot.c.s.
and ratio at LHC
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
Last Year : Present. by K. Igi
• We analyzed crossing-even forward scatt.
• 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
E710/E811 data input : green lines
CDF datum input : blue lines;
K. Igi presented in ’05. 4
The predictions of tot and at LHC energy , s 14TeV
Last Year presented by Prof. Keiji Igi.
tot s 14TeV s 14TeV
Large systematic difference between E710/E811
input and CDF input.
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
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).
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).
dk tot k tot
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’ .
Brief survey of our prev.work on
sat.of Froissart -Martin bound:
K.Igi and M.Ishida,PRD66(2002)034023
• As is well known,
increase above 70GeV.
• It had not been known,
however, if this increase
bound before 2002.
• We proposed to use rich
inf.of in low energy
reg. to investigate high
• FESR(1),(2) as constraints
• Searched for best fit to
above 70 GeV.
• We can conclude that our
analysis strongly favours
Main Topic :Derivation of new
• 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
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 .
n= -1 mt. FESR, free from unphysical region.
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.
Contribution to Constraint(2)
from unphysical region: negligible
• Why? Ave. of im. part from boson poles
below p p threshold
= smooth extrapolation of t-ch q q exch.
contribution from high energy to M
due to FESR duality.
• Since Im Fq q Im F ,
d 0 Im F d
of LHS of
Im Fq 2
d k o
Im f pp
Im f pp
0 2 4
1 2nd term
k tot k dk 3403 20GeV
of LHS of
So, resonance contributions from unphysical regions
is less than 0.1% of resonance contributions from phys. reg.
Therefore : negligible
• The ratio = the ratio of the real to
imaginary part of F
: subtraction const. 16
• 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%).
• 3 Parameters :
• We search for simultaneous best fit to
the data of tot k and k to determine
values of params. giving the least .
• fitted energy region: and below
• We can predict and
in Tevatron energy and in LHC
energy, using these params.
We use rich data
to evaluate the
of cross sections
Evaluation of integrals:
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)
Based on these phenom. fits,
For Constraint(1): (N1,N2,ν1)=(10,70,40) GeV
Error is less than 1%, negligible.
For Constraint(2): N=10GeV.
Integral region is devided into two parts:
polygonal line graph↑ ↑phenom. fit
Consistent with prev. estim.:
Error is less than 1%, negligible. 22
• Constraint(1) : (N1,N2,ν1)=(10,70,40)GeV
(normalized : 0.104βP’+c0+4.41c1+19.1c2=7.22)
• Constraint(2) : N = 10 GeV
(normalized : 0.367βP’+c0+2.04c1+4.26c2=8.85)
• When and data points
are listed at same value of k,
we make data
points by averaging them.
27(11) points for up to
• 5(1) points in SPS energy region:
• In summary, we have
32(12) points of
in . 24
• Both tot and
(actually Re F ) data in
(SPS) are fitted simult. .
• Constraints(1),(2) are used.
• The fitting is done by three parameters:
input E. region
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
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
Prediction at Tevatron
• We have investigated a possibility to resolve
the discrepancy among the
E710, E811 and CDF at 1.8TeV.
• 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.
Prediction at LHC(14TeV)
Based on the result of analysis of crossing-even
amplitude, we obtain the prediction at LHC,
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:
• Analyses of both even and odd
amplitudes make it possible to use all
the data, not selected for
• Number of fitted data is about 5 times
compared with the analysis of even-
• Using the crossing-odd property,
• 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-
• Number of parameters is 4: ,
determined by the fit.
Prediction of and at LHC. 34
Value of αV
• Using two constrs,
20GeV < k < SPS.
: fixed in the following
Result of the analysis of
both even and odd
amplitudes using two
constraints. Data in
10GeV < k < SPS are
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.
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
• Consistent with the result of analysis of
# of data becomes 5 times, but error does not reduce
so largely, since the added data are below 2100GeV.
Comparison with other groups
• Predicted central values of
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)
tot 111.5 1.2syst 2.1stat mb,
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.
• 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
resolve Fermilab discrepancy of at 1.8TeV.
• It is also important to investigate pp
in extremely high-energy regions 5 1019 eV . 42
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.
• 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,
• It should be checked if their values of
parameters satisfy duality constraints or not.
• 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
• Conjecture: Universality of B (coeff. of (log
ν)2 ) should be checked through the re-
analysis of πp in our case.
• Since amplitude is crossing-even,we have
2c0 c2 2 c1 log log
i M M
2M 2 e i
c2 log 2 log 2
e i M P' M P'
M sin P'
and so obtain
Re R c1 2c2 log ,
2M 2 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
• Fitted energy regions are changed:
10,20,30,50,70GeV < k (below SPS).
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
2. T.K.Gaisser et al.(GSY)
3. M.M.Block et al.(BHS)
Nikolaev claims the tot is
larger than BHS’s results.
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.
• 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
• 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.