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11
Universal Rise of Hadronic Total Cross Sections based on
Forward π p , Kp and , pp Scatterings
Keiji IGI (RIKEN)
May 28, 2010 at NTU SeminarPhys.Lett.B670(2009)395
Phys.Rev.D79(2009)096003
In collaboration with Muneyuki Ishida
pp
ー
2
Introd. of new sum rule(~1962)
Tot. cross sec.
σ(k) ≈ σ(∞) + βkα-1
Private letter from Gell-Mann to Chew
(If α< 0 is proved, theorysimple)
• Many young active physicists attempted to prove this but failed.
• I derived new sum rule to prove the above conjecture empirically.
σ(k)
σ(∞)k
α(s)
P
P’
s (GeV2)
f2(1275)
3
Such a kind of attemptK.IGI.,PRL,9(1962)76
• The following sum rule has to hold under the assumption that there is no sing.with
vac.q.n. except for Pomeron(P).
Evid.that this sum rule not hold pred. of the traj. with
and the f meson was discovered on the
'P ' 0.5P
'P
2
01
N
tot tot
f Na dk k
M M
0.0015 -0.012 2.22 1
VIP: The first paper which predicts high-energy from low-energy ( FESR1 )
44
SCR(Super Conv.Rel.) Fubini et al
'
'
'
' '
Im1( ) 0
1
Im 0
FF d
faster than as
Then d F
'
'
1 1
0f
This work was one of the highlight at Be rkeley Conf, 1966.Gell-Mann was the first speaker but he gave a summary talk in the beginning.This was an interesting work but the application was limitted to only one.
55
FESR and DualityIgi-Matsuda (1967)
Dolen-Horn-Schmid (1967)
'
' ' '
10
Im 0
Assume F F as
f F F faster than
Then d f
Since satisfies SCR. 'f
66
Using the original physical amp., we obtain
0
1 1Im 0
N
N
d F F d
Igi, Matsuda, PRL(1967)
Average of ImF = low-energy extension of Im
F
77
n nR t R sV
n s n t
Veneziano amplitude -- String Theory -- Super String Theory
88
Brief Survey on the Main Topic • Increase of has been shown to be at most by Froissart (1961) using Analyticity and Unitarity.• Squared log behaviour was confirmed by using the duality
constraint of FESR . Keiji Igi & M.I. ’02. Block & Halzen, ’05.• COMPETE collab.(PDG) further assumed
for all hadronic scattering to reduce the number of adjustable
parameters.
tot 2log s
20logtot B s s Z
99
Particle Data Group(by COMPETE collab.)
The upper side : σ
The lower side : ρ - ratio
B (Coeff. of 20log )s sAssumed to be universal
Theory:Colour Glass Condensate of QCD sug.
Not rigidly proved from QCDTest of univ. of B:necessaryeven empirically.
This is the main point of today’s talk.
10
I ncreasing tot.c.s.
• Consider the crossing-even f.s.a.
with
• We assume
at high energies.
2
pp ppf fF
Im4totk
F
'
''2
0 1 22
Im Im Im
log logP
P
P
F R F
c c cM M M M M
M : proton massν, k : incident proton energy, momentum in the laboratory system
11
The ratio
• The ratio = the ratio of the real to imaginary part of
F
'
'
Re ReRe
Im ImImP
P
R FF
R FF
'
0.5
1 22 2 log 02
4
0 .
P
tot
c c FM M M M
k
F subtraction const
*
F F
1212
How to predict σ and ρ for pp at LHC based on duality?
• We searched for simultaneous best fit of σ and ρ up to some energy(e.g.,ISR) in terms of high-energy parameters constrained by FESR.
• We then predicted and in the LHC regions.
tot
(as an example)
1313
• Both and data are fitted through two formulas simultaneously with FESR as a constraint.
• FESR is used as constraint of
and the fitting is done by three parameters:
giving the least .• Therefore, we can determine all the parameters
tot ReF
' ' 0 1 2, ,P P
c c c
2 1 0, ,c c and c
2 ic
'2 1 0, , , , 0P
c c c F
These predict at higher energies including LHC energies , 14s TeV
1414
ISR(=2100GeV)
(a) All region tot
(b)
(c) : High energy region tot
LHCLHC
LHC
Predictions for and
( 62.7 )s GeVThe fit is done for data up to ISR
2100 ( )GeV lab As shown by arrow.
1515
Pred. for and at LHC• Predicted values of agree with pp exptl. data
at cosmic-ray regions within errors.• It is very important to notice that energy range of
predicted is several orders of magnitude larger than energy region of the input.
• Now let us test the universality of B.
It turns out that FESR duality plays very important roles.
tot
tot
,tot
1616
Main Topic:Universal Rise of σtot ?• B (coeff. of (log s/s0)2) : Universal for all hadronic scatterings ?• Phenomenologically B is taken to be universal in the
fit to π p , Kp , , pp ,∑ p ,γ p , γγ forward scatt.
COMPETE collab. (adopted in Particle Data Group)
• Theoretically Colour Glass Condensate of QCD suggests the B universality.
Ferreiro,Iancu,Itakura,McLerran’02
Not rigourously proved yet only from QCD. Test of Universality of B is Necessary even
empirically.
pp
ー
1717
Particle Data Group(by COMPETE collab.)
The upper side : σ
The lower side : ρ - ratio
B (Coeff. of 20log )s sAssumed to be universal
Theory:Colour Glass Condensate of QCD sug.
Not rigidly proved from QCDTest of univ. of B:necessaryeven empirically.
1818
• We attempt to obtain
for
through search for simultaneous best fit to experimental .
B values( ), ,pp p p p Kp scatterings
tot and ratios
19
New Attempt for
• In near future, will be measured at LHC energy. So, will be determined with
good accuracy.• On the other hand, have been measured
only up to k=610 GeV. So, one might doubt to obtain for (as well as ),
with reasonable accuracy.
• We attack this problem in a new light.
,p Kp
pptot
ppB: 1.8pp s TeV
ptot
B p Kp p KpB B
: 26.4p s GeV
: 24.1Kp s GeV
20
Practical Approach for search of BTot. cross sec.= Non-Reggeon comp. + Reggeon(P’) comp.
• Non-Reg. comp. shows shape of parabola as a fn. of logν with a min.
• Inf. of low-energy res. gives inf. on P’ term. Subtracting this P’ term from σtot
(+), we can obtain the dash-dotted line(parabola).
Fig.1 pp, pp -
40
50
100
Fig.1 pp, pp -
1 10 102 103 104 105 106 log ν(GeV)
●
●
●
●
●
●Tevatron s = 1.8 TeV√
SPS s = 0.9 TeV
√
s = 60 GeV √ISR
σtot (mb)100
50
40Non-Regge comp.(parabola)
• We have good data for large values of log ν compared with LHS, so (ISR, SPS, Tevatron)-data is most important for det. of c2(pp)(or Bpp) with good accuracy.
21
Fig.2 πp
• σtot measured only up to
s = 26.4 GeV (cf. with pp, pp).√ -
• So, estimated Bπp
may have large uncertainty.
• The πp has many res. at low energies, however.
So, inf. on LHS of parabola obtained by subtracting P’ term from σtot(+)
is very helpful to obtain accurate value of B(πp).
(res. with k < 10GeV).
•( Kp : similar to πp ) .
25
30
35
40
45
50
55
60
Fig.2 πp
●
1 10 102 103 logν(GeV)
highest energy s = 26.4 GeV
kL = 610 GeV
√
σtot (mb)
Resonances
↓
50
40
30Non-Regge comp(parabola)
s
2222
Fitting high-energy data with no use of FESR
For pp scatterings We have data in TeV.
σtot = B pp (log s/s0)2 + Z (+ ρ trajectory) in high-energies. parabola of log s
B pp = 0.273(19) mbestimated accurately.Depends the data with the highest energy. (CDF D0)
pp
ー
ISR Ecm<63GeV
SPSEcm<0.9TeV
Tevatron Ecm=1.8TeV
CDFD0
σtot
Fitted energy region
pp
ー
pp- , pp Scattering
2323
π p , Kp Scatterings
• No Data in TeV
Estimated Bπ p , B Kp have large uncertainties.
πp π+
p
No Data
K -
p
K +
p
No Data
2424
Test of Universality of B• Highest energy of Experimetnal data: : Ecm = 0.9TeV SPS; 1.8TeV Tevatron
π- p : Ecm < 26.4GeV
Kp : Ecm < 24.1GeV No data in TeV B : large errors.
B pp = 0.273(19) mb
Bπ p = 0.411(73) mb B pp =? Bπ p =? B Kp ?
B Kp = 0.535(190) mb No definite conclusion
• It is impossible to test of Universality of B only by using data in high-energy regions.
• We attack this problem using duality constraint from
finite-energy sum rule (FESR).
ppー
2525
Kinematics• ν : Laboratory energy of the incident particle
s =Ecm2 = 2Mν+M2+m2 ~ 2Mν
M : proton mass of the target. Crossing transf. ν ー ν
m : mass of the incident particle
m=mπ , m K , M for π p; Kp;pp ; k = (ν2 – m2)1/2 : Laboratory momentum ~ ν
• Forward scattering amplitudes fap(ν): a = p ,π+, K +
Im fap (ν) = (k / 4 π) σtotap : optical theorem
• Crossing relation for forward amplitudes:
f π- p (-ν) = fπ+ p (ν)* , f K - p (-ν) = f K + p (ν)*
pp ppf f
pp
2626
Kinematics• Crossing-even amplitudes : F(+)( ー ν)=F(+)(ν)*
average of π- p , π+ p; K - p , K + p; pp ,
Im F(+)asymp(ν) = β P’ /m (ν/m)α’(0)
+(ν/m2)[ c0+c1log ν/m +c2(log ν/m)2] β P’ term : P’trajecctory (f2(1275) ): α P’ (0) ~ 0.5 : Regge Theory
c0,c1,c2 terms : corresponds to Z + B (log s/s0)2
c2 is directly related with B . (s ~ 2M ν)
• Crossing-odd amplitudes : F(-)( ー ν)= ー F(-)(ν)*
Im F(-)asymp(ν) = βV /m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5
β P’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies.
2ap apF f f
2ap apF f f
pp
' 0P
27
FESR Duality• Remind that the P’ sum rule in the introd..
• Take two N’s(FESR1)• Taking their difference, we obtain
has open(meson) ch. below ,and div. above th.• If we choose to be fairly larger than we have no difficulty. ( : similar)
No such effects in .
2 2
1 1
2
1 2Im
2
N N
tot asymp
N N
d k k d F
1 2 2 1, ( )N N N N N N
2 20 0
1 2Im .
2
N N
asympdk k d F constk
LHS is estimated from Low-energy exp.data.
RHS is calculable from The low-energy ext. of Im Fasymp.
pp pp
1N m K p
p
28
Average of
in low-energy regions should coincide with
the low-energy extension of the asymptotic
formula
• This relation is used as a constraint between
high-energy parameters:
Im 1 4 totF k k
Im .asympF
' 2 1 0 ., ,P
c c c
2929
Choice of N1 for π p Scattering• Many resonances Various values of N1
in π- p & π+ p• The smaller N1 is taken,
the more accurate
c2 (and Bπ p ) obtained.
• We take various N1
corresponding to peak and
dip positions of resonances.
(except for k=N1=0.475GeV)
For each N1,
FESR is derived. Fitting is performed. The results checked.
Δ(1232)N(1520)
N(1650,75,80)
Δ(1700)
Δ(1905,10,20)
-
3030
Analysis of π p Scattering• Simulateous best fit to exp. data of
σtot for k = 20~370GeV(Ecm=6.2~26.4GeV)
and ρ ( = Re f(k) / Im f(k) ) for k > 5GeV
in π- p , π+ p forward scatterings.
• Parameters : c2,c1,c0,βP’,βV, F(+)(0) (describing ρ)
• FESR N2=20GeV fixed. Various N1 tried.
Integral of σtot estimated very accurately.
Example : N1=0.818GeV 0.872βP’+6.27c0+25.7c1+109c2=0.670 (+ ー 0.0004 negligible)
βP’ = βP’ (c2,c1,c0):constraint. fitting with 5 params
3131
N1 dependence of the resultN1(GeV) 10 7 5 4 3.02 2.035 1.476
c2(10-5) 142(21) 136(19) 132(18) 129(17) 124(16) 117(15) 116(14)
χtot2 149.05 149.35 149.65 149.93 150.44 151.25 151.38
N1(GeV) 0.9958 0.818 0.723 (0.475) 0.281 No SR
c2(10-5) 116(14) 121(13) 126(13) (140(13)) 121(12) 164(29)
χtot2 151.30 150.51 149.90 148.61 150.39 147.78
• # of Data points : 162.• best-fitted c2 : very stable.
• We choose N1=0.818GeV as a representative.• Compared with the fit by 6 param fit with No use of FESR(No SR)
3232
Result of the fit to σtotπ p
c2=(164±29) ・ 10-5 c2=(121±13) ・ 10-5
Bπ p= 0.411±0.073mb Bπ p= 0.304±0.034mb
No FESR FESR used
Fitted region
Fitted region
FE
SR
integralπ- p
π+
pmuch improved
3333
Choice of N1 in Kp , , pp
• Exothermic K - p open at thres. meson-channels
• Exotic K + p , pp steep decrease at Ecm~2GeV ?
We take N1 larger than π p . N1=5GeV.
K -
p
K +
p
pp
pp
ー
ppー
ppー
3434
Result of the fit to σtotKp
c2=(266±95) ・ 10-4 c2=(176±49) ・ 10-4
B Kp= 0.535±0.190mb B Kp= 0.354±0.099mb
large uncertainty much improved
Fitted region
No FESR
Fitted region
FESR
integral
FESR used
3535
Result of the fit to σtotpp , ppー
c2=(491±34) ・ 10-4 c2=(504±26) ・ 10-4
B pp= 0.273±0.019mb B pp=0.280±0.015mb
Improvement is not remarkable in this case.
No FESR FESR used
Fitted regionlarge
FESR
integral Fitted regionlarge
3636
Test of the Universal Rise • σtot = B (log s/s0)2 + Z
B (mb)
πp
0.304±0.034
Kp
0.354±0.099
pp
0.280±0.015
B(mb)
0.411±0.073
0.535±0.190
0.273±0.019
FESR used No FESR Bπ p ≠ ? B pp =? B Kp
No definite conclusion in this case.
Bπ p = B pp = B Kp within 1σUniversality suggested.
3737
Concluding Remarks
• In order to test the universal rise of σtot ,
we have analyzed π±p ; K ±p ; , pp independently.
• Rich information of low-energy scattering data constrain, through FESR, the high-energy parameters B to fit experimental σtot and ρ ratios.
• The values of B are estimated individually for three processes.
pp
3838
• We obtain Bπ p = B pp = B Kp .
Universality of B
suggested.
Use of FESR is essential
to lead to this conclusion.
• Universality of B suggests
gluon scatt. gives dominant cont. at very high energies( flav. ind. ).
• It is also interesting to note that Z for
approx. satisfy ratio predicted by quark model.
Kpπ p pp
, ,p Kp pp pp2:2:3
39
• Our results predicts at 14TeV
102.0(1.7)mb at 10 TeV
96.0(1.4)mb at 7 TeV
• Our Conclusions will be tested by LHC TOTEM.
0.283(15)ppB mb108.0(1.9) .LHC
pp mb
4040
• Finally, let us compare our pred. at LHC with other pred.
ref.
Ishida-Igi (this work)
Igi-Ishida (2005)
Block-Halzen (2005)
COMPETE (2002)
Landshoff (2007)• Pred. in various models have a wide range.
pptot mb
108.0 1.9106.3 5.1 2.4sys stat
107.3 1.2
115.5 1.2 4.1
125 25
The LHC(TOTEM) will select correct one among these approaches.Ishida-Igi gives pred., not only for pp but also for π p and Kp,althoughexperiments are not so easy in the near future. This is very import. point.