Universal Rise of Hadronic Total
Cross Sections based on
ー
Forward πｐ, Ｋｐ and ｐｐ, ｐｐ
Scatterings
Muneyuki Ishida(Meisei University)
Phys.Lett.B670(2009)395-398.
arXiv:0903.1889[hep-ph]
In collaboration with Keiji Igi
Rise of Total Cross Sections
• Total Cross Sections σtot rise in high-energy
regions logarithmically.
• σtot = B (log s/s0)2 + Z
COMPETE collab.
Pomeron contrib. ：dominant in high-energies in Regge theory.
B : New term introduced consistently with Froissart unitarity bound.
better in low energies than soft-Pomeron fit,
σtot =βP (s/s0)0.08 by Donnachie Landshoff
• Squared log behaviour was confirmed by using
the duality constraint of FESR .
Keiji Igi & M.I. ’02. Block & Halzen, ’05.
Universal Rise of σtot ?
• B (coeff. of (log s/s0)2) :
Universal for all hadronic scatterings ?
• Phenomenologically B is taken to be universal in
ー
the fit to πｐ,Ｋｐ, ｐｐ,ｐｐ,∑ｐ,γｐ, 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 only from QCD.
 Test of Universality of B is Necessary.
Experimental Data Situations
σtot
Forー
pp scatterings
We have data in TeV.
Tevatron
Ecm=1.8TeV
σtot = Bｐｐ (log s/s0)2 + Z
(+ ρ trajectory)
in high-energies.
parabola of log s
SPS
ー
ｐｐ
ISR
Ecm<0.9TeV
Ecm<63GeV
CDF
D0
ｐｐ
Fitted energy region
Bｐｐ = 0.273(19) mb
estimated accurately.
Depends the data with
the highest energy. (CDF
D0)
πｐ , Ｋｐ Scatterings
No
Data
πーｐ
π+ｐ
No
Data
Ｋ-ｐ
Ｋ+ｐ
• No Data in TeV
 Estimated Bπｐ , BＫｐ have large uncertainties.
Test of Universality of B
• Highest energy of Experimetnal data:
ー : Ecm = 0.9TeV SPS; 1.8TeV Tevatron
ｐｐ
π-ｐ ： Ecm < 26.4GeV
Ｋｐ ： Ecm < 24.1GeV No data in TeV B : large errors.
Bｐｐ = 0.273(19) mb
Bπｐ = 0.411(73) mb  Bｐｐ =? Bπｐ =? BＫｐ ?
BＫｐ = 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).
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 , mＫ , M for πｐ, Ｋｐ,ーｐｐ,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)* , fＫ-ｐ(-ν) = fＫ+ｐ(ν)*
ｐｐ
f (-ν) = fｐｐ(ν)*
Kinematics
• Crossing-even amplitudes : F(+)(ーν)=F(+)(ν)*
F(+)(ν) = ( f -ap(ν) + fap(ν) )/2
ー
average of π-ｐ, π+ｐ； Ｋ-ｐ, Ｋ+ｐ；ｐｐ,
ｐｐ
Im F(+)asymp(ν) = βＰ’ /m (ν/m)α (0)
+(ν/m2)[ c0+c1log ν/m +c2(log ν/m)2]
Ｐ’
βＰ’ term : P’trajecctory (f2(1275) ): α (0) ~ 0.5
c0,c1,c2 terms : corresponds to Z + B (log s/s0)2
Ｐ’
: Regge Theory
c2 is directly related with B . (s～2M ν)
• Crossing-odd amplitudes : F(-)(ーν)= ーF(-)(ν)*
F(-)(ν) = ( f -ap(ν) ー fap(ν) )/2
Im F(-)asymp(ν) = βV /m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5
βＰ’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies.
Finite-energy sum rule(FESR)
• πｐ Forward Scattering
～
F(ν) = F(+)(ν) ー F(+)asymp(ν) ~ ν -1.5  ～
F(N) = ~0
～
∞
～
Re F(mπ) = (P/π)∫-∞ dν’ Ｉm F(ν’)/(ν’-m)
N high-energy , but finite.
～
∞
= (2P/π)∫0 dν’ Im F(ν’) ν’/ k’2
 (2P/π) ∫0N dν Im F(+)(ν) ν/k2 ーRe ～
F(mπ)
= (2P/π) ∫0N dν Im F(+)asymp(ν) ν/k2 Igi 1962
• moment Sum Rules
∫0N dν Im F(+)(ν)
Igi Matsuda; Dolen Horn Schmid 1967
νn = ∫0N dν Im F(+)asymp(ν) νn
n=1,3,… contribution from higher-energy regions is enhanced.
FESR Duality
•  ‘Average’ of Im F(+)(ν)( = k/4π σtot(+)(k) )
in low-energy regions should
This shows many peak and dip structures of resonances.
coincide with the low-energy extension of
the asymptotic formula Im F(+)asymp(ν) .
• Taking two N’s : N1 in resonance-energy,
N2 in asmptotically high energy. Taking their difference
N
 ∫N- 2 σtot(+)(k) dk /2π2  estimated from low-energy exp. data.
1
N2
= 2/π ∫N1 dν Im F(+)asymp(ν) ν/k2
 calculable
Relation between high-energy parameters βP’,c2,1,0 : A constraint
Choice of N1 for πｐ Scattering
• Many resonances
in π-ｐ & π+ｐ
• The smaller N1 is taken,
the more accurate
c2 (and Bπｐ) obtained.
Various values of N1
Δ(1232)
N(1520)
N(1650,75,80)
• We take various N1
corresponding to peak and
dip positions of resonances.
(except for k=N1=0.475GeV)
Δ(1905,10,20)
For each N1,
Δ(1700)
FESR is derived. Fitting is performed. The results checked.
Analysis of πｐ 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
N1 dependence of the result
N1(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
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)
148.61
Result of the fit to σtotπｐ
No FESR
Fitted region
FESR integral
Fitted region
π-ｐ
FESR used
π+ｐ
much improved
c2=(164±29)・10-5
Bπｐ＝0.411±0.073mb
c2=(121±13)・10-5
Bπｐ＝0.304±0.034mb
ー
Choice of N1 in Ｋｐ, ｐｐ,ｐｐ
ー
ｐｐ
Ｋ-ｐ
ｐｐ
Ｋ+ｐ
• Exothermic Ｋ-ｐ
open at thres. ∑-ｐ, Λｐ
• Exotic Ｋ+ｐ, ｐｐ
ー
ｐｐ
meson-channels
steep decrease at Ecm~2GeV ?
 We take N1 larger than πｐ. N1=5GeV, representat.
Result of the fit to σtot
No FESR
FESR used
Fitted region
FESR integral
Fitted region
c2=(266±95)・10-4
BＫｐ＝0.535±0.190mb
large uncertainty
Ｋｐ
c2=(176±49)・10-4
BＫｐ＝0.354±0.099mb
much improved
ー
ｐｐ,ｐｐ
Result of the fit to σtot
No FESR
FESR used
FESR integral
Fitted region
large
Fitted region
large
c2=(491±34)・10-4
c2=(504±26)・10-4
Bｐｐ＝0.273±0.019mb
Bｐｐ＝0.280±0.015mb
Improvement is not remarkable in this case.
Test of the Universal Rise
• σtot = B (log s/s0)2 + Z
B (mb)
πｐ
B(mb)
0.304±0.034
0.411±0.073
Ｋｐ 0.354±0.099
0.535±0.190
ｐｐ
0.273±0.019
0.280±0.015
FESR used
Bπｐ= Bｐｐ= BＫｐ within 1σ
Universality suggested.
No FESR
Bπｐ ≠? Bｐｐ =? BＫｐ
No definite conclusion
in this case.
Concluding Remarks
• In order to test the universal rise of σtot
(A common value of B in σtot=B (log s/s0)2 + Z )
in all the hadron-hadron scatterings, we have
analyzed π±ｐ,Ｋ±ｐ, ー
ｐｐ,ｐｐ independently.
• Rich information of low-energy scattering data
constrain, through FESR, the high-energy
parameters to fit experimental σtot and ρ ratios.
• The values of B are estimated individually for
three processes.
Concluding Remarks
• We obtain
Bπｐ= Bｐｐ= BＫｐ.
Universality of B
suggested.
Ｋｐ
πｐ
ｐｐ
• Use of FESR is essential to lead this conclusion.
• Our result, Bｐｐ = 0.280(15) mb, predicts
σｐｐLHC=108.0(1.9) mb .
• Our Conclision will be checked by LHC TOTEM.
•
Colour Glass Condensate of QCD
In high-energy scattering, proton is not composed of three
valence quarks, but a large number of gluons with small
momentum fraction x.
• The gluon density of the target proton drastically increases
with small x ( = large s)  gluon condensation.
BFKL, BK eq. derived from pQCD.
 strong absorption of incident beam : black disc
Its radius R increases like ~log s : depends upon soft physics.
 σtot ~ 2 π R2 ~ B (log s)2
• CGC: B=0.446mb(αs=0.1)
rNLO BFKL eq
Itakura, lectures in Dec.2008
 B(LO) = π/2 (ω α- s/mπ)2=2.09mb
• Bpp(exp) = 0.280±0.015mb ~ 0.3 mb
( << B(Martin) = π/mπ2=62mb )
高エネルギー散乱での陽子の振る舞い
Lecture by Itakura
深非弾性散乱でみた陽子の内部構造
パートン：クォークとグルオンの総称
陽子
各パートンの分布関数
g*
1/Q
transverse
longitudinal
1/xP+
Q2 = qT2 : transverse resolution
x =p+/P+ : longitudinal mom. fraction
パートンの持つ運動量比ｘ
・ 陽子は単純な３つのヴァレンスクォークの集まりでは「ない」
・ 陽子は小さな運動量比（ x < 10 -2 ）を持つ膨大な数のグルオンからなる
・ そのグルオンは高エネルギー散乱（ x ~ Q2/(Q2+W2) 0 ）で見えてくる
同様のことは、全てのハドロンや原子核にあてはまる
Result of the fit to ρ ratios
π-ｐ π+ｐ
Ｋ-ｐ Ｋ+ｐ
ー ｐｐ
ｐｐ