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ー. Universal Rise of Hadronic Total Cross Sections based on Forward π p , 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 . - PowerPoint PPT Presentation
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Universal Rise of Hadronic Total Cross Sections based on
Forward π p , 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
ー
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
the fit to π p , Kp , pp , 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 only from QCD. Test of Universality of B is Necessary.
ー
Experimental Data Situations
ISR Ecm<63GeV
SPSEcm<0.9TeV
Tevatron Ecm=1.8TeV
CDFD0
σtot
Fitted energy region
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
pp
ー
ー
π p , Kp Scatterings
• No Data in TeV Estimated Bπ p , B Kp have large
uncertainties.
π ー
p π+ p
K -
p
K+ p
No Data
No Data
Test of Universality of B• Highest energy of Experimetnal data: pp : 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 K
p ? 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).
ー
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 , 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)* f pp (-ν) = f pp (ν)*
-
ー
Kinematics• Crossing-even amplitudes : F(+)( ー ν)=F(+)(ν)* F(+)(ν) = ( f ap(ν) + fap(ν) )/2 average of π- p , π+ p; K - p , K + p;pp , p
p Im F(+)
asymp(ν) = β P’ /m (ν/m)α P’ (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)* 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
β P’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies.
-
-
ー
Finite-energy sum rule(FESR)• π p Forward Scattering F(ν) = F(+)(ν) ー F(+)
asymp(ν) ~ ν -1.5 F(N) = ~0
Re F(mπ) = (P/π)∫-∞∞dν’ I m 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 Igi Matsuda; Dolen Horn Schmid 1967
∫0N dν Im F(+)(ν) νn = ∫0
N 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
∫N1
N2 σtot(+)(k) dk /2π2 estimated from low-energy exp. data.
= 2/π ∫N1
N2 dν Im F(+)asymp(ν) ν/k2 calculable
Relation between high-energy parameters βP’,c2,1,0 : A constraint
--
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)
-
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
N1 dependence of the resultN1(GeV) 10 7 5 4 3.02 2.035 1.476c2(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)
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
FESR integralπ- p
π+
pmuch improved
Choice of N1 in Kp , pp ,pp
• Exothermic K - p pp open at thres. ∑- p , Λ p meson-
channels• Exotic K + p , pp steep decrease at Ecm~2GeV ? We take N1 larger than π p . N1=5GeV,
representat.
ー
K -
p
K+ p
pp
pp
ー
ー
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
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
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.
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 , 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.
ー
Concluding Remarks• We obtain Bπ p = B pp = B Kp .
Universality of B suggested.
• Use of FESR is essential to lead this conclusion.• Our result, B pp = 0.280(15) mb, predicts σ pp
LHC=108.0(1.9) mb .• Our Conclision will be checked by LHC TOTEM.
Kpπ p pp
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)
B(LO) = π/2 (ω αs/mπ)2=2.09mb• Bpp(exp) = 0.280±0.015mb ~ 0.3 mb ( << B(Martin) = π/mπ
2=62mb )
rNLO BFKL eqItakura, lectures in Dec.2008
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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
Q2 = qT2 : transverse resolution
x =p+/P+ : longitudinal mom. fraction
1/Q
1/xP+
g*
transverse
longitudinal
高エネルギー散乱での陽子の振る舞い
陽子
各パ
ート
ンの
分布
関数
パートンの持つ運動量比x
Lecture by Itakura
Result of the fit to ρ ratios
π- p π+ p K - p K + p pp pp
ー