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

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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.

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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).

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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)*

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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 , ｐｐ

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.

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-

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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)

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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 1.476

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)

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

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.

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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 ｐｐLHC=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/mπ2=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

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