Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003

40
1 Universal Rise of Hadronic Total Cross Sections based on Forward π , Kp and , pp Scatterings Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003 In collaboration with Muneyuki Ishida

description

pp. ー. Universal Rise of Hadronic Total Cross Sections based on Forward π p , Kp and , pp Scatterings. Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003 In collaboration with Muneyuki Ishida. 1. Introd. of new sum rule(~1962). - PowerPoint PPT Presentation

Transcript of Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003

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

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

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

ISR Ecm<63GeV

SPSEcm<0.9TeV

Tevatron Ecm=1.8TeV

CDFD0

σtot  

Fitted energy region

pp

pp- , pp Scattering

2323

π p , Kp Scatterings

• No Data in TeV

Estimated Bπ p , B Kp have large uncertainties.

πp π+

No Data

K -

K +

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 -

K +

pp

pp

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.