The odderon and the spin dependence of proton-proton scattering

T.L. TruemanBrookhaven National Lab

Talk given at RBRC workshop on Odderon Searches at RHICSeptember 27, 2005

0.005 0.01 0.015 0.02 0.025 0.03

0.01

0.02

0.03

0.04

0.05

0.06

ANN

-t GeV2

equal mixture

pure pomeron

pure odderon

Leader & Trueman PRD 61 077504

The proton-proton helicity amplitudes

Two important initial state polarization asymmetries

Buttimore, ibid.

Amplitudes are normalized so that

The singular electromagnetic amplitudes are

The parameters in the hadronic amplitudes have s dependence determined from unpolarized scattering

Regge pole couplings: cf. Berger et al PRD 17, 2971

βRnfβR

nf → βRf βR

nf

βRnfβR

nf → βRf βR

f

for single flip amplitude, replace

for double flip amplitude, replace

I use βRf = τR

√−t/m2 βR

nf τRwith constant

φhad1 = −

1

8πΣRβR

nfβRnf (s/s0)

αR(exp (−iπαR) + SR)/ sin παR

Class 1 Class 2 Class 3

τ = P = C τ = −P = −C τ = −P = C

φ+, φ5, φ2−φ4 φ−

φ2 + φ4

IP, O, ρ, ω, f, a2 a1 π, η, b

Table 1: Classification of pp amplitudes by exchange symmetries and the associated Reggepoles.

1

Regge Classification

cf. Buttimore et al PRD 59, 114010

T.L.Trueman 11/9/04

The Model

• is based on Regge fit to pp scattering over wideenergy range (cf. Cudell et al) which fixes non-flip parameters for the Pomeron (simple or multiplepole), a C = −1 vector meson (mainly ω) and aC = +1 tensor meson (mainly f2).

• The non-flip amplitude is

g0(s, 0) = gP (s) + gf(s) + gω(s),

where the functions gR(s) have enery dependenceand phase determined by standard Regge theory.

• The corresponding flip amplitude is determined bythree real, energy independent constants

g5(s, t) = τ(s)

√−t

mg0(s, t)

=

√−t

m{τP gP (s) + τf gf(s) + τω gω(s)}.

3

T.L.Trueman 11/9/04

so

τ(s) = {τP gP (s) + τf gf(s) + τω gω(s)}/g0(s, 0)

• The two constants in τ(21.7) = −0.213 −0.054i determines two relations between the threeconstants τP , τf , τω

• We need one more measurement to fix their values.If one measures the “shape” of the raw asymmetryover the CNI region without knowing the valueof P at that energy one can obtain the neededinformation:

S(pL) =PIm[τ(pL)]

P (κ/2 − Re[τ(pL))]=

Im[τ(pL)]

κ/2 − Re[τ(pL)]

4

0.01 0.02 0.03 0.04 0.05

0.01

0.02

0.03

0.04

21.7 GeV data and E950 fit with known

P(upper) and Regge prediction based on fit to

100 GeV data and 21.7 GeV shape (lower)

new 100 GeV data, best fit with known

P(lower) and Regge prediction based on E950 fit and 100 GeV shape (upper)

0.01 0.02 0.03 0.04 0.05

0.01

0.02

0.03

0.04

Checking Regge model

T.L.Trueman 11/9/04

I = 1 couplings and proton-proton elastic

scattering

• Because pp scattering involves the exchange of ofI = 1 Regge poles, the ρ and the a2 in particular,we cannot simply use the results above to makepredictions for this case. But we can use thebeautiful new p-jet data and a couple of reasonableassumptions to achive this. We assume (1) atthese energies the proton-proton and neutron-protonunpolarized scattering amplitudes are approximatelyequal and (2) the two I = 1 Regge poles aredegenerate with the corresponding I = 0 Regge poleof the same Charge Conjugation parity, C = −1 forω, ρ and C = +1 for f, a2. Then we can describepp scattering in terms of 3-parameters: τ+, τ− andthe pomeron coupling τP . Since we already knowτP , in some sense, from the pC analysis and wecan determine two parameters from the real andimaginary parts of τ obtained by fitting the p-jetdata, we are in business.

13

! from p-jet fit = -0.0625 -0.011 i

! from E704 fit = 0.185 + 0.024 i

0.002 0.004 0.006 0.008 0.01

0.01

0.02

0.03

0.04

0.05

New p-jet data

-t

AN

Regge residues in 1/GeV

non-flip

flip from Trueman, Spin2004non-flip from Cudell et al PR D61:034019

flip

pomeron

C=+

C=-

6.95± 0.10

12.73±0.23

9.65±0.43

0.626±0.10

-4.12±0.25

10.22±0.77

Odderon pole form assumed for intercept of 1

Threshold t factors explicitly shown here. These destroy the CNI enhancement.

φO2 =

t

m2

1

8πβO

f βOf (s/s0)

0.005 0.01 0.015 0.02 0.025 0.03

-0.002

-0.001

0.001

0.002

0.003

0.004

Regge pole model prediction for ANN

odderon flip = 0 or = pomeron flip

odderon flip = ! flip

ANN

s=200

To estimate Regge cut associated with odderon use absorption model of Henyey et al, PR 182, 1579

φO2 cut = −

sσtot

4πb(1 − iρ)

(βOf )2

bm2

The most important feature of this formula is that the suppression factor of -t has been replaced by the inverse of the slope b .

Cut properties

1. Power behavior very close to pole case, up to logs. Phase the same as pole.

2. C and signature the same as pole3. Both parities present

for pomeron-pole cut

4. No factorization of residues-<++|M|--> so is not forced to vanish at t=0

0.005 0.01 0.015 0.02 0.025 0.03

0.01

0.02

0.03

ANN

ANN for odderon cut with Regge pole

background, odderon flip = ! flip

assumed

-t

s=200

s=200x200

0.01 0.02 0.03 0.04 0.05

0.001

0.002

0.003

0.004

ANN from ! absorptive cut, no odderon at all

ANN

-t

s=200

s=200x200

2 4 6 8 10

0.005

0.01

0.015

0.02

0.025

odderon flip coupling

s=500x500s=200 s=200x200

2 4 6 8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

2 4 6 8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Height of Ann peak at t=-0.001 as a function of the Odderon

spin flip coupling at three different energies

Unsuppressed rho cut included here and in following

Ann(Odderon flip)-Ann(0)/Ann(0)

s GeV2 Rho flip

200 0.0256 6.849

200x200 0.148 39.70

500x500 0.251 67.08

Pomeron flip

1. If the odderon exists and has a “reasonable” spin-flip coupling there is a good chance of observing it in ANN.

Conclusions

2. The odderon-pomeron cut is essential to producing an observable signal in the small-t region.

3. There is a significant background mainly arising from the -pomeron cut. If the odderon spin-flip is small,

but larger than the pomeron spin-flip, measurements of ANN over a wide energy range will be needed

to separate out the signal

ρ