The Θ+ Photoproduction in a Regge

Model

Herry Kwee

College of William and Mary

Cascade 2005

Jefferson Lab

December 3, 2005

1

1. Introduction

2. Photoproduction

• γn → Θ+K− for spin-1/2 Θ+

• γp → Θ+K0 for spin-1/2 Θ+

• γn → Θ+K− for spin-3/2 Θ+

3. Results

• Total cross sections (σ)

• Differential cross sections ( dσdt

)

• Photon asymmetries (Σ)

• Decay angular distributions

4. Conclusions

2

Pentaquarks (q4q)

QCD does not prohibit pentaquarks

Known: Baryon (qqq) and Meson (qq)

Other possibility: (qqq)(qqq), (qq)(qq), glueball,

and Pentaquark (q4q)

Questions:

Where should we look?

How can we distinguish pentaquark from (qqq)

resonance?

and How stable? (width)

Theoretical prediction: Diakanov et. al.

(hep-ph/9703373)

• lightest pentaquark: antidecuplet

• no ordinary baryon (qqq) has s = +1

• mass around 1530 MeV

• width ' 15 MeV

3

Hidden Strangeness:

JJ

J

JJ

JJ

JJ

JJ

JJ

JJ

JJ

J

r r r r

r r r

r r

r

uuddsΘ+(1530)

P5 =√

23|uudss〉 +

√

13|uuddd〉

Σ+5 =

√

23|uudsd〉 +

√

13|uusss〉

ddssu uussd

Ξ−−

5(1860?)

Ξ+5

Figure 1: SU(3)F Antidecuplet for Pentaquark

• naively, m(Ξ+5 ) − m(Θ+) = ms ≈ 150 MeV.

• Mixing: effect only on N5 and Σ5.

4

Experimental Evidence

• First experimental sign: Nakano et. al. (LEP collaboration,

hep-ex/0301020)

– mass 1540 ± 10 MeV

– width less than 25 MeV

• Other experiment: V.V. Barmin et al. (hep-ex/0304040) →very narrow width (≤ 9 MeV)

• Current experimental status:

– positive signal: 10 published experiments

– non-observing experiment: 11 published results

• LEP still see signal (see talk by Nakano at International

Conference on QCD and Hadronic Physiscs, Beijing).

5

Quantum Numbers

• SU(3)F : 3 ⊗ 3 ⊗ 3 ⊗ 3 ⊗ 3 → many possibilities (multiplets).

• Θ+ (s = +1) can be member of 35plet, 27plet or antidecuplet.

• Antidecuplet??

isospin = 0 (searches for θ++ → no result, J. Barth et al.,

hep-ex/0307083)

• result has been ruled out by new CLAS g11 experiment

• also STAR see signal for Θ++ (see talk by Huang at

International Conference on QCD and Hadronic Physics,

Beijing).

• The spin and parity of the Θ+ is currently unknown

6

Θ+ Photoproduction

γ(q) + N(pN ) −→ K(pK) + Θ+(pΘ)

s ≡ (pN + q)2, t ≡ (q − pK)2, and u ≡ (q − pΘ)2

(M. Guidal, HJK, M.V. Polyakov, M. Vanderhaeghen,

hep-ph/0507180)

7

γn → K−Θ+ : JP

Θ = 1

2

+(

1

2

−

)

8

MK = (−i) e gKNΘ · PKRegge(s, t) · εµ(q, λ)

×[

FK(t) · (2pK − q)µ · Θ γ5 N

− FΘ(u) · (t − m2K) · Θ γµ (γ · pu + MΘ)

u − M2Θ

γ5 N

+ 2pµK · (F (s, t, u) − FK(t)) · Θ γ5 N

−(

t − m2K

u − m2Θ

)

· 2pµΘ · {F (s, t, u) − FΘ(u)} · Θ γ5 N

]

• difference between (+) and (-) parity: only in γ5 and (±, i)

• t- and u-channel reggeized the same way (required by gauge

invariance)

• the last 2 terms: required contact terms

9

Θ Decay and KNΘ Vertex

JP = 1

2

− ⇐⇒ l = 0, S-wave

JP = 1

2

+ ⇐⇒ l = 1, P-wave

JP = 3

2

+ ⇐⇒ l = 1, P-wave

JP = 3

2

− ⇐⇒ l = 2, D-wave

10

KNΘ+ vertex and Θ+ Width

• JP = 12

+

LKNΘ = i gKNΘ

“

K†Θγ5N + Nγ5ΘK

”

ΓΘ→KN =g2

KNΘ

2π

|pK |MΘ

„

q

p2K + M2

N − MN

«

• |pK | ' 0.267 GeV

• with ΓΘ→KN = 1 MeV → gKNΘ ' 1.056

• JP = 12

−

LKNΘ = gKNΘ

“

K†ΘN + NΘK

”

ΓΘ→KN =g2

KNΘ

2π

|pK |MΘ

„

q

p2K + M2

N + MN

«

• ΓΘ→KN = 1 MeV → gKNΘ ' 0.1406

11

K Regge Exchange

-2 0 2 4 6

-2

0

2

4

α

m2

K

t (GeV2)

K (0.490)

K1 (1.270)

K2 (1.770)

K3 (2.320)

K4 (2.500)

• imagine K as a tube (rod) with 2 quarks at

the end → rotate → higher angular

momentum

• higher-spin excitation of Kaons lie on K

Regge trajectory

• K Regge trajectory for t-channel:

αK(t) = α0K + α′

K · t

• trajectories can be either non-degenerate or

degenerate

12

Non-degenerate Regge Trajectory

1

t − m2K

=⇒ PKRegge(s, t) =

„

s

s0

«αK(t)πα′

K

sin(παK(t))

× SK + e−iπαK(t)

2

1

Γ(1 + αK(t))

• s0 ' 1 GeV

• standard linear trajectory for K is:

αK(t) = 0.7 (t − m2K)

• as t → m2K , αK → 0 and PK

Regge(s, t) → 1t−m2

K

• Γ(1 + α(t)) suppresses propagator poles in the unphysical region

• JP = 0−, 2−, 4−, ... correspond with S = +1

• JP = 1+, 3+, 5+, ... correspond with S = −1

13

Degenerate Regge Trajectory

Degenerate trajectory: obtained by adding or subtracting the two

non-degenerate trajectories with the two opposite signatures

1. a constant (1) phase

2. a rotating (e−iπα(t)) phase: strong degeneracy assumption

1

t − m2K

=⇒ PKRegge(s, t) =

„

s

s0

«αK(t)πα′

K

sin(παK(t))

× e−iπαK(t) 1

Γ(1 + αK(t))

14

• Regge vertex functions (Regge residues) away from the pole

position, need form factors FK(t), FΘ(u), and F .

• We choose monopole forms for FK and Fu :

FK(t) =

(

1 +−t + m2

K

Λ2

)−1

,

FΘ(u) =

(

1 +−u + M2

Θ

Λ2

)−1

,

• where cut-off Λ is chosen to be 1 GeV,

• and F = FK(t) + FΘ(u) − FK(t) · FΘ(u) to cancel the poles in

the contact terms.

15

γp → K0Θ+ : JP

Θ = 1

2

+(

1

2

−

)

MK∗ = (i) e fK∗0K0γ fK∗NΘ · PK∗

Regge(s, t) · εµ(q, λ)

× εµνλα qν (q − pK)λ Θ

»

i σαβ (q − pK)β

MN + MΘ

–

γ5 N (1)

• fK∗0K0γ is determined from radiative K∗ decay experiment

• similar term contributes to γn → Θ+K− if K∗ is considered

16

K∗NΘ+ vertex

• JP =“

12

+” “

12

−”

LK∗NΘ = −i fK∗NΘ Θ

»

i σµν pνK∗

MN + MΘ

–

γ5 N · Vµ(pK∗) + h.c.

• Using SU(3) symmetry for the vector meson couplings within the

baryon octet and SU(6) symmetry between the baryon octet and

antidecuplet

gρ0pp + fρ0pp =7

10

„

V0 +1

2V1

«

+1

20V2 = 18.7(input)

gφpp + fφpp = − 1

10

„

V0 +1

2V1

«

+7

20V2 = 0(input)

fK∗0Θ+p =3√30

„

V0 − V1 −1

2V2

«

17

• gV NN (fV NN ): vector (tensor) coupling constants

fK∗0pΘ+ = (gρ0pp + fρ0pp)3√

3√10

4/5 − r

r + 2

• where r ≡ V1/V0 ' 0.35 is obtained in the chiral quark soliton

model

fK∗NΘ ≡ fK∗0pΘ+ ' 5.9

• rescaling r, corresponding to Θ+ width of 1 MeV yields:

fK∗NΘ ≡ fK∗0pΘ+ ' 1.1

18

K∗ Regge Exchange

1

t − m2K∗

=⇒ PK∗

Regge =

„

s

s0

«αK∗ (t)−1πα′

K∗

sin(παK∗(t))

× S∗K + e−iπαK∗ (t)

2

1

Γ(αK∗(t))

• Regge trajectory: αK∗(t) = α0K∗ + α′

K∗ · t

• standard linear trajectory for the K∗(892) is :

αK∗(t) = 0.25 + α′K∗ t

• where α′K∗ = 0.83 GeV−2

• strong degeneracy assumption

19

γn → K−Θ+ : JP

Θ = 3

2

+(

3

2

−

)

• additional contact term: to preserve

gauge invariance

• Θ → Θα (spin-3/2): change the vertex

and propagator structure

20

MK =(−) (i) e gKNΘ

mK

· PKRegge(s, t) · εµ(q, λ)

׈

FK(t) · (2pK − q)µ · (pK − q)α · Θα γ5N

− FΘ(u) · (t − m2K) · Θα γ

αβµ (γ · pu + MΘ)

u − M2Θ

·Sβν · γνσρ (pK)σ · (pu)ρ

M2Θ

γ5N

+ (t − m2K) · Θα γ

αµν (FΘ(u) · pK + FK(t) · pΘ)ν

MΘγ

5N

+ 2pµK · (F (s, t, u) − FK(t)) · p

αK · Θ γ

5N

−„

t − m2K

u − m2Θ

«

· 2pµΘ · {F (s, t, u) − FΘ(u)} · p

αK · Θ γ

5N

–

with

Sβν = gβν − γβ γν

3− (γβ (pu)ν − γν (pu)β)

3MΘ− 2 ((pu)β(pu)ν)

3M2Θ

• again difference between (+) and (-) parity: only in γ5 and (±, i)

21

KNΘ+ vertex

• JP = 32

+

LKNΘ =gKNΘ

mK

n

ΘαgαβN

“

∂βK

”

+ NΘαgαβ

“

∂βK

†”o

ΓΘ→KN =g2

KNΘ

2π

|pK |MΘ

|pK |23m2

K

„

q

p2K + M2

N + MN

«

• with ΓΘ→KN = 1 MeV → gKNΘ ' 0.4741

• JP = 32

−

LKNΘ =gKNΘ

mK

n

Θαγ5Ngαβ

“

∂βK

”

+ Nγ5Θαgαβ

“

∂βK

†”o

ΓΘ→KN =g2

KNΘ

2π

|pK |MΘ

|pK |23m2

K

„

q

p2K + M2

N − MN

«

• ΓΘ→KN = 1 MeV → gKNΘ ' 3.558

22

Cross Section for γn → Θ+K−

0

0.05

0.1

0.15

0.2

0.25

σ tot (n

b)

K Regge

K* Regge

K + K* Regge

K - K* Regge

0

2

4

6

8

10

12

5 10 15

s (GeV2)

0

0.5

1

σ tot (n

b)

5 10 15

s (GeV2)

0

10

20

30

40

50

60

1/2-

1/2+

3/2+

3/2-

• Note the uncertainty in the sign of the coupling fK∗0pΘ+

23

Cross section for γp → Θ+K0

0

0.03

0.06

0.09

0.12

σ tot(n

b) K* Regge

5 10 15

s (GeV2)

0

0.05

0.1

0.15

0.2

σ tot(n

b)

1/2-

1/2+

24

Differential Cross section for γn → Θ+K−,

s= 8.4 GeV2

0

0.05

0.1

0.15

0.2

0.25

dσ/d

t (nb

/GeV

2 )

K Regge

K* Regge

K + K* Regge

K - K* Regge

0

5

10

0 0.2 0.4 0.6 0.8 1

-t (GeV2)

0

0.2

0.4

0.6

0.8

dσ/d

t (nb

/GeV

2 )

0 0.2 0.4 0.6 0.8 1

-t (GeV2)

0

10

20

30

40

1/2-

1/2+

3/2+

3/2-

• spin-1/2− has peak around −t ' 0.2 GeV2

25

Differential Cross section γp → Θ+K0,

s = 8.4 GeV2

0

0.01

0.02

0.03

0.04

dσ/d

t (nb

/GeV

2 )

K* Regge

0 0.2 0.4 0.6 0.8 1

-t (GeV2)

0

0.01

0.02

0.03

dσ/d

t (nb

/GeV

2 )

1/2-

1/2+

• γp has peak(from K∗)around −t ' 0.4− 0.5GeV2

26

Photon Asymmetries for γn → Θ+K−,

s= 8.4 GeV2

-1

-0.5

0

0.5

1

Σ

K Regge

K* Regge

K + K* Regge

K - K* Regge

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

-t (GeV2)

-1

-0.5

0

0.5

Σ

0 0.2 0.4 0.6 0.8

-t (GeV2)

0

0.2

0.4

0.6

0.8

1/2-

1/2+

3/2+

3/2-

• Σ =σ⊥−σ‖

σ⊥+σ‖

• K(K∗) exchange dominant, Σ go to -1(1)

27

Photon Asymmetries for γp → Θ+K0,

s = 8.4 GeV2

0

0.2

0.4

0.6

0.8

1

Σ K* Regge

0 0.2 0.4 0.6 0.8 1

-t (GeV2)

0

0.2

0.4

0.6

0.8

Σ

1/2-

1/2+

28

Decay Angular Distribution

29

• angular distribution of Θ+ → K+n

W (θ, φ) =∑

sf ,s′f;sθ,s′

θ

Rsf ,sθρsθ,s′

θ(Θ+)R∗

s′f,s′

θ

=∑

sθ,s′θ

{

R−

12,sθ

ρsθ,s′θR∗

−12,s′

θ

+ R 12,sθ

ρsθ,s′θR∗

−12,s′

θ

+R 12,sθ

ρsθ,s′θR∗

12,s′

θ

+ R−

12,sθ

ρsθ,s′θR∗

12,s′

θ

}

• The transition operator

Rsf ,sθ≡ 〈n, sf ,Pθ − p′|t|Θ+, sθ,Pθ = 0〉

• ρsθ,s′θ: the photon density matrix elements in the Θ+

production

30

1.0 in

0

0.05

0.1

0.15W

0

0.05

0.1

W

0

0.05

0.1

W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

3/2+

3/2-

φ = 0 φ = π/4 φ = π/2

Figure 2: γn → Θ+K−, α = 0, s= 8.4 GeV2, t = −0.1 GeV2

31

1.0 in

0

0.05

0.1

0.15

0.2W

0

0.05

0.1

0.15

W

0

0.05

0.1

0.15

W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

0.15

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

3/2+

3/2-

φ = 0 φ = π/4 φ = π/2

Figure 3: γn → Θ+K−, α = 1, s= 8.4 GeV2, t = −0.1 GeV2

32

1.0 in

0

0.05

0.1

0.15

0.2W

0

0.05

0.1

0.15

W

0

0.05

0.1

0.15

W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

0.15

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

3/2+

3/2-

φ = 0 φ = π/4 φ = π/2

Figure 4: γn → Θ+K−, α = 2, s= 8.4 GeV2, t = −0.1 GeV2

33

1.0 in

0

0.05

0.1

0.15

0.2W

0

0.05

0.1

0.15

W

0

0.05

0.1

0.15

W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

0.15

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

3/2+

3/2-

φ = 0 φ = π/4 φ = π/2

Figure 5: γn → Θ+K−, α = 3, polγ = −1, s= 8.4 GeV2, t = −0.1 GeV2

34

1.0 in

0

0.05

0.1

0.15

0.2W

0

0.05

0.1

0.15

W

0

0.05

0.1

0.15

W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

0.15

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

3/2+

3/2-

φ = 0 φ = π/4 φ = π/2

Figure 6: γn → Θ+K−, α = 3, polγ = 1, s= 8.4 GeV2, t = −0.1 GeV2

35

1.0 in

0

0.05

0.1

0.15W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

φ = 0 φ = π/4 φ = π/2

Figure 7: γp → Θ+K0, α = 3, polγ = −1, s = 8.4 GeV2, t = −0.1 GeV2

36

1.0 in

0

0.05

0.1

0.15W

-1 -0.5 0 0.5 1Cos θ

0

0.05

0.1

W

-0.5 0 0.5 1Cos θ

-0.5 0 0.5 1Cos θ

1/2-

1/2+

φ = 0 φ = π/4 φ = π/2

Figure 8: γp → Θ+K0, α = 3, polγ = 1, s = 8.4 GeV2, t = −0.1 GeV2

37

Some observable of the W (decay angular distribution):

• more variation in:

– γn → K−Θ+

– circular polarization

• hard to miss spin-3/2 characteristic

• φ = π/2 mostly not interesting

38

Conclusion

• theoretically both parity assignment of the Θ+ are allowed

• the spin of the Θ+ can be 1/2 or 3/2

• photoproduction experiments: good place to determine the

spin and parity of the Θ+

• if statistics allow, decay angular distribution is even better to

determine the spin and parity of the Θ+

39