Nuclear Physics A468 (1987) 669-682

North-Holland. Amsterdam

pp + HYPERON-ANTIHYPERON TRANSITION POTENTIAL

IN THE QUARK MODEL*

S. FURUI and Amand FAESSLER

Institui fCr Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, 7400 Tiibingen, F. R. Germany

Received 30 October 1986

(Revised 5 January 1987)

Abstract: The transition potential of pp to the hyperon-antihyperon (fi) system is calculated in the

constituent quark model with the ‘PO model and with the %, model. The 3P0 model allows the

tensor coupling between the pp and YY states while the ‘S, model restricts that the pp and YP

have the same orbital angular momentum. To reproduce the relative ratios of pp annihilation into

hyperon anti-hyperon qualitatively, the tensor interaction is important and the 3P0 model compares

better with the experiment than the ‘S, model.

1. Introduction

The transition from pp to hyperon-antihyperon is conventionally assumed to

occur through the exchange of K, K* and/or K** mesons. Since the masses of these

mesons are relatively large and the range of the potential is almost the same as the

size of hadrons, one could ask about an alternative description of the potentiai in

the quark model. The simplest mechanism for the transition is one uii or dd pair

annihilation and one SS pair creation. Genz and Tatur ‘) studied pp annihilation

into hyperon-antihyperon pair production with an assumption that the annihi-

lated/created qq pair has the quantum number of a gluon. Since a gluon has the

quantum number of 3S1 we call the model the 3S1 model. They found that except

for the ;I;+E*+ production the experimental data compared well with the experiment

in the region of laboratory momentum p = 2-6 GeV/c region. Rubinstein and Snell-

man ‘) also studied hadronic processes including a creation of an SS pair in the 3S,

model.

The annihilated/created qq pair has, in the 3P0 model, the quantum number of

the vacuum. This model was successful in explaining meson decays, meson-baryon

coupling constants and pp annihilation into mesons 3*4). In the analysis of pp

annihilation into rTT+rr- and into K+K-, we found that the ss pair creation should

be suppressed as compared to uii or dd pair creation by about a factor 4 [ref. “)I.

In the analysis of pp annihilation into A ++A++ and into E+F in the 3S1 model I),

the SS pair creation was found to be suppressed as compared to uii pair creation

* Supported by the Deutsche Forschungsgemeinshaft.

0375-9474/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

670 S. Furui, A. Fuessler / pp --, w

by a factor 21. The transition to hyperon-antihyperon systems in the 3P0 model is different from that in the 3S, model and consequently the phenomenological sup- pression factor for the 3P0 model will be different. We study in this paper the transition potential in the 3P0 model and in the 3S1 model and compare with the K

and K* exchange potentials.

2. The 3Po model

We describe hadrons as products of three constituent quarks which are assumed to be given by gaussian functions. In momentum space a nucleon and a hyperon wave function are

where

NN= (3R$7~)~‘*,

NY=(3R&%r)3’2.

The transition operator in the 3P0 model is

O=hzVL6,B~,(qs-q6)15(q3+q~)~7-8v*~T”(q7-q~)8(q7+q8)~(-1)‘+~~(-l)‘+~,

where A is a strength parameter for qq pair creation. A strength parameter for SS pair creation is denoted by A, in order to distinguish a SS pair creation from a uii or dd. These parameters will be fitted from the expe~mental data. qi with i = 1 to 6 are the quark momenta of the pp system, and qj with i = 1, 2, 7,4, 5, 8 are those of the yii system. We calculate the transition amplitude for each partial wave of the pp by sandwiching the operator between the wave functions I,!+?= 41y&x and lcfpp = #p#px where x represents spin, flavour, colour and the relative orbital wave function, and performing the integral over the phase space of quarks. The matrix elements are calculated by using the SU(6) wave functions for the hadrons. They give a good approximation to the true wave functions near the threshold. We perform partial wave expansion and express the amplitudes in the form

T= d3q,.. I

* d3g, ~~~0~~~~ fI qi ( >

~(q~+qz+q?~q4+qs+q~) i=l

ZZ c S(O)?;&c’, k)Y:‘,,,(ff)Y’,:,,(~)([CJ ~a]‘~‘>, _YJLL’SS’M

where YisM (i) is the vector spherical harmonics and ([u x a](“‘) is the reduced spin-flavour matrix element. In the 3Po model, there appear scalar (L?? = 0) and tensor

S. Furui, A. Faessler / pji+ fl 671

(Z= 2) coupling terms. The final form of the transition amplitude for the total angular momentum f, pp relative angular momentum L, YY relative angular momen- tum L’, pp relative momentum k and fi relative momentum k’ reads

C I?,,,.( k’, k)([a x w](~‘) 9

2R2,

3(R2,+ R:) )

X {( l- 2R2,

3(R&+R;) 3(R&+R$)+ >

R: R”N 3(R2,+R;)

x C agiiti’ 1 L 1

E=O,2 ( > 0 0 0

xexp -- [

: RRyR2 (k2+kr2) , N Y 1

where i = m and the amplitude aSwT J specifies the relative strength of the tensor (9 = 2) and the central (2 = 0) component for the spin singlet (S) and for the spin triplet (T) channel. The last term has the same structure as that of the 3S1 model, and the amplitude asT can be calculated by performing the Fierz transformation 1*5). We calculate the amplitudes azT for both _Y = 0 and S = 2 by using the technique of the coeflkient of fractional parentage (cfp). In the 3Po model there appears also couplings between singlet and quintet state. The transition amplitude for this case becomes

672 S. Fun& A. Faessier / p@+ yr’

We estimate the strength parameter A* from the value fitted in the analysis of the pp annihilation into two mesons. The optical potential for the process shown in fig. 1 has the form:

V(k’,k,E)=h6 q2 dq g(q, k)g*(q, k’)

&ME - EI - &I

and A6 has the dimension of [GeV2]. If time ordered pe~urbation theory is applied for this diagram, four energy denominators should appear in the expression. We took the relativistic phase space for mesons, but the 3P0 vertex was defined for

Fig. 1. The contribution to the optical potential for pp scattering with VT or K+K- in the intermediate

state.

S. Furui, A. Faessler / pp+ w 673

nonrelativistic kinematics. Consequently the factor 1/E,E2 should be compensated

to compare with the original ‘PO model. We interpret A6 = y6 * (E,E,)/(E~E~), where

l/& and l/E, are the energy denominator that appear in the intermediate state.

In the transition amplitude for pp + fi, we interpret A2 = r*/(EJ, where l/E, is

the energy denominator for the intermediate qqG state. If all the energy scales are

of the order of 1 GeV, we obtain A,h = A2/4.4[GeV-‘1, where 4.4 is the suppression

factor estimated from the branching ratios of pp+ K+K- and pp+ rrcrTT-.

3. The 3S1 model

One considers in the 3S, model transitions from a uii or dd to a gluon in the

s-channel and transition from the gluon to a sS. One could consider also an interaction

between a uu or dd and a SS by a Coulomb-type gluon. For the process with a gluon

in the s-channel, the transition amplitude for a qq pair annihilation into a gluon

and subsequent qq pair creation is given by

1 t = g, - iiy,vl7y,u )

s

where s is the center-of-mass energy squared of

impulse approximation the amplitude reduces to

the gluon: s = (p + 4)‘. In the

where N is a constant normalizing the spinor of the quarks. For simplicity, one

usually chooses the first term only.

When the gluon is of Coulomb type, then the factor l/s changes to l/( Q*+ m’)

where Q is the momentum of the gluon and m is a screening mass which is estimated

to be about 0.6 GeV [ref. I”)].

In both cases the operator for the transition can be expressed in the form

Q= ; 4m,A:603,66(q3-q,-Q)A;8a7,p8(q,-q8-Q)- ij=l

The transition amplitude for pp to ti in the 3S, model for a gluon in the s-channel

becomes

fiL,(kf, k)([u x a](‘)) = 4m,N2,N2 ‘(R,;R:)“2(f)“2(3~R~~R~~~‘2 1

X7 imLjL (

4 R&R;

2T -j_

3 R2,+Rtkk’ >

ILL'

x exp -; P;yj2 (k2+ k”)] ;$. N Y

674

Fig. 2(a). Transition from pp to fi in the 3Po model. (b) Transition from pp to fi in the 3S, model. The wiggly line represents a gluon.

Although the qq gluon vertex is effective for spin 1 and color 8, on the baryon level the operator is effective for both spin triplet and spin singlet states. In the case of Ax the spectator quarks are coupled to spin 0 and consequently the transition occurs only for the triplet state. When the second term of the quark-antiquark gluon vertex is included, then the coupling between the singlet and the triplet state becomes possible due to the difference of the s-quark and u-quark masses. In the present model, the spin matrix element for the 3S1 model is the same as the last term of the 3P,, model. A main difference between our 3S1 model and that of ref.8) is that our potential is nonlocal and state dependent.

4. The transition amplitude p#i + Y%!

In table 1 we show spin-flavour matrix elements for p( a)p( CT) + Y( a’)Y( 8) where cr and (T’ denote the spin projection of the proton and the hyperon respectively, and C? and ii’ are those of anti-proton and anti-hyperon respectively. Spin-flavour matrix elements for s and for A++A++ creation are also shown. Here we defined

(s’a’S’#, S’M’J O,]saScF, SM)

~(-1) s’+b’+S+@-, s s (’

-r s S’ cl @’ -M’ >

e-w”‘+“’

> (-l)s-s+M( Y(d) F(cqPx P]‘“‘~p(u)jqtF)>,

where S and S’ denote the total spin of the initial and the final states respectively, and M and M’ denote the magnetic quantum number of the corresponding states quantized along the beam axis. The operator 0, contains [a x cr](‘) and the operator OZ contains [a x a] (‘) The amplitudes for 2 = 0 are the same as the results of Genz . and Tatur ‘) except the colour matrix element f, which is replaced by 1 in the 3Po model. We think the value for the Z’x” production in ref. ‘) is not correct.

In table 2 the cross sections estimated with the statistical weight 3 and 1 for the initial pp3S1 and ‘So states are shown. From the table we find that the tensor coupling

S. Furui, A. Faessler / p@ -$ ‘uii 615

Fig. 3. Half off-shell transition potential (A,?(E’)]T(E)]pp(E)) in units of [GeV-*] for the ‘S, and the

‘D, diagonal and the %, to 3D, and the ‘D, to 3S, off-diagonal transitions for the the initial antiproton

laboratory momentum pr = 1.5 GeV/c as a function of the off-shell center of mass relative Ax momentum

in GeV/c. The on-shell final relative momentum in the c.m. system is indicated by the crosses (p,,~ = 0.155 GeV/c). The 3P, and the 3S, models are compared with the K, K* meson exchange model. The

‘P,, results are shown by the solid line, the %, is given by the dashed-dotted curve. The exchange of the

K* alone is represented by the dashed line, while the dashed-dot-dotted line results from the exchange

of K and K*.

between spin singlet pp and spin quintet .X+-Z*’ or ?X*’ is large and the agreement with the experiment in the 3Po model becomes better than that of the 3S1 model. The experimental A+‘A++ production cross section at Iab momentum p = 5.7 GeV/c is 1.28 mb [ref. “)I, while the 3S1 model predicts that is about 60 kb [ref. “)I. There- fore in the 3S, model one expects a suppression factor of about 21 for a SS creation. In the ‘PO model, if the L5’= 2 component is added with simple statistical weight the cross section becomes 367 pb. We ignored corrections to ac3’ due to the .Y = 2 component since it is expected to be smaller than S = 0 component. Coupling between the spin singlet and spin quintet state could be important. We found in a coupled channel calculation of pp scattering and annihilation a strong coupling from 2T+‘T2S+1LJ = ‘rDZpp to ‘?S2 virtual Ad [ref. “)I. If the spatial part of the matrix element is taken into account, the above estimate of the Ad cross section may be

676 S. Funk, A. Faessler / pfi +- ti

considered as an upper limit, and the suppression factor 3.5 as a lower limit. On the other hand, the model underestimates the Z+F production also by about a factor 3.5. It is clear that the final state interaction is important for the total cross section and we cannot rely too much on the suppression factor derived from the Born term. In any case, the 3P0 model predicts a smaller suppression factor than that of 3S, model and it is not inconsistent with the estimation from the pp annihilation into two mesons, which yields a suppression factor of 4.4’.

5. Comparison with K, K* and K** meson exchange

The transition from pp to Ax in the meson exchange model was calculated by several authors 6*8). In the high energy region there exists an analysis with Regge pole paramet~zation “). Although relative importance of the three mesons at low energy is not well known, the analysis shows that the Regge trajectories of K*(892 MeV) and its exchange degenerate partner K**( 1430 MeV) play an important role and K-meson exchange is negligible in the high energy region. The importance of K”” suggests that the 2 = 2 term should exist in the transition potential.

Since the Z NK coupling constant is much smaller than the /INK coupling constant r*) we expect the K-meson does not play an important role in the X? production. In order to compare the two models we show in fig. 3,4 and 5 the half off-shell transition matrices for 3S, and 3D1 channels of pp + A_& pp + A3 and pp+ Z+F in the quark model and in the meson exchange model. The on-shell points are marked by crosses. The momentum of the anti-proton in laboratory system are 1.5 GeV/c, 1.7 GeV/ c and 1.9 GeV/c respectively. In the 3S1 model, one-gluon exchange in the s-channel is considered here. The absolute strength of the potential from the quark model is arbitrary. The meson exchange potentials contain static central, spin-spin, spin-orbit and tensor terms, and in order to take into account the mass difference of a nucleon and a hyperon, we adopt the prescription of Durso et al. 13). We introduced a monopole cut-off of 1.2 GeV/c on the meson-baryon vertex. We observe that the transition potential in the D-wave is strong in the ‘PO model. The meson exchange transition potential between 3S, and 3D1 of pp+ AA is strong due to coherence of K(494 MeV) and K*(X92 MeV) contributions.

The differences in these models could manifest themselves in differential cross sections and polarization observables. These values are, however, sensitive to the initial state interaction and the final state interaction. The interactions in the hyperon- antihyperon channels are theoretically and also experimentally not well known. In order to see qualitative features, we adopt a procedure originated by Sopkovich 14) and extended by Tabakin and Eisenstein 6), and calculate the modified T matrix

“J T L’s’,Ls = &3Jt~s~T&s~,Ls~ .

Here T is the transition matrix from the Born approximation and the S are the S-matrices describing the initial and final state elastic scattering. The S-matrices

S. Fwui, A. Faessler f pp -f ti 677

01 0;s 1.0

3DS, ~-----..,,~

: w z

0 0.5 1.0 NC1

Fig. 4. Half o&hell transition potential of 3S, and 3D, channels of pp+ A3 for pL = 1.7 GeV/c. For a more detailed description, see fig. caption 3. The on-shell relative momentum in the final state indicated

by a cross is 0.133 GeV/c.

Fig. S. HaIf off-shell transition potential of 3S, and 3D, channeis of pp-r Zi-3 for pre = 1.9 G&/c For more details see fig. caption 3. The crosses indicate the relative on-shell momefltum of Z+ and 3 in

the final state of 0.102 GeVic.

TABLET

The

oret

ical

sp

in-f

lavo

ur

mat

rix

elem

ents

(s

’oS’

a’,

S’M

’lO

~lso

, SC

?, SM

) de

fine

d in

eq.

(4

.1)

for

p(c?

)p(a

)+

B(c?')B(o'),

whe

re

Crc

r and

o’

b’

indi

cate

th

e sp

in

proj

ectio

ns

of t

he

initi

al

prot

on-a

ntip

roto

n an

d th

e fi

nal

bary

on-a

ntib

aryo

n st

ates

al

ong

the

beam

ax

is,

resp

ectiv

ely.

(a’;

S’M

’Iu;

SM

)

6aT

6a

z 6a

: 6a

s

BB

(+

; 11

1+;

11)

(-;

lOI-

; 10

) (+

; lo

t-;

10)

(-;

OO

I-;

00)

(+;

111+

; 11

) (-

; lO

I-;

10)

(+;

lOI-

; 10

) (-

; 20

1-;

00)

AA

9

912

912

0 0

on

-2J5

-5

&/2

&

I2

-3A

-k

-‘

J3

-5

0 Y

E0

713

1316

116

2

-l/3

-l

/3

-l/3

0

P+P+

-1

413

-131

3 -l

/3

-4

213

213

213

0

z*+

z+

-2&

j3

-&I3

-&

I3

0

a/3

4Jzl

3

-2fi

/3

2Jz

i++,

++

8&

4lf

i/6

a/6

46fi

/6

-1oJ5

-118

fi/6

-A

/3

-58f

i/3

The

to

tal

spin

an

d sp

in

proj

ectio

ns

are

in

the

initi

al

stat

es

S=O

, 1;

M

= -

1,

0,

1 an

d in

th

e fi

nal

S’,

M’.

T

he

colu

mns

gi

ve

the

diff

eren

t ab

ove

mat

rix

elem

ents

in

th

e no

tatio

n (a

’;

S’M

’la;

SM

).

The

y ca

n be

ex

pres

sed

by

the

para

met

ers

a, S,

T i

n th

e fo

llow

ing

way

: (g

, g;

lll

O,l$

$,

2;

11)

= 6a

$;

(ff,f

-f;O

OlO

,lf~,

f;00

)=6a

~;

(fi,$

-f;2

01O

&f,f

-f;0

0)=

6ag.

TABLE

2

The

th

eore

tical

cr

oss

sect

ions

su

mm

ed

over

po

lari

zatio

ns

for

pp+

BE

BB

(3)

(To

(1)

co

(5)

02

vw+

(1)

O;ll

eor

uexp

flt

theo

r fle

x,

pr

= 3.6

GeV

/c

pL =

5.1

GeV

/ c

&I

24314

0

0

24314

24314

77in

put

77*1

8

31in

put

31*6

Lsn

9

2114

0

6314

6314

19.0

33.5

* 10

7.6

5

14.5

*3

,0x0

49112

1

0

61112

61112

5.8

0

SlO

2.5

9

Fz+

4913

4

0

6113

6113

23.3

2

30*4

9.8

8

34*4

s+

s+

213

0

10

213

3213

10.0

5

12*3

4.8

0

d++

d +

+ 96

529118

13456118

125.4

716.9

829.9

367.0

12805 1

10

The

ex

peri

men

tal

data

fo

r pL

= 3

.6 G

eV/c

(c

olum

n 8)

and

fo

r P

L =

5.7

GeV

/c

(col

umn

10)

r’)

are

com

pare

d w

ith

the

pred

ictio

ns

(col

umn

7 an

d 9)

. T

he

colu

mn

c I)

+

is t

he

sum

of

am

plitu

de

squa

red

mul

tiplie

d by

th

e st

atis

tical

w

eigh

t fo

r th

e ?S

, m

odel

an

d th

e co

lum

n (T

(3)+

(‘)+

(5)

is t

he

corr

espo

ndin

g va

lues

for

the

3Po

mod

el.

The

th

eore

tical

cr

oss

sect

ions

ar

e es

timat

ed

for

the

3Po

mod

el,

by

mul

tiply

ing

the

phas

e sp

ace

fact

or

to t

he

sum

of

the

am

plitu

de

squa

red.

S. Fun& A. Faessler / pfj + w 679

are parametrized by using a Frahn-Venter model “) in which a central and a spin-orbit piece of the potential is contained. In the case of pp transition to A& we choose parameters for the initial state interaction same as the ref. “) and fit parameters in the final state. The sign of the spin-orbit potential and the magnitude of the diffuseness of the potential d in the final state is crucial for the polarization and the shape of the differential cross section. Tabakin and Eisenstein “) fixed the sign of the spin-orbit potential from the fit to the experimental data of Plab = 6 GeV/ c,

Fig. 6. Angular distribution du/dL? [ pb/,,] and polarization PY for plRb= 1.5 GeV/c calculated in the

3P0 model. The angle 9 is the angle between the beam axis and the direction of the momentum of the

x in the center of mass system. Experimental data are from ref. 16).

680 S. Furui, A. Faessler / p~5 + *

1 I T. / I

/' \' \

O- \

i

i __----___ \ ,A=- 7_---__-___

\ /= .I'* i CR i L .’ i ‘\ !

I /“( t / I

~

_

-0.5 /I” \ ::,z

‘\ ; ‘_.

/’ \ \

\ ./

‘\._./

-1 I I I I -1 -0.5 0 0.5 1

cosa

Fig. 7. Spin correlation Cij for p,ab = 1.5 GeV/c calculated in the %$ model. The solid curve represents C,,, the long dashed curve represents C,,,,, the short dashed curve represents C,, and the dash-dotted

curve represents C,,.

but the data of pp elastic scattering show that the sign could be energy dependent. We choose the parameters R, d, p+, p_ and E for the AA channel as 2.52 fm, 0.20 fm, -1.211, 0.866 and 0.01 respectively. Fig. 6 shows the angular distribution and polarization for pp -+ AA process at Plab = 1.5 GeV/ c calculated in the 3P0 model, and they are compared with the recent experimental data from LEAR 16). The fourth power of the parameter for the quark pair production A4 is chosen to be 5 in this model. Although the fitted value of A depends on parameters of the final state interaction, we observe that the value is suppressed as compared to that for u- or d-quark pair creation which is y4/(Ez) = 3.44 with (E,) = 1 GeV. Fig. 7 shows the behaviour of the spin correlation parameters C,j.

In the meson exchange model, we observe that the prescription of Durso et al. 13) makes the effective mass of the meson heavier, and the angular distribution from the K-meson exchange model becomes more oscillating, which reflects the fact that the s-wave component is more suppressed for heavier meson exchange. In the 3S1 model, the angular distribution in the backward region becomes more suppressed than the 3P0 model due to smaller D-wave amplitudes.

6. Discussion and conclusions

Rubinstein and Snellman ‘) discussed that due to the heavier mass of the s-quark as compared to those of u-, d-quarks, the SS creation could be treated perturbatively

S. Furui, A. Faessler / pfi + ti 681

in the 3S1 model or one-gluon-exchange model. We showed that the model is in

conflict with the experiment for the X+X*+ production and that the 3P0 model

improves the agreement with the experiment through additional tensor coupling

terms. A comparison of the 2+2+ production and the A++A++ production suggests

that a SS pair creation is suppressed as compared to a uiI pair creation and a

comparison of the K+K- creation and the ~T+Y creation also suggests that a SS

pair creation is suppressed as compared to a UU and dd pair creation if the

annihilation model dominates over the rearrangement model in the pp annihilation

into two mesons “). In the S-wave pp annihilation at rest into mesons, the isospin

mixing effect is expected to enhance the I = 0 component, which makes the ratio

of ~T+T- production and K+K- production worse 4*‘8). The suppression of SS pair

creation and the isospin mixing mechanism will explain consistently the branching

ratios of m+v-/K+K- and KLKs/K+K- from S-wave pp annihilation. A 3P,, model

with suppression for a SS pair creation through tunneling effects was recently

proposed by Dosch and Gromes in a framework of the strong coupling lattice

QCD [ref. “)I. The value of the suppression factor should depend on models of

hadrons. With a suppression factor for the SS creation as compared to uii or dd

creation, we obtained the transition potential of pp + AA which has almost the same

range as the sum of spin-spin and tensor component of K and K” meson exchange,

and with a reasonable parameter set for the final state, we could reproduce the

preliminary experimental data of LEAR. The 3S, model gives also the potential of

almost the same range if the s-channel one-gluon exchange is calculated. A Coulomb

type gluon exchange makes the range longer. Unlike the meson exchange model,

the 3S1 model does not give a tensor coupling term and shows different patterns in

the differential cross section. As pointed out by several authors, the final state

interaction is, however, crucial for the angular distribution and for the spin observ-

ables. To answer the question, whether the 3P,, model or the meson exchange model

or the combination of meson exchange and the 3S1 model describe the nature best,

it is necessary to have more information on the final state interaction and solve the

coupled equation for the pp and fi systems. Such investigation is in progress. This

will allow a direct comparison with new data from LEAR.

We thank Prof. R. VinhMau for useful discussions and K. Brluer for his help in

the calculation of spin observables.

References

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2) H.R. Rubinstein and H. Snellman, Phys. Lett. 165B (1985) 187

3) A. Le Yaouanc et aZ., Phys. Rev. D8 (1973) 2223; D9 (1974) 1415; Dll (1975) 1272

4) C.B. Dover, Medium energy nucleon and antinucleon scattering, Proc. of the Int. Symp. held at Bad

Honnef, June, 1985, ed. H.V. von Geramb (Springer, Berlin, 1985);

S. Furui, Z. Phys. A325 (1986), 375, (E) A327 (1987)

682 S. Furui, A. Faessler / pji + ti

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