Constraints in vector meson photoproduction

PHYSICAL REVIEW C, VOLUME 61, 015501

Constraints on vector meson photoproduction spin observables

W. M. KloetDepartment of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855-0849

Frank TabakinDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

~Received 23 August 1999; published 17 December 1999!

Extraction of spin observables from vector meson photoproduction on a nucleon target is described. Startingfrom density matrix elements in the vector meson’s rest frame, we transform to spin observables in thephoton-nucleon c.m. frame. Several constraints on the transformed density matrix and on the spin observablesfollow from requiring that the angular distribution and the density matrix be positive definite. A set ofconstraints that are required in order to extract meaningful spin observables from forthcoming data areenunciated.

PACS number~s!: 24.70.1s, 25.20.Lj, 13.60.Le, 13.88.1e

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I. INTRODUCTION

Availability of polarized beam, target, and recoil polarimetry, at, for example, Jefferson Lab, has opened enhaninterest in spin dependent measurements, such as photduction of vector mesonsg1N→V1N8. The dynamics ofthis reaction is strongly dependent on baryonic and mesresonances@1–4#. Detailed knowledge of the properties onucleon excited states provides tests of QCD inspired mofor the strong interaction.

In an earlier paper@5#, we emphasized that the anguldistribution of the pseudoscalar mesons that arise fromdecay of photoproduced vector mesons does not depenthe vector meson’s vector polarization, but only on its tenpolarization and that standard single and double spin obsables need to be defined in the overall photon-nucleon ceof mass frame. It was also found that a simple descriptionthe decay angular distribution in thegN c.m. frame is ob-tained by using the angle between the decay meson’s veity difference vector and the direction of the photoproducvector meson. The main purpose of this paper is to formua procedure for extracting meaningful spin observables frthe analysis of forthcoming vector meson photoproductdata to allow one to examine conventional spin observabThese spin observables are subject to known rules concing their forward and backward angular behavior@6#. Thenodal structure of spin observables@7–9#, e.g., their produc-tion angle dependence, might reveal important underlydynamics such as baryonic and mesonic resonances.

Here we show how to extract meaningful spin observabunder the assumption that analysis of the photoproductiovector mesons will yield a vector meson rest-frame denmatrix. Since new data are not yet available, we invokolder 1968 Aachenet al. information @10# and found thatsome of their vector meson rest-frame density matrix resuwhen transformed to spin observables in the photon-nuccenter of mass frame, violated basic constraints and thereneed to be rejected. The grounds for that rejection wassome of their elements, even including their stated uncert

0556-2813/99/61~1!/015501~7!/$15.00 61 0155

edro-

ic

ls

eonr

rv-terr

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g

sofyd

s,nreatn-

ties, yielded nonpositive, and therefore unacceptable angdistribution functions. That observation, which we subsquently found to be related to a set of constraints deducedothers earlier@11,12#, led us to examine the various constraints based on the positivity of the density matrix.

In Sec. II, we analyze the limits on the tensor polarizatiprovided by the simple requirement that the angular distrition of the decay mesons be a positive definite functiSimple limits on the tensor polarizations follow from evalating the decay angular distribution at selected anglesSec. III, the limits on observables due to positivity of thdensity matrix are discussed. In Sec. IV, constraints onvector meson’s density matrix due to Daboul’s analysis@11#using Schwarz inequalities, are invoked and analyzed.Schwarz inequalities described by Daboul can be repressed as four separate conditions on spin observaHowever, all but two of these conditions are already cotained in the simple requirement that the angular distributshould be positive definite. The two remaining conditioinvolve not only the tensor polarization, but also the vecmeson’s vector polarization. These conditions could alsoused to limit the vector meson’s vector polarization. Finalin Sec. V the method for extracting spin observables fractual data is outlined. Use of these basic constraints shbe included in the fitting procedure to assure that genrequirements concerning the angular dependence of spinservables, especially at forward and back angles, are satiand could then be used to deduce interesting new dynam

II. THE DECAY DISTRIBUTION

In photoproduction of a vector meson (r,f) on a nucleontarget the final vector meson decays into two pseudoscmesons. The angular distribution of the decay providesformation about the spin-state of the vector meson. Howeonly information about the tensor polarizationsT20

V ,T21V ,T22

V

can be obtained. The angular distribution of the pseudoscdecay mesons is given by@5#

©1999 The American Physical Society01-1

yr

nifiar

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wnion.sony total-.

W. M. KLOET AND FRANK TABAKIN PHYSICAL REVIEW C 61 015501

WV~ u,f !51

4pjV~ u !@12A2T2m

V ~Q,F!C2m* ~ uf !#

51

4pjV~ u !F12A1

2T20

V ~3 cos2 u21!

1A3T21V sin 2u cosf2A3T22

V sin2u cos 2fG .~1!

Here C2m* [A(4p/5)Y2m* is a spherical harmonic function

and the anglesu,f refer to the direction between the velocitvector difference,DvW [vW 12vW 2 , and the momentum vectoof the produced vector meson;vW 1 andvW 2 refer to the velocityvectors of the two decay mesons in the overall photonucleon center of mass frame. Use of these angles simplthe expression for the angular distribution in the overphoton-nucleon center of mass frame in which spin obse

01550

-esllv-

ables are defined. Note that the spin observablesT2mV depend

on the vector meson production anglesQ,F, as well as onthe total c.m. energy. The factorj, which arises from de-scribing the vector meson decay in the overall center of msystem and from a density of state factor, is given by

jV~ u !51

@sin2u1~Er /mr!2 cos2u #5/2, ~2!

wheremr ,Er are the vector meson’s mass and energy.The decay angular distributionWV( u,f) does not depend

on the vector meson’s vector polarization and as shoabove includes only the vector meson’s tensor polarizatOnce the angular distribution is measured and vector merest frame density matrices are provided, it is necessarmap that data over to the anglesu,f. One can then projecout the vector meson’s tensor polarization from the normized ratioWnorm

V ( u,f)[WV( u,f)/j, as described in Sec. V

TABLE I. Linear constraints forT20,T21,T22; uc5arccos(1/A3).

Constraints W>0 X†rX>0 Schwarz

12A2T20>0 u50 r00>0 r00>0

111

A2T20>0 u5

p

2; f5

p

4r11>0 r11>0

112

A3T21>0 u5 uc ; f5

p

4122A2Rer10>0

122

A3T21>0 u5 uc ; f5

3p

4112A2Rer10>0

112

A3T22>0 u5 uc ; f5

p

2112r121>0

111

A2T201A3T22>0 u5

p

2; f5

p

212r0012r121>0 ur121u2<r11r2121

111

A2T202A3T22>0 u5

p

2; f50 12r0022r121>0 ur121u2<r11r2121

221

A2T202A3T22>0

a

221

A2T201A3T22>0 u5

p

4; f5

p

2

121

2A2T201A3

2 T21>0 u5p

4; f5

p

411r0024Rer10>0

121

2A2T202A3

2 T21>0 u53p

4; f5

p

411r0014Rer10>0

112A2

A3T212

2

A3T22>0 u5 uc ; f50 124Rer1022r121>0

122A2

A3T212

2

A3T22>0 u5 uc ; f5p 114Rer1022r121>0

aThis condition follows fromx1<1.

1-2

th

sr

min

eni

hdrib

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a-

nsity

CONSTRAINTS ON VECTOR MESON PHOTOPRODUCTION . . . PHYSICAL REVIEW C61 015501

The tensor polarization must take on values that allowangular distribution functionWV( u,f) to be positive defi-nite. By selecting the anglesu,f one can use that obvioucondition to extract allowed limits for the tensopolarization.1 In the first and second columns of Table I a listis given of specific choices of anglesu,f and the resultingconditions onT20

V ,T21V ,T22

V . Such constraints also arise frodirect conditions on the density matrix, as will be seenSecs. III and IV.

Thus from Table I, we see that the simple requiremthat the decay angular cross section be positive yields limon the possible tensor polarization. We now consider otways to recognize constraints on the spin observables anassociated density matrix. In the next section, we descthe constraints onT20

V ,T21V ,T22

V that follow from the positivityof the density matrix.

III. LIMITS ON OBSERVABLES FOR A POSITIVEDEFINITE DENSITY MATRIX

Recall that for a general observableV the classical en-semble average is

^V&5

(a

vaâuVua&

(a

va

, ~3!

where va is the positive definite probability for findingbeam particle pointing in the direction stipulated by the Euangle labela. Note that the above is a classical average, wthe quantum effects isolated into the expectation valueeach beam particleauVua&.

The spin density matrix of the vector meson is defined

r5(a

ua&vaâu, ~4!

01550

e

ttserthee

rhr

s

whereva is non-negative. The helicity matrix elementsthe density matrix are

rll85(a

^lua&vaâul8&. ~5!

The classical ensemble average for observableV is now ob-tained from the density matrixr as

^V&5Tr@rV#

Tr@r#. ~6!

The density matrixr is positive definite, which can beshown as follows. Let us define@11# a set of vectorsvl by itselements ina space

vla[âul&Ava. ~7!

The elementsrll8 of the density matrix can now be writteas dot products ina space of the set of vectorsvl ,

rll85~vl ,vl8!. ~8!

Now for any vectorX

X†rX5 (l,l8

Xl* rll8Xl85S (l

Xlvl ,(l8

Xl8vl8D>0,

~9!

which displays the positive definiteness ofr.At this point we explore the linear constraints on the m

trix elements ofr implied by Eq.~9!. The density matrixr isa 333 matrix with elementsrl,l8 where the helicityl takesthe values 1,0,21. Since the production of the vector mesooccurs via a parity conserving mechanism, the spin denmatrix elements satisfy the symmetries

rll85rl8l* , and rll85~21!l2l8r2l2l8 . ~10!

The density matrix in any frame takes the form@13,14#

pure state

r5S ~12r00!/2 Rer101 i Imr10 r121

Rer102 i Imr10 r00 2Rer101 i Imr10

r121 2Rer102 i Imr10 ~12r00!/2D . ~11!

In the language of spin observablesPyV ,T20

V ,T21V ,T22

V the density matrix can be written in the c.m. frame as

r51

3 F I 13

2SW •PW V1t•TVG , ~12!

where SW is the spin-1 operator andt is the symmetric traceless rank-2 operator with Cartesian componentst i j 532 (SiSj

1SjSi)22d i j . In matrix form this becomes

1The allowed ranges for the tensor polarization can also be deduced by considering the spin-state occupation amplitudes in thelimit.

1-3


r51

3 S 11A1

2T20

V 3

2A1

2~2 iPy

V!2A3

2T21

V A3T22V

3

2A1

2~ iPy

V!2A3

2T21

V 12A2T20V 3

2A1

2~2 iPy

V!1A3

2T21

V

A3T22V 3

2A1

2~ iPy

V!1A3

2T21

V 11A1

2T20

V

D . ~13!

n

doniza

-oes-tio

o

e-seioth

e

rix

-al

di-

ns

col-inedain

-an

an-to

uct.led

pina

If the vector X in Eq. ~9! is such that the combinatioXl* Xl8 is symmetric under the exchange ofl and l8, theobtained constraints are similar to the constraints derivethe previous section for a positive decay angular distributiThe constraints in that case involve only the tensor polartionsT20

V , T21V , andT22

V , and not the vector polarizationPyV .

This is because symmetric combinationsXl* Xl8 only pickout the symmetric part of the density matrixrll8 , i.e., in thiscase the part that has even rank. The antisymmetric rankpart ofr, which is due to the vector polarization, then givno contribution toX†rX. This is exactly the symmetry selection made if one considers the decay angular distribuof Sec. II, and which was discussed in detail in Ref.@5#. Theresultant linear constraints are listed in columns 1 and 3Table I.

Even relations involving the vector polarizationPyV can be

obtained from Eq.~9!, if the above symmetry restrictions arnot invoked onXl* Xl8 . However, the resulting linear constraints involvingPy

V are only of academic interest becauPy

V cannot be measured from the decay angular distributThese additional constraints are therefore not listed inpaper.

IV. SPIN OBSERVABLE LIMITSFROM SCHWARZ INEQUALITIES

Additional constraints on the density matrix are obtainusing Schwarz inequalities as described in Ref.@11#.Namely, from the previously derived Eq.~8! and

u~vl ,vl8!u<uvluuvl8u, ~14!

follows

urll8u<Arllrl8l8. ~15!

Similar constraints exist for differences or sums of matelementsrll8 . Such constraints were exploited in Ref.@11#.In this case, two additional inequalities can be derived

urll81r2ll8u<A2~rll1rl2l!rl8l8 ~16!

and

urll82r2ll8u<A2~rll2rl2l!rl8l8. ~17!

From Eqs. ~15!–~17! one finds several quadratic constraints. Using Eq.~10! and the property that the diagonmatrix elementsr1,1, r0,0, andr21,21 are non-negative or

01550

in.-

ne

n

f

n.is

d

11A1

2T20

V >0, ~18!

12A2T20V >0, ~19!

some of the Schwarz inequalities collapse to linear contions

11A1

2T20

V 1A3T22V >0, ~20!

11A1

2T20

V 2A3T22V >0. ~21!

However, one also finds two very useful quadratic conditiothat involve the squares ofPy

V andT21V ,

9~PyV!2112~T21

V !2<8~12A2T20V !S 11A1

2T20

V D , ~22!

9~PyV!2112~T21

V !2<4~12A2T20V !S 11A1

2T20

V 2A3T22V D .

~23!

The resulting restrictions on the observables are listed inumns 1 and 4 of Tables I and II. As one can see, the obtalinear rules are equivalent to those discussed earlier. Agwe omit from Table I linear conditions that includePy

V .

V. DATA ANALYSIS METHOD

The Aachenet al. Collaboration@10# measured the reaction g1p→r01p using an unpolarized photon beam andunpolarized proton target. The final proton recoils when armeson is produced and ther subsequently decays intop1

andp2 mesons, both of which are detected. Hence, thegular pion distribution in the final state is measured. A fitthis angular distribution in ther meson rest-frame yieldsthree density matrix elements from which we can reconstrtheir pion angular distribution in ther meson rest frameThis reconstructed decay pion angular distribution is calWV(u,f), whereu,f define the direction ofp1 in ther restframe.

The above pion angular distribution depends on the sstate of the producedr meson, which is described by333 spin density matrixrl,l8 . The angular distributiononly depends on the three real elementsr00, Rer10, r121.

1-4


TABLE II. Quadratic constraints forPy ,T20,T21,T22.

Constraints Schwarz inequality Other

9Py2112T21

2 <8~12A2T20!S 111

A2T20D ur10u2<r11r00

9Py2112T21

2 <4~12A2T20!S 111

A2T202A3T22D ur102r210u2<2ur112r121ur00 x2>0

9Py2112T21

2 <8S 111

A2T20D S 22

1

A2T201A3T22D x3<1

9Py2112T21

2 112T222 16T20

2 <12 Tr@r2#<1

rdd

oehe

etthduexit

ne

catere

tic

ed

the

n of

avetheysig.

on

f

Values of these elements in the rest frame of ther meson arepublished by Aachenet al. @10# for a set of several photonbeam energies and vector meson production angles. In oto study reaction mechanisms, however, we are interestespin correlations~single and double spin observables! thatare defined in the overall center of mass frame. How done obtain spin correlations from these previously publisdensity matrix elements?

Our aim is to use the Aachenet al. data in the form ofWV(u,f) to obtain the pion angular distribution in thphoton-nucleon center of mass frame and reanalyze iterms of the spin correlations. If one or more particles inreaction are polarized@5#, such future data can be analyzein a similar way to extract meaningful spin correlations. Oprocedure is therefore preparation for analysis of futureperimental results from Thomas Jefferson Laboratory wpolarized photons@15#.

The first step is to obtain the angular distributioWV(u,f) in the r meson rest frame from the values of thelementsr00, Rer10, r121 using

WV~u,f!53

4p F12r00

21

3r0021

2cos2u

2A2 Rer10sin 2u cosf2r121 sin2u cos 2fG .~24!

Then one constructs the angular distributionWV( u,f) in theg-nucleon center of mass using

WV~ u,f !51

@sin2u1~Er /mr!2 cos2u #3/2WV@u~ u !,f#.

~25!

In the c.m. frame the variables areu and f, whereu,f arethe angles between the relative velocity of the two depions in theg-nucleon c.m. frame. These angles are relato the angles of the decay meson in the vector meson’sframe by

u5arctanS Er

mrtanu D , ~26!

01550

erin

sd

ine

r-

h

ydst

and f5f.The next step involves including the known kinema

factor j( u) in WV( u,f) see Eq.~2!. Aside from the overallfactor of j( u), the remaining angular behavior is expressas a series in spherical harmonics:

WV~ u,f !51

4pj~u !F12A1

2T20

V ~3 cos2u21!

1A3T21V sin 2u cosf2A3T22

V sin2 u cos 2fG .~27!

Next we defineWnormV ( u,f) using Eq.~27! and WV( u,f)

[j( u)3WnormV ( u,f).

Finally, we project out the three spin observables, i.e.,three tensor polarizations of the vector meson,T20

V ,T21V ,T22

V ,

from WnormV ( u,f) using spherical harmonicsYlm( u,f)’s.

These spin observables can then be studied as functiophoton energy and the vector meson production anglesQ,F.

Once the spin observables are properly defined and hcorrect production angle dependence, one can visualizerole of the tensor polarization in three-dimensional displaof the decay angular distribution. Examples are given in F1 for the case ofQ50°,180° and positiveT20 and in Fig. 2for the case ofQ50°,180° and negativeT20; with in bothcasesT215T2250. In Fig. 3, a more realistic case, based

FIG. 1. Forward/backW for T20.0. This is the distribution in

terms of the anglesu,f with the up direction being the direction othe produced vector meson’s momentum.

1-5

e

outs

ub-e of

ints

ho-cay

me,

ne

the-

m

reles

lor-

t

le

in-

s.le Ipre-

f

o


Ref. @10# for Eg53 GeV andQ570° is shown; namely, forT20520.72, T21520.21, T2250.19. One can thereforassociate a shape ofW with each point in the allowedT20,T21,T22 space.

In the process of carrying out these steps, we found fcases of the Aachen data that do not satisfy the constrain

FIG. 2. Forward/backW for T20,0. This is the distribution in

terms of the anglesu,f with the up direction being the direction othe produced vector meson’s momentum.

FIG. 3. Shape ofW for T20520.72, T21520.21, T2250.19.

This is the distribution in terms of the anglesu,f with the updirection being the direction of the produced vector meson’s mmentum.

01550

rin

Tables I and II. Therefore those sets had to be rejected. Ssequently we found another author had also rejected somthe data@11# using similar general constraints.

It would therefore be best to incorporate these constradirectly into the data analysis.

VI. CONCLUSION

For analysis of experimental data for vector meson ptoproduction one should describe the resulting angular dedistribution in the overallgN c.m. frame instead of in thecommonly used vector-meson rest frame. In the c.m. fraone should use the anglesu,f of the relative velocity vectorDvW 5vW 12vW 2. In the transformation from the vector-mesorest frame to thegN c.m. frame, and due to the use of thanglesu,f, a kinematical factorj of Eq. ~2! needs to beincluded. Furthermore, constraints should be satisfied byobservablesT20

V ,T21V ,T22

V . All of these constraints can be derived from the positivity of the density matrix.

We have also explored restrictions which follow fropositivity of the eigenvalues of the spin density matrixrsuch as the conditions that detr>0 and Tr@r2#<1, see Ref.@12#. The relations obtained from these restrictions arespectively cubic and quadratic in the spin observaband involve the vector meson’s vector polarizationPy

V .A more complete set of relations can be obtained by exping the explicit forms of the rootsx1 ,x2 ,x3 of the eigen-value equation of r. We impose the conditions tha0<xi<1, x11x21x351, and x1

21x221x3

2<1. For ex-ample, one root isx15 1

3 @11(1/A2)T20V 1A3T22

V #. From 0<x1<1 relations follow that are similar to the rules in TabI. The other two roots are x25 1

3 @12(1/2A2)T20V

2 (A3 /2)T 22V # 2 1

2 APy21 4

3 T 212 1@(1/A2)T202(1/A3)T22#2

and x3512x12x2. Both x2 and x3 involve PyV and the

above root rules lead to quadratic constraints that arecluded in Table II.

The linear and quadratic constraints onT20V ,T21

V ,T22V mean

FIG. 4. Allowed domain in the space spanned byT20V and T21

V

independent of the value ofT22V . Dashed lines are upper bound

Solid lines are lower bounds. Linear constraints are from Taband quadratic constraints are from Table II. The shaded area resents the allowed region.-

1-6

th

-seTse

aabll

rstcles

a-ntri-isob-erv-ctly

is

astion

s.lep

s.le Ipre-


that the allowed domain in the three-dimensionalT20V 2T21

V

2T22V space is confined. As examples Figs. 4–6 show

allowed areas for the two-dimensional subspacesT20V 2T21

V ,T20

V 2T22V , andT22

V 2T21V , respectively. The dashed lines in

dicate the various upper bounds and the solid lines reprethe lower bounds associated with the various constraints.allowed regions are shaded. Figures 4–6 refer in each catwo-dimensional subspaces and only those constraintsshown that are independent of the values of the observthat would play the role of the third dimension. The futhree-dimensional representation of constraints is richer.

We close with some remarks about double spin obseables. For a polarized photon beam again the angular dibution of the decay mesons can be measured. This dedistribution now depends on the single spin observabT20

V ,T21V ,T22

V as well as on the double spin correlationCx20

gV ,Cx21gV ,Cx22

gV ,Cy21gV ,Cy22

gV ,Cz21gV ,Cz22

gV ~Ref. @5#!. ~Again the


V


Solid lines are lower bounds. Linear constraints are from Taband quadratic constraints are from Table II. The shaded area resents the allowed region.

ai,

C

01550

e

nthe

torele

v-ri-ays

vector polarization and correlations with the vector polariztion cannot be measured.! Similar to the method described iSecs. II and III and based on positive decay angular disbutions and positivity of the complete density matrix for thcase, linear and quadratic relations involving double spinservables can be derived. The constraints on the spin obsables presented in this paper should be incorporated direinto the analysis of forthcoming data.

ACKNOWLEDGMENTS

The authors wish to thank Mr. Wen-tai Chiang for hhelp at an early stage of this study. One author~W.M.K.!thanks the University of Pittsburgh, another~F.T.! thanksRutgers University for warm hospitality. This research wsupported, in part, by the U.S. National Science FoundaPhy-9504866~Pitt! and Phy-9722088~Rutgers!.

Ire-


V


Solid lines are lower bounds. Linear constraints are from Taband quadratic constraints are from Table II. The shaded area resents the allowed region.

o-

D.C.1;

@1# W. Roberts and A. Rakotovao, inProceedings of the FourthCEBAF/INT Workshop on N* Physics, Seattle, WA, 1996, ed-ited by T.-S. H. Lee and W. Roberts~World Scientific Sin-gapore, 1996!.

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@6# M. Pichowsky, C. Savkli, and F. Tabakin, Phys. Rev. C53,593 ~1996!.

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2430 ~1992!.@8# B. Saghai and F. Tabakin, Phys. Rev. C53, 66 ~1996!.@9# C. Savkli, F. Tabakin, and S.N. Yang, Phys. Rev. C53, 1132

~1996!.@10# Aachen-Berlin-Bonn-Hamburg-Heidelberg-Munchen Collab

ration, Phys. Rev.175, 1669~1968!.@11# J. Daboul, Nucl. Phys.B4, 180 ~1967!.@12# P. Minnaert, Phys. Rev.151, 1306~1966!.@13# K. Schilling, P. Seyboth, and G. Wolf, Nucl. Phys.B15, 397

~1970!.@14# Wen-tai Chiang and F. Tabakin, Phys. Rev. C55, 2054~1997!.@15# P. Cole, R. Whitney, and J. Connelly, Report No. E-94-109;

Tedeschi, P. Cole, and J. Mueller, Report No. E-98-109;Marchand, M. Anghinolfi, and J. Laget, Report No. E-93-03J. Napolitano and D. Weygand, Report No. E-93-033.

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Constraints in vector meson photoproduction