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Russian Physics Journal, Vol. 56, No. 9, January, 2014 (Russian Original No. 9, September, 2013)

DIRECT MECHANISMS OF PION PRODUCTION IN THE REACTION A(γ, πNN)B

I. V. Glavanakov and A. N. Tabachenko UDC 539.172

A model of the pion photoproduction on nuclei with emission of two nucleons is presented that takes account of NΔ correlations due to the virtual transitions NN N NN→Δ → in the ground state of nuclei.. The main

components of the model are the two-particle density matrix and the NγΔ → π and 'N Nγ → π transition operators. Direct reaction mechanisms are considered which follow from the structure of the density matrix. An analysis is made of the experimental data of the reaction 12 C( , )p+γ π .

Keywords: photoproduction of pions, A(γ, πNN)B reaction, isobar configurations, ΔN correlations.

INTRODUCTION

Processes of electromagnetic knockout of two nucleons from nuclei are an effective instrument for studying the interaction of nucleons in nuclei. At small and intermediate distances the dynamics of interaction is associated with non-nucleon degrees of freedom in nuclei: mesons, isobars, and quarks. Within the framework of the independent particle model, knockout of two nucleons from a nucleus is possible via the two-particle operator corresponding to meson exchange currents and isobar currents. Another mechanism of nucleon knockout is based on the model of the nucleus, in which correlation pairs of nucleons are present in the nucleus, whose interaction departs from the independent particle model. The process of nucleon knockout in this model is due to the action of the single-particle operator. These two approaches to a description of nucleon knockout are widely used at the present time in studies of the ( , )e e NN′ reaction oriented for the most part toward an examination of short-range correlations caused by the repulsive part of the nucleon-nucleon potential at short distances [1]. Another type of two-particle correlation in the nucleus is associated with the virtual transitions NN N NN→Δ → in the ground state of the nucleus. The interaction of the incoming particle with the correlated system NΔ by way of the single-particle operator can also lead to knockout of two nucleons. These NΔ correlations correspond to components of the nucleon-nucleon potential at intermediate distances.

The role of the three indicated competing mechanisms of knockout of a pair of nucleons from nuclei is different for different reactions and kinematics. The peculiarity of manifestations in the nuclear reactions of NΔ correlations consists in the fact that knockout of a Δ-isobar leads to the decay NΔ → π with production of a pion. Therefore, reactions of the type ( , )NNγ π by virtue of the composition of the particles in the final state are more sensitive to manifestations of correlations of this type.

In the present paper we carry out an analysis of direct mechanisms of the process ( , )A NN Bγ π with account of NΔ correlations in the ground state of the nucleus. The method of the analysis consists in an extension of the approach

developed in [2, 3] for the process ( , )Nγ π at large momentum transfers to the case of knockout of two nucleons.

Physical-Technical Institute at National Research Tomsk Polytechnic University, Tomsk, Russia, e-mail: ant1936@tpu.ru. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 99–107, September, 2013. Original article submitted June 17, 2013.

1064-8887/14/5609-1083 ©2014 Springer Science+Business Media New York

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THEORY

We write the differential cross section of the ( , )A NN Bγ π reaction in the laboratory coordinate system in the form

2

1 21 2 3 3 3 32 ( ) ,

4 (2 ) (2 ) (2 ) (2 )fi n n R

T n n RT d d d d

d E M E E E EE E

πγ π

γ πσ = π δ + − − − −

π π π π

p p p p (1)

where (Eγ, pγ), (Eπ, pπ), (En1, pn1), (En2, pn2), and (ER, pR) are the 4-momenta of the photon, pion, two nucleons, and the residual nucleus B, MT is the mass of the nucleus A, fiT is the matrix element of the transition from the initial state,

including the photon and the nucleon A, to the final state, including the pion, two nucleons in the free state, and the residual nucleus B.

We will carry out our analysis of the ( , )A NN Bγ π reaction within the framework of the formalism developed in [4] to describe the ground state of nuclei and used in [2, 3] in a consideration of the ( , )A N Bγ π reaction − photoproduction of a pion with emission of one nucleon. According to [4], the baryons in the nucleus, in addition to the spatial r, spin s, and isospin t coordinates ( ,s t x, ≡r ), are also characterized by the internal coordinates m ( ,x m X≡ ). The nuclear wave function ( )1,..., AX XΨ in its general form is represented as a superposition of different configurations

( ) ( )1 1 1,..., ( ,..., ) ,...,nA n n A A

nX X A m m x xβ βΨ = φ Ψ∑ ,

where 1( ,..., )nAx xβΨ is a wave function describing the state A of the baryons in the ordinary, spin, and isospin spaces,

and ( ,..., )n m m1 Αφ is a wave function describing the internal state of the baryons. The subscript 1,..., Aβ ≡ β β characterizes the spatial, spin, and isospin states A of the particles, and the index 1,..., An n n≡ enumerates the internal states A of the particles. For example, the index 1 2, ,..., An N N N= describes the internal state A of the nucleons; the internal state of the system in which the first particle is a Δ-isobar and the rest are nucleons is described by the index

1 2, ,..., An N N= Δ . The operator nA is the antisymmetrization operator of the wave function. In this approach, we represent the matrix element fiT in the form

*1 1 2 1 2 1 1 1 2( ', , ,..., ) ( ', ,..., ) ' ( , ,..., ).fi A F A T AT A d X X X X X X X X t X X X Xγπ= Ψ Ψ∫ (2)

Here TΨ and FΨ are the wave functions of the nucleus A and the system F comprising the free nucleons and the residual nucleus B, tγπ is the pion photoproduction operator, and the integral sign indicates summation over the discrete variables and integration over the continuous variables.

Representing the wave function FΨ in the form of an antisymmetrized product of the wave function of two free nucleons

1 2α αϕ in the states 1α and 2α and the wave function of the residual nucleus fΨ

1 21 2 12;3... 1 2 3( , ,..., ) ( , ) ( ,..., )F A A f AX X X A X X X Xα αΨ = ϕ Ψ ,

where 12;3...AA is the antisymmetrization operator, we obtain the following expression for fiT :

1 2

1/2* *

1 1 1 2 3 42 ( ', ,..., ) ( ', ) ( , ,..., )

1fi A f AAT d X X X X X X X X

A α α⎛ ⎞ ⎡= ϕ Ψ⎜ ⎟ ⎣−⎝ ⎠ ∫

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1 2

1 2

1 2

* *3 2 1 4

* *1 3 2 4

* *3 4 1 2

( , ) ( ', ,..., )

( ', ) ( , ,..., )

( , ) ( ', ,..., )

f A

f A

f A

X X X X X

X X X X X

X X X X X

α α

α α

α α

−ϕ Ψ

− − − − − − − − − − − − − − − −

−ϕ Ψ

− − − − − − − − − − − − − − − − −

+ϕ Ψ

− − − − − − − − − − − − − − − − −

+1 2

* *1 3 1 2 1 1 1 2( , ) ( ,..., ', ) ' ( , ,..., )A A f T AX X X X X X t X X X Xα α − γπ⎤ϕ Ψ Ψ⎦ .

The direct reaction mechanisms include those reaction mechanisms in which an active baryon (the baryon with coordinate X1) goes as a result of interaction with a photon to a free state. Among the ( 1) / 2A A− terms of this expression, 1A− terms satisfy this condition. Their sum is equal to

( )

1 21/2 * *

1 1 2 1 2 3 4

1 1 1 2

2 ( 1) ( ', , ,..., ) ( ', ) ( , ,..., )

' ( , ,..., ).

d A f A

T A

T A A d X X X X X X X X X

X t X X X X

α α

γπ

= − ϕ Ψ

× Ψ

∫

Applying the condition of completeness of the final states of the residual nucleus, we can represent the square of the modulus of the direct amplitude Td, summed over the states of the residual nucleus, as

1 2 1 2

21 1 2 1 1 2

*1 2 1 1 1 2 1 2 1 1 1 2

2 ( 1) ( ', , , ', , )

( ', ) ' | | ( , ; , ) | | ' ( ', ),

df

T A A d X X X X X X

X X X t X X X X X X t X X X+α α γπ γπ α α

= −

×ϕ < > ρ < > ϕ

∑ ∫

where

*1 2 1 2 3 4 1 2 3 1 2 3( , ; , ) ( , ,..., ) ( , , ,..., ) ( , , ,..., )A T A T AX X X X d X X X X X X X X X X Xρ = Ψ Ψ∫

is the two-particle density matrix. In our model, we take into account two internal configurations of the wave function ΨT: the configuration in

which all of the particles are nucleons, and the isobar configuration, in which one particle is a Δ-isobar and the remaining particles are nucleons:

NT T T

ΔΨ = Ψ +Ψ .

Here NTΨ and T

ΔΨ are the wave functions of the nucleon and isobar configurations. We are interested in manifestations of ΔN correlations in the ground state of the nucleus; therefore, we will

consider the two-particle density matrix due to isobar configurations of the wave function of the nucleus

*1 2 1 2 3 4 1 2 3 1 2 3( , ; , ) ( , ,..., ) ( , , ,..., ) ( , , ,..., )A T A T AX X X X d X X X X X X X X X X XΔ Δ Δρ = Ψ Ψ∫ . (3)

Assuming that only two nucleons of the nucleus take part in the process of excitation of internal degrees of freedom of the nucleon, we can represent the wave function of the isobar configuration 1( ,..., )T AX XΔΨ in the form of

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a superposition of products of wave functions [ ] 1 2( , )i jN X XΔβ βΨ of the system ΔN including the isobar and the second

nucleon, which is a participant in the transition NN → ΔN, and the wave function of the nucleon core 1 3( )( ,..., )

i j

NAX X−β β

Ψ , which describes the state of the remaining А – 2 nucleons:

11 12;3... [ ] 1 2 3( )( ,..., ) ( , ) ( ,..., )

i j i j

N NT A A A

ijX X A X X X X−

Δ Δβ β β β

Ψ = Ψ Ψ∑ . (4)

Substituting the wave functions of the isobar configurations, represented in the form given by Eq. (4), into Eq. (3), and carrying out the integration over the variables 3 ,..., AX X , we obtain, according to [2],

1 2 1 2( , ; , ) N N C NC CN C CCX X X XΔΔ Δ Δ Δρ = ρ + ρ +ρ +ρ + ρ +ρ +ρ .

Here the two subscripts on the terms of the density matrix represent the states of particles 1 and 2. The subscripts Δ, N , and C indicate, respectively, that the particle is a Δ-isobar, a nucleon of the system ΔN , or that it belongs to the nucleon core.

Due to orthogonality of the wave functions Nφ and Δφ the direct amplitudes corresponding to the matrices

NΔρ and CΔρ are equal to zero. In the amplitude corresponding to the matrix CCρ , the system ΔN enters into the composition of the residual nucleus and does not manifest itself dynamically; therefore, we will not consider it. Let us consider the reaction mechanisms corresponding to the four remaining terms of the density matrix ,NΔρ ,C NCΔρ ρ ,

and CNρ :

* *1 2 1 2 1 2 1 2 1 2 1 2

1( , ; , ) ( , ) ( , ) ( , ) ( , )( 1) i j i j

N NN N N

ijX X X X m m x x x x m m

A AΔ Δ

Δ Δ Δ⎡ ⎤ ⎡ ⎤β β β β⎣ ⎦ ⎣ ⎦

⎡ ⎤ρ = φ Ψ Ψ φ⎢ ⎥

−⎣ ⎦∑ , (5)

1 2 1 2

* * *1 2 3 1 3 2 2 1 3 1 2

,

( , ; , )

1( , ) ( ) ( , ) ( ) ( ) ( , ) ( , ),( 1) k ki j i j

C

N NN N

ij k ij

X X X X

m m d x x x x x x x m mA A

Δ

Δ ΔΔ β β Δ⎡ ⎤ ⎡ ⎤β β β β⎣ ⎦ ⎣ ⎦≠

ρ

⎡ ⎤= φ Ψ Ψ Ψ Ψ φ⎢ ⎥

−⎣ ⎦∑ ∫

1 2 1 2

* * *1 2 3 3 1 2 2 3 1 1 2

,

( , ; , )

1( , ) ( ) ( , ) ( ) ( ) ( , ) ( , ),( 1) k ki j i j

NC

N NNN NN

ij k ij

X X X X

m m d x x x x x x x m mA A

Δ Δβ β⎡ ⎤ ⎡ ⎤β β β β⎣ ⎦ ⎣ ⎦≠

ρ

⎡ ⎤= φ Ψ Ψ Ψ Ψ φ⎢ ⎥

−⎣ ⎦∑ ∫

1 2 1 2

* * *1 2 3 3 2 1 1 3 2 1 2

,

( , ; , )

1( , ) ( ) ( , ) ( ) ( ) ( , ) ( , ).( 1) k ki j i j

CN

N NNN NN

ij k ij

X X X X

m m d x x x x x x x m mA A

Δ Δβ β⎡ ⎤ ⎡ ⎤β β β β⎣ ⎦ ⎣ ⎦≠

ρ

⎡ ⎤= φ Ψ Ψ Ψ Ψ φ⎢ ⎥

−⎣ ⎦∑ ∫

To each term of the density matrix there corresponds a definite reaction mechanism that depends on the structure of the matrix and the photoproduction operator 1 1' | |X t Xγπ< > . The correspondence between the terms of

the density matrix and the reaction mechanisms is simple: according to formula (2), a photon interacts with baryon 1, which together with baryon 2 goes to a free state. A graphic illustration of the reaction mechanisms corresponding to matrices ,NΔρ ,C NCΔρ ρ , and CNρ is given in Fig. 1. Let us consider the mechanisms of the ( , )A NN Bγ π reaction

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that contribute in the kinematic region where at least one of the nucleons in the final state has momentum significantly greater than the Fermi momentum.

The matrix NΔρ corresponds to the reaction mechanism in which knockout of the system ΔN by a photon takes place. The nucleon produced during the transition NγΔ → π , and the nucleon of the system ΔN goes to a free state.

We express the wave function 1 2( , )i j

N x xΔ⎡ ⎤β β⎣ ⎦

Ψ in expression (5) in terms of its Fourier transform

[ ] 1 2( , )i jN y yΔβ βΨ , where ,y s t≡ ,p , and we write [ ] 1 2( , )

i jN y yΔβ βΨ in the form of an expansion over states of the system

ΔN with definite values of the total angular momentum J, isospin T, orbital angular momentum l, total spin s, and their projections MJ, MT, ml и ms ( , , , , , ,s J T sJ M T M s m lβ ≡ ) :

[ ] 1 2 [ ] , 1 2 1 2,

( , ) ( , ) ( ) ( , , , ),i j i j s l s

s l

N Nl m

my y p Y s s t tΔ Δ

β β β β β ββ

Ψ = Ψ Ω∑ P p

where P is the momentum of the system ΔN, p is the relative momentum, and sβ

Ω is the spin-isospin wave function of

the system ΔN. As a result, we can write an expression for the square of the modulus of the amplitude T NΔ associated

with the matrix NΔρ , summed over the states of the residual nucleus and the spin states of the nucleons, in the form

[ ( )

( )( )( )

1 1 2 21 2

2 2 1 1

1 2 2

* 62 2 2, , , ,

, , ,

1 1 1, , , ,

2 1, ,

TT (2 ) Sp ( ) ( , ; , )

+Sp ( ) ( , ; , )

Sp Sp ( , )

n ns s n n s sn n s s

n ns s n n s s

n ns s n n

NN N m mNf m m

NN N m m

N N m m

O U O p p

O U O p p

O V O

+ ΔγΔ→π γΔ→πβ β τ τ β βΔσ σ β β

+ ΔγΔ→π γΔ→πβ β τ τ β β

+γΔ→π γΔ→πβ β τ τ

= π ρ

ρ

− ρ

∑ ∑ p P P

p P P

p p

( )( ) ]

1

2 1 1 2

2 1, ,

1 2 1 2, , , ,

( , ; , )

Sp Sp ( , ) ( , ; , ) .

s s

n ns s n n s s

N

NN N m m

p p

O V O p p

Δβ β

+ ΔγΔ→π γΔ→πβ β τ τ β β− ρ

P P

p p P P

Here the matrix T and the matrix T in formula (1) are related by the expression

31 2(2 ) ( )Tn n RT γ π= π δ − − − −p p p p p ,

NOγΔ→ π is the matrix of the transition NγΔ → π ,

( ) ( )1 2 1( )n N n NM M M MΔ Δ= − − +p p q p , ( ) ( )2 1 2( )n N n NM M M MΔ Δ= − − +p p q p ,

Fig. 1. Diagrams illustrating the direct mechanisms of photoproduction of pions on atomic nuclei corresponding to the matrices NΔρ (a), CΔρ (b), NCρ (c), and CNρ (d) in the ( , )A NN Bγ π reaction.

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, , , , , *1 2 , 1 , 2, ; , 3 2, ;1 2, 3 2, ;1 2,, ; , , ; ,

,( , ) C C C C ( ) ( ),n J s J s

l ll s n nn s s l sl l

m m J M s m J M s ml m l ml m s m m m m mm m l m s m

m mV Y YΔ

Δ ΔΔ

σ σσ σ σ σσ σ β β

= ∑p p p p

,,, , ; ,( ) ( , )n

s s n s sn

m mm mm m

mU V ΔΔ Δ

Δ

σ σσ σβ β σ σ β β

σ= ∑p p p ,

1 21 2

, , **1 2 [ ] 1 2 3 2, ;1 2, 3 2, ;1 2,, , , [ ]( , ; , ) ( , ) ( , ) C CT T

i j s n nn n s s i j s

T M T MN N Nm m m mm m

ijp p p p

Δ Δ

Δ Δ Δβ β β τ τ τ ττ τ β β β β βρ = Ψ Ψ∑P P P P .

Pion production as a result of reaction mechanisms corresponding to the matrices CΔρ and NCρ takes place similarly as in the previous case, at the interaction of a photon with the system ΔN. The difference consists in the fact that the passive particle of the ΔN system remains in the bound state.

In the kinematic region, where the momentum of nucleon 1 is significantly greater than the Fermi momentum, the amplitudes T CΔ and TNC of the ( , )A NN Bγ π reaction, associated with the matrices CΔρ and NCρ , can be expressed in terms of the amplitude of the ( , )A N Bγ π reaction. Thus, we can represent the square of the modulus of

the amplitude T CΔ as follows:

2 2

1 2 1

2* 32 ,

, , ',TT (2 ) ( ) T

n T n Tn n T n

m n M m M CCf m m M f mWτ τ ΔΔσ σ σ

= π ρ∑ ∑ ∑p . (6)

Here TTM CΔ

is the amplitude of direct knockout of an isobar in the ( , )A N Bγ π reaction at the interaction of a photon

with the system ΔN in the state MT,

2 2

2 3 2,( ) ( ) , ( ) (2 ) exp( ) ( ),

n k n k k k

A

m m mk

d i−τ τ τ β β βρ = δ Ψ Ψ = π − Ψ∑ ∫p p p r pr r

( )kβ

Ψ r is the spatial part of the wave function of a nucleon bound in the nucleus, 2 2, ,( 2 ) /

T TM m M mW A n Aτ τ= − , and

2,T nM mn τ is the number of single-particle states kβ with 2k nm mτ = τ liberated as a result of the transition NN N→Δ .

The mechanisms of the ( , )A N Bγ π reaction due to isobar configurations were analyzed in [3]. According to [3], the square of the modulus of the amplitude of direct knockout of an isobar T

TM after summing over the states of the

residual nucleus and the spin states can be represented in the form

( )1 2 211

2 32 2 2 ,, , ,

',T (2 ) Sp ( ) ( , ; , )aT n T Ts s n n s sn nn

n s s

NM N M Mm mNCf m

d O U O p p+ ΔγΔ→π β β τ τ β βγΔ →πΔσ β β

= π ρ δ∑ ∑∫ P p P P .

Production of a pion-nucleon pair via the reaction mechanism corresponding to the matrix CΔρ takes place at the interaction of a photon with an isobar of the system ΔN as a result of the transition NγΔ → π .

In the case of the reaction mechanism associated with the matrix NCρ , the relation between the amplitudes of the ( , )A NN Bγ π and ( , )A N Bγ π reactions is analogous to relation (6). The structure of the expression for the square

of the modulus of the amplitude TTM NC

is also similar:

( )1 1 2 21

2 32 2 2 2 ,, , ,

',T (2 ) Sp ( , ) ( , ; , )

T n n T Ts s n n s sn nn s s

NM N N N N M Mm mNCf m

d O S O p p+ Δγ →π γ →πβ β τ τ β β

σ β β= π ρ δ∑ ∑∫ P p p P P .

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Here 'N NOγ → π is the matrix of the transition 'N Nγ → π ,

, ,, , ; ,( , ') ( , ')n n n

s s n s s

m m m mm m

mS V Δ

ΔΔ

σ σ σ σβ β σ σ β β

σ= ∑p p p p .

Production of a pion-nucleon pair takes place in this case at the interaction of a photon with the nucleon of the system ΔN.

Let us consider the mechanism of the ( , )A NN Bγ π reaction corresponding to the matrix CNρ . In this mechanism a pion is formed as a result of the interaction of a photon with a nucleon from the nucleon core, which goes to a free state. The square of the modulus of the amplitude, summed over the states of the residual nucleus and the spin states of the nucleons, can be written in the form

[ ( ) ( )( ) ( )

1 2

1 1 1 1 2 2

2 2 2 2

* 6

, ,

2 2 1 1 , 2 2 2 2, , , ,

1 1 2 2 ,,

TT (2 )

Sp Sp ( , ) ( , ) ( , ; , )

+Sp Sp ( , ) ( , )

n n s s

n n n T ns s n n s s

n n n T ns s

CNf m m

NN N N N m m k k M m m m m

N N N N m m k k M m m

d

O O S W p p

O O S W

π π

π

Δσ σ β β

+ Δγ →π γ →π τ + τ τ + τβ β τ τ β β

+γ →π γ →π τ + τ τ + τβ β

= π

× ρ ρ

ρ

∑ ∑ ∫ p

p p p p P P

p p p p1 1 1 1 1 1, , , ( , ; , )

n n s s

Nm m p p

πΔτ τ β βρ P P

( )( ) ]

1 1 2 12 1

2 2 1 21 2

2 1 1 2 , 2 2 1 1, , , ,

1 2 2 1 , 1 1 2 2, , , ,

Sp ( , ) ( , ) ( , ; , )

Sp ( , ) ( , ) ( , ; , )

a a n T ns s n n s sn n

a a n T ns s n n s sn n

Nm m k k M m m m mN N N N

Nm m k k M m m m mN N N N

O S O W p p

O S O W p p

π π

π π

+ Δτ + τ τ + τβ β τ τ β βγ →π γ →π

+ Δτ + τ τ + τβ β τ τ β βγ →π γ →π

− ρ ρ

− ρ ρ

p p p p P P

p p p p P P ,

where I nIΔ= +P p p , kI nI π γ= + −p p p p , and *1 2 , 1 2

1( , ) ( ) ( )2n k n k km m m

kτ τ τ β βρ = δ Ψ Ψ∑p p p p .

Depending on the isospin state of the reaction products, different terms of this expression will contribute to the cross section. If the isospin states of two nucleons differ, then in the photoproduction of charged pions either the first or the second term will be nonzero whereas in the production of neutral pions the sum of the first two terms will be nonzero. If the free nucleons are of one type, then all of the terms of this expression will contribute.

RESULTS AND DISCUSSION

At present, experimental data on the ( , )A NN Bγ π reaction are lacking. However, according to the data obtained in experiments [5–7] that examined photoproduction of pions with emission of one nucleon, the excitation spectrum of the residual nuclear system B produced in the ( , )A N Bγ π reaction extends far beyond the limits of the energy threshold of emission of a second nucleon. Therefore, the experimental cross sections of the ( , )A N Bγ π reaction, measured by simultaneous recording of a pion and a nucleon can contain a contribution from the reaction

( , )A NN Bγ π . This conclusion was indeed confirmed experimentally in work performed at the Tomsk synchrotron [8]

which examined the reactions 12 C( , )p−γ π and 12 C( , ).pp−γ π The data of an experimental study [9] in which the

experimental cross section of the reaction 16 O( , )p+γ π at large momentum transfers to the residual nuclear system was satisfactorily explained as the total contribution of processes of pion production with emission of one or two nucleons also agree with this conclusion. Thus, the experimental data on the ( , )A Nγ π reaction can be a source of inproduction with regard to the ( , )A NNγ π reaction.

In connection with the absence of data on the ( , )A NN Bγ π reaction, consider the experiment [6] performed at

the Mainz Microtron, which measured the cross section of the reaction 12 C( , )p+γ π . Measurements of the differential

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cross section are plotted in Fig. 2 as a function of pion energy. These data are interesting in that the differential cross sections contain fine structure which was interpreted in [10,11] as a manifestation of isobar-nuclear states – highly excited states of nuclei decaying with emission of a pion and a nucleon. An alternative interpretation of the data on the 12 C( , )p+γ π reaction may be built around a contribution of isobar configurations of the ground state of the nucleus. A test of the possibility of such an explanation of the Mainz data [6] is one of the goals of the present work.

The referenced experimental data are quantities that have been averaged over a wide range of momentum transfers to the residual nuclear system, including also the region of small momentum transfers. Therefore the data contain a substantial contribution from reaction mechanisms in which p+π pairs are produced by way of the processes

( , )n+γ π and 0( , )pγ π with subsequent charge-exchange rescattering of a neutron into a proton or 0π into +π . Curve 5 in Fig. 2 is a cross section calculated in [6] using a model that takes into account such mechanisms of production of pion-nucleon pairs, an important element of which is the interaction in the final state. As can be seen, curve 5 reproduces the general trend of the dependence of the differential cross section on pion energy. However, at energies greater than 80 MeV the calculated cross section is 2–3 times smaller than the experimental value. The maximum of the cross section at ~90 MeV has also not been explained.

To explain the Mainz data [6], we considered contributions to the cross section of production of p+π pairs

from the 12 C( , )p B+γ π and 12 C( , )p NB+γ π reactions associated with the manifestation of isobar configurations.

Curve 3 in Fig. 2 plots the cross section of the 12 11C( , ) Bep+γ π reaction. The differential cross section was calculated using a model [2] that takes account of direct and exchange mechanisms of pion production with emission of one nucleon which are due to ΔN correlations in the ground state of the 12C nucleus. As can be seen, the calculated cross section of the 12 11C( , ) Bep+γ π reaction comprises not more than 10% of the experimental cross section of production

of p+π pairs.

Fig. 2. Differential cross section of the reaction 12 C( , )p+γ π as a function of pion energy: experimental data from [6]; curve 5 is for calculated data (see text) of [6], curve 3 is for cross section of the reaction 12 11C( , ) Bep+γ π , curves 1, 2, and 4 are for contributions to the cross section of production of p+π pairs

from the reaction mechanisms of the 12 10C( , ) Bep n+γ π and 12 10C( , ) Lip p+γ π reactions corresponding

to the matrices NΔρ , CΔρ , and CNρ , respectively, and curve 6 is for sum of contributions of the considered reaction mechanisms.

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According to the referenced model, photoproduction of p+π pairs in the considered kinematic region is

possible as a result of three direct mechanisms of the 12 10C( , ) Bep n+γ π reaction corresponding to the matrices NΔρ ,

CΔρ , and CNρ , and one mechanism of the 12 10C( , ) Lip p+γ π reaction corresponding to the matrix CΔρ . Curve 1 in Fig. 2 plots the results of a calculation of the contribution to the cross section of the mechanism of

the 12 10C( , ) Bep n+γ π reaction corresponding to the matrix NΔρ . In the calculation of the density matrix the wave

function of the ΔN system [ ]i jNΔβ βΨ was found by solving the Schrödinger equation for the potential due to the exchange

of the π and ρ mesons [4, 12]. Following [4, 12], it was assumed that the ΔN system, formed as a result of the transition NN → ΔN, is found in the state J = 0, T = 1, l = s = 2. The state of the nucleon core was described using the the oscillator shell model with a parameter reproducing the root-mean-square charge radius of the nucleus. The operator of the NγΔ → π transition was found with the help of the S-matrix approach used to describe Nγ + Δ → + π processes and was represented in the form of an expansion over products of four spin and three isospin independent structures with expansion coefficients that depend on the coupling constants and the magnetic moments [2]. The interaction in the final state was taken into account within the framework of the optical model in the eikonal approximation. The amplitude corresponding to the matrix NΔρ includes the interaction of a photon with NΔ isobars of the systems n++Δ

and p+Δ . The contribution to the amplitude of the reaction from the system n++Δ dominates here.

The sum of contributions to the cross section of the 12 10C( , ) Bep n+γ π and 12 10C( , ) Lip p+γ π reaction

mechanisms corresponding to the matrix CΔρ is plotted in Fig. 2 by curve 4. In these mechanisms a pion-nucleon pair

is formed as a result of the transition p++ +γΔ → π . This determines the shape of the energy dependence of the cross section, which reflects, similarly as in the previous case, the dependence of the Fourier transform of the wave function

[ ]i jNΔβ βΨ on the isobar momentum.

Curve 2 in Fig. 2 plots the contribution to the cross section of the 12 10C( , ) Bep n+γ π reaction mechanism

corresponding to the matrix CNρ . In this case, production of a pion takes place as a result of the interaction of a photon

with a nucleon in the process p n +γ → π . For the operator of the 'N Nγ → π transition we used the nonrelativistic

Blomqvist−Laget operator [13]. In the production of p+π pairs, the following NΔ systems contribute to the cross

section: p+Δ , 0 pΔ , and p−Δ . The energy dependence of the cross section in this reaction mechanism reflects mainly the change in the phase volume with increasing pion energy.

The sum of contributions of the considered reaction mechanisms is plotted in Fig. 2 by curve 6. In the pion energy region above 80 MeV, taking ΔN correlations into account increases the cross section of photoproduction of

p+π pairs by not less than ~40%, which improves the agreement between the experimental and calculated cross sections. However, the second maximum in the energy dependence of the cross section of the reaction at ~90 MeV cannot be explained within the framework of models taking into account ΔN correlations in the ground state of the 12С nucleus.

It should be noted that the conclusion made regarding the magnitude of the contribution of ΔN correlations to the cross section of production of p+π pairs does not agree with the results of [14], in which the calculated cross

section of the 12 11C( , ) Bep+γ π reaction, shaped by isobar configurations in the ground state of the 12С nucleus, is almost an order of magnitude greater. The reasons for such a discrepancy are many since the implemented models of production of pion-nucleon pairs differ in many respects. But the main difference is that Fix et al. [14] used the independent particle model as their model of the nucleus and considered quasi-free knockout of an isobar, i.e., they neglected the correlation between the isobar and the nucleon of the ΔN system, which substantially suppresses the reaction cross section. The approach used in the present work, based on taking the ΔN correlations into account, is physically more soundly based.

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CONCLUSIONS

A study of the behavior of nucleon resonances in a nuclear medium is an important question in the physics of intermediate energies. In this work we have touched on one of the aspects of this problem – the influence of ΔN correlations associated with virtual transitions of the type NN N NN→Δ → in the ground state of nuclei on the process of photoproduction of pions accompanied by emission of two nucleons. The main components of the model of the ( , )A NN Bγ π reaction are the two-particle density matrix and the NγΔ → π and 'N Nγ → π transition operators. We have considered four direct reaction mechanisms which follow from the structure of the density matrix. The model was applied to an interpretation of the data of the 12 C( , )p+γ π reaction [6]. We calculated the differential cross sections

of the reactions 12 10C( , ) Bep n+γ π and 12 10C( , ) Lip p+γ π , whose contribution to the cross section of production of

p+π pairs improves the agreement between the calculated and experimental cross sections in the high-energy region of the pion spectrum. The maximum of the energy dependence of the differential cross section at pion energies ~90 MeV, which in [10, 11] was interpreted as a manifestation of a quasi-bound isobar-nuclear state, cannot be explained within the framework of the proposed model by a contribution of cross sections of production of pions with emission of two nucleons.

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