Momentum space 3N Faddeev calculations of hadronic and electromagnetic reactions with proton-proton Coulomb and three-nucleon forces included

D OI 10.1140/epja/i2011-11030-7

T E

P ^HYSICAL J ^OURNAL A

R egular A rticle - T h eo retical Physics

Momentum space 3N Faddeev calculations of hadronie and electromagnetic reactions with proton-proton Coulomb and three-nucleon forces included

H. W ita ła 1,a, R. S kibiński1, J. G o lak 1, an d W . G löckle2

1 M. Smoluchowski In stitu te of Physics, Jagiellonian University, PL-30059 K rakow , Poland 2 In s titu t für theoretische P hysik II, R uh r-U n iv ersität Bochum , D-44780 Bochum , G erm any

Received: 15 D ecem ber 2010 / Revised: 24 Ja n u ary 2011 P ublished online: 2 M arch 2011

© T he A uthor(s) 2011. T his article is published w ith open access a t Springerlink.com C om m unicated by M.C. Birse

Abstract. We extend our approach to incorporate th e p ro to n -p ro to n (pp) Coulom b force into th e three- nucleon (3N) m om entum space Faddeev calculations of elastic pro to n -d eu tero n (pd) scatterin g and breakup to th e case w hen also a three-nucleon force (3NF) is acting. In addition, we form ulate th a t approach in th e application to electron- and y-induced reactions on 3He. T he m ain new ingredient is a 3-dim ensional screened pp Coulom b i-m a trix obtained by a num erical solution of a 3-dim ensional Lippm ann-Schw inger equation (LSE). T he resulting equations have th e sam e stru c tu re as th e Faddeev equations w hich describe pd scatterin g w ith o u t 3NF acting. T h a t shows th e practical feasibility of b o th presented form ulations.

1 Introduction

T h e long-range n a tu re of th e C oulom b force prevents th e ap p licatio n of th e s ta n d a rd techniques developed for sh o rt-ra n g e intera ctio n s in th e analysis of nuclear reac­

tions involving two p rotons. O ne proposal to avoid the difficulties including th e C oulom b force is to use a screened C oulom b in te ra c tio n an d to reach th e p u re C oulom b lim it th ro u g h ap p licatio n of a ren o rm alisa tio n p rocedure [1-4].

E lastic p d sc a tte rin g first calculations, w ith m odern nuclear forces a n d th e ex act C oulom b force in coordinate re p resen ta tio n included, have been achieved in a varia­

tio n al hyperspherical h arm onic approach [5]. Recently, the inclusion of th e C oulom b force was u n d e rta k e n also for the p d b re a k u p reactio n using a screened p p C oulom b force in m o m en tu m space and in a partial-w ave basis [6]. To get th e final predictions w hich can be com pared to th e d a ta , th e lim it to th e unscreened situ a tio n has been perform ed num erically applying a ren o rm alizatio n to th e resulting 3N on-shell am p litu d es [6,7].

O ne m ain concern in such ty p e of calculations is th e ap p licatio n of a partial-w ave decom position to th e long- ran g ed C oulom b force. E ven w hen screening is applied, it seems reasonable to tr e a t from th e beginning th e screened p p C oulom b t-m a trix w ith o u t partial-w ave decom position because th e req u ired lim it of vanishing screening leads nec­

essarily to a d ra stic increase of th e n u m b er of partial-w ave a e-mail: w i t a l a @ i f .u j . e d u . p l

sta te s involved [8]. In consequence, th is leads to an explo­

sion of th e nu m b er of 3N p a rtia l waves req u ired for con­

vergence. T he very successful app ro ach to include th e pp C oulom b force in to th e 3N F addeev calculations of refs. [6] an d [7] revealed a fast convergence in th e screening ra ­ dius using a tw o-nucleon partial-w ave basis of large size. It ap p ears th a t an in d ep en d en t calcu latio n al schem e should be carried th ro u g h w here th e tre a tm e n t of th e C oulom b p a rt to ta lly avoids a partial-w ave decom position an d th u s providing an in d ep en d en t check of th e results o b tain ed in [6,7].

Therefore, we developed in [9,10] a novel app ro ach to include th e p p C oulom b force into th e m o m en tu m space 3N Faddeev calculations. It is based on a s ta n d a rd for­

m u latio n for sh o rt-ran g e forces an d relies on th e screening of th e long-range C oulom b in teractio n . In order to avoid all u n certain ties connected w ith th e ap p licatio n of th e partial-w ave expansion, in ad e q u a te w hen w orking w ith long-range forces, we used d irectly th e 3-dim ensional pp screened C oulom b t-m a trix . We d e m o n stra te d in [9, 10]

th e feasibility of th a t ap p ro ach in th e case of elastic p d sc a tte rin g and b reak u p using a sim ple dynam ical m odel for th e nuclear p a rt of th e in teractio n . In th is first study, we applied th e m ost sim ple expo n en tial screening of th e C oulom b force w ith pow er n = 1 for which th e 3-dim ensional m o m en tu m space m a trix elem ent can be ob­

ta in e d analytically. A p proxim ating th e 3-dim ensional pp screened C oulom b t-m a trix by th e p o te n tia l allowed us to avoid th e tim e-consum ing, m any-dim ensional interpo-

Page 2 of 6 T he E u ro p e an Physical Jo u rn al A

lations w hen solving th e Faddeev equations. In addition, w hen calcu latin g th e observables, we neglected in th e tr a n ­ sition am p litu d e th e la st te rm of eq. (62) of ref. [9]. How­

ever, in fu tu re ap plications to d a ta analysis an d p articu lar- ily w hen th e com parison to th e ap p ro ach of ref. [7] will be perform ed, b o th these approxim ations m u st be rem oved.

In th e p resen t p a p e r we ex ten d th a t app ro ach to in­

clude a 3N F into th a t form ulation. Also, we show how th a t form ulation can be applied to electrom agnetic pro­

cesses induced by electrons or 7’s on 3He.

In sect. 2, for th e convenience of th e reader, we sh o rtly describe th e m ain points of th e form alism o u tlin ed in de­

ta il in [9,10] for th e case of 3N Faddeev calculations w ith pairw ise forces only an d ex ten d th e corresponding equa­

tions to th e case w hen a 3N F is also acting. In sect. 3 we ap p ly th a t form ulation to electrom agnetic reactions on 3He. T he su m m ary is given in sect. 4.

2 Faddeev equations with screened pp Coulomb force

W hen only pairw ise forces are actin g we use th e Faddeev eq u atio n in th e form [1 1,1 2]

T\ E) = t P \ E) + t P G 0T\ P), (1)

\pqa) = p q (ls)j ( A ^ I (j I )J Ct2 T (2)

p2dpq2dq \pqa) (pqa\ =

P ro je c tin g eq. (1) for T \ P) on th e \pqa) an d \pqß) states one gets th e following system of coupled in teg ral equa­

tions:

(pqa\T\&) = (pqa\ t N+CP\&)

+ (pqa\tN+cPGo p l2d plql2dql \plqla l)(plq'a'\T\<E) a

+ (pqa\tN+cPGo ^ / p l2d p lql2dql \plqlß l)(p lqlß l \T \P),

(4) ( p q ß \T\$) = ( p q ß \t RP\ $)

+ (pqß\tc P G o ^ / p l2dplql2dql \plqla l)(plqla l \T\<P)

w here th e p e rm u ta tio n o p e ra to r P is defined in term s of tra n sp o sitio n o p e rato rs P j of nucleons i an d j , P = P 12P 23 + P1 3P2 3, G 0 is th e free 3N p ro p ag ato r, an d \P) is th e in itial s ta te com posed of a d e u tero n s ta te and a m o m en tu m eig en state of th e p ro to n . K now ing T\ P) the b rea k u p as well as th e elastic p d sc a tte rin g am plitudes can be gained in th e s ta n d a rd m an n er [11]. T he physical co n ten t of eq. (1) is revealed a fter ite ra tin g it. T he re su lt­

ing m u ltip le-sc atterin g series contains all possible re sc a t­

te rin g co n trib u tio n s induced by interactio n s of th re e nu­

cleons an d free p ro p ag atio n in betw een.

We use our sta n d a rd m o m e n tu m space partial-w ave basis \pqa)

+ ( p q ß \ t R P G o Y J p12dplql2dql \plqlß l)(plqlß l \ T \$), (5) ß' J

w here t N + c an d t R are t-m a tric e s g en e rate d th ro u g h a LSE by th e in teractio n s VN + V cR an d VCR , respectively. Namely, for sta te s \a) w ith a tw o-nucleon sub sy stem to ta l isospin t = 1 th e corresponding t-m a trix elem ent (pa\tN+c ( E -

43mq2)\pla l) is a linear com bination of th e pp, tRp+c, and th e n eu tro n -p ro to n (np), t np, t = 1 t-m atrices, w hich are gen erated by th e in terac tio n s V p Ton9 + VR an d VßpTon9, respectively. T he coefficients of th a t com b in atio n d epend on th e to ta l 3N isospin T an d T l of th e sta te s \ a) and

\aß [9,13]:

t = 1T = 1 \tN+c\tl = 1T l = 1¹ ) = \ _ 1 tnp + 1 22 t R 2 / = 3 tnp + 3 lpp+cl

3 \ 2 1

= 1T = 2 \t N +c\t = 1T = 2 ^ = 3 t np + 3 t pp+c,

t = 1T = 2 \tN+c\tl = ,R Ul _ 11 rpl _ 3 \ _ T l = 2 ) = ^ ( t n p - tRp+c),_!_R bN+c\

2 t N + c \

t = 1T = ^ \t N + c \t l = 1t1 = 2 ) = ^ ( t n p - tRp+c). (6)

an d d istin g u ish betw een th e partial-w ave sta te s \pqa) and

\pqß). T he \pqa) are sta te s w ith to ta l 2N a n g u lar m om en­

tu m j below some value j max: j < j max, in w hich the nuclear, VN , as well as th e p p screened C oulom b in terac­

tion, VcR (in isospin t = 1 sta te s only), are acting. In the sta te s \pqß) w ith j > j max, only VcR is actin g in th e pp subsystem . T h e sta te s \pqa) a n d \ pqß) form a com plete set of state s

For th e isospin t = 0, in w hich case T = T l = 2

( t = 0T = 2 \tN+c\tl = 0T l = 1 ^ = t np. (7)

In th e case of t R only th e screened p p C oulom b force VcR is acting.

T he th ird te rm on th e rig h t-h a n d side of (5) is pro ­ p o rtio n a l to ( pqß\tRP G 0\plqlß l)( plqlß l \tR. A d irect calcu­

la tio n of its isospin p a rt shows th a t in d ep en d en tly of th e value of th e to ta l isospin T it vanishes [9].

In sertin g ( p q ß \ T \E) from (5) in to (4) one gets (pq a \ T \ E) = (pqa\tN+cP\&) + (pqa\ t N+cPGot R P\<P) p 2dp q2dq ( ^ \pqa)(p q a \ + ^ \pqß)(p q ß \ j = L (3) - ( p q a \ t N +cP G o ^ J p l2dplql2dql \plqla l)(plqla l \ t RP\ $)

ß

a

+ (pqa\tN+cPGo J p ,2dp' q,2dq' \ p' q' a' )( p' q' a' \ T\ $) a

+ (p q a \t N+cP G0t R P G ( ^ j p ' 2 d p ' q 2d q \ p q a ) (p q a \ T \<P) a

- ( p q a \ t N +c P G o ^ J p ' 2dp'q'2dq' \p' q ' a ' ) ( p' q ' a' \ t RPGo a

X Z ) J p''2dp''q''2dq''\p''q''a'')(p''q''a''\T\&). (8) a"

T his is a coupled set of in teg ral equations in th e space of th e sta te s \a) only, w hich in co rp o rates th e c o n trib u tio n s of th e p p C oulom b in teractio n from all partial-w ave sta tes u p to infinity. It can be solved by ite ra tio n an d P ad e sum ­ m atio n [9,11].

W h en com pared to our s ta n d a rd tre a tm e n t w ith o u t screened C oulom b force [11] th e re are tw o new leading term s: (pqa\tN+cP G o t RP\&) a n d - ( p q a \ t N + cP G o \ a ' ) X (a'\ t RP\&). T he first te rm m u st be calcu lated using di­

rectly th e 3-dim ensional screened C oulom b t-m a trix t R, while th e second te rm requires only th e p a rtia l-w a v e- p ro je cted screened C oulom b t-m a trix elem ents in the

\a) channels. T he kernel also contains tw o new term s:

th e te rm (pqa\tN+cP G 0t RP G 0\ a' )( a' \T\ P) m u st again be c alcu lated w ith a 3-dim ensional screened C oulom b t - m a trix t R , while th e second one, - ( p q a \ t N + cP G 0\a') X (a' \tRPG0\ a'' )(a' ' \T\&) , involves only th e p artia l-w a v e- p ro je cted screened C oulom b t -m a trix elem ents in th e |a) channels. T h e calcu latio n of those new term s w ith th e p artia l-w a v e-p ro je c te d C oulom b t -m atrices follows our s ta n d a rd p rocedure [11]. Namely, th e tw o sub-kernels t N +CP G 0 an d t R P G 0 are applied consecutively to th e corresponding s ta te . T he d eta ile d expressions how to cal­

culate th e new te rm s w ith th e 3-dim ensional screened C oulom b t-m a trix are given in ap p en d ix A of ref. [9].

T he tra n sitio n am p litu d e for b reakup, ( $ 0\U0\P), is given in te rm s of T\ P) by [1 1,1 2]

($o\Uo\$) = ( ^ ( 1 + P )T\<P), (9) w here \&0) = \ p q m 1m 2m 3v 1v 2v3) is th e free s ta te and th e Jacobi m o m en ta p an d q specify com pletely a p a r­

tic u la r exclusive b re ak u p configuration of th re e o u tgo­

ing nucleons. T he p e rm u ta tio n s actin g in m om entum , spin-, an d isospin-spaces can be applied to th e b ra -sta te (&0\ = ( p q m 1m 2m 3v 1v 2v3 \ changing th e sequence of nucleons spin an d isospin m agnetic q u a n tu m num bers m 2 an d v 2 an d leading to well-known linear com bina­

tions of th e Jacobi m o m e n ta p and q. T h u s ev alu at­

ing (9), it is sufficient to reg a rd th e general am plitudes (p q m 1m 2m 3v 1v 2v3 \T\P) = ( p q \ T \ P ) . U sing eq. (5) and th e com pleteness re la tio n (3) one gets:

(pq \ T\<P) = (pq \ ^ J p ' 2dp'q'2dq' \p' q'a') (p'q'a'\T\<P) a

- W \ p '2d p 'q '2dq' \p'q'a') (p' q' a' \ t RP\<P) a

- ( Pp \ J p ' 2 dp'q'2dq' \p'q'a') {p'q' a'\tR P Gq

a

X p " 2d p "q "2dq" \p''q''a'') (p''q''a''\ T\&) a"

+ (pq \ t R P \ $ ) + (pq \ t R P Gq

x ^ J p ' 2dp'q'2dq' \ p ' q ' a ' ) ( p ' q ' a ' \ T \ P ) . (1 0) a

It follows th a t, in a d d itio n to th e am p litu d es (pqa\T\&), also th e p artia l-w a v e-p ro je c te d am p litu d es (pqa\tRP\ P) an d (pqa\tR P G 0\ a' )( a' \T\ P) are required.

T he expressions for th e c o n trib u tio n s of these th re e term s to th e tra n sitio n am p litu d e for th e b reak u p reactio n are given in ap p en d ix B of ref. [9].

T he la st two te rm s in (10) again m u st be calcu­

la te d using d irec tly th e 3-dim ensional screened C oulom b t -m atrices. In ap p en d ix C of ref. [9] th e expression for ( p q \ t RP\ P) is given an d in ap p en d ix D for th e last m a­

trix elem ent ( p q \ t RP G 0\a' )(a' \T\&).

T he tra n sitio n am p litu d e for elastic scatterin g , U , con­

ta in s in ad d itio n to all re scatterin g s P T also a d irect ex­

change te rm P G - 1 an d is given by

(P\U\<P) = ( P \ P G - 1 + P T \ P ) , (11) w here th e outgoing p ro to n -d e u te ro n s ta te \ P ) differs from

\P) by th e directio n of th e relative p ro to n -d e u te ro n m o­

m entum . I t can be o b ta in ed by q u a d ra tu re using (10).

It was show n in [9] th a t th e elastic p d sc a tte rin g am ­ p litu d e has a w ell-defined screening lim it an d does n o t require ren o rm alisatio n . To get th e physical b re a k u p am ­ p litude, however, it is unavoidable to perform th e renor­

m alisatio n of th e p p half-shell t-m atrices [10]. Such a b e­

havior of th e elastic sc a tte rin g a m p litu d e in our schem e is in c o n tra d ictio n to th a t in th e screening an d renorm aliza­

tio n m e th o d derived in [1,2] an d applied in [6,7]. T here th e screening of th e elastic sc a tte rin g a m p litu d e is u n ­ avoidable as was d e m o n stra te d in [14]. Very p ro b ab ly it reflects different form s of th e e q u atio n s to be solved. Nev­

ertheless it m u st b e checked if th e ap p ro x im atio n s used in our feasibility calculation of ref. [9], m entioned in th e in tro d u ctio n , does n o t in tro d u ce a w rong behavior. T he resolution of th a t co n trad ictio n aw aits to be solved.

W h en on to p of pairw ise forces th re e nucleons in te ra c t also th ro u g h a 3N F ad d itio n al re scatterin g s g en erated by th a t 3N F a p p e a r in th e m u ltip le-scatterin g series an d th e F addeev eq u atio n for th e s ta te T\ P) changes to [15]:

T \ $ ) = t P \ $ ) + ( 1 + t G o ) v ( 1] (1 + P M + t P G o T \ p + ( 1 + tG o)V4(1)(1 + P ) GqT \ p . (12) T he 3N force v 4 is n a tu ra ly sp lit in to 3 p a rts

V4 = V4(1) + V4(2) + V4(3), (13) w here V4(i) is sy m m etrical u n d e r th e exchange of nucle­

ons j an d k ( i , j , k = 1, 2, 3, i = j = k). Such a sp littin g

Page 4 of 6 T he E u ro p e an Physical Jo u rn al A

is always possible an d in th e case of th e n -n exchange 3NF corresponds to th e th re e possible choices of th e nu­

cleon undergoing off-shell nN sc atterin g . E q u a tio n (12) contains tw o new term s: one leading te rm an d one in th e kernel. T h ey reflect a d d itio n al co n trib u tio n s to the m u ltip le-sca tte rin g series caused by th e 3NF.

Perform ing analogous step s as for (1) an d s ta rtin g w ith th e p ro jectio n of (1 2) on th e \pqa) an d \pqß) sta te s one gets

(pqa\T\&) = (pqa\tN+cP\&)

+ (p q a \( 1 + t N+cG 0) V ) ( 1 + P ) \^ ) + (pqa\tN+cPGo\a' ) ( a' \ T \ $) + ( pqa\tN+cPGo\ß' ) ( ß ' \ T \$)

+ ( p q a \ ( 1 + t N + c G o V 1 ( 1 + P )G o \a') (a' \T\P) + (pqa\(1+ tN+cGo)V( 1 (1 + P )Go\ß') ( ß ' \ T \ $ ) , (14) and

(p q ß \ T \$) = (pqß\ t R P \ $ )

+ (pqß\ (1+ t R Gq)V4( 1 ) ( 1 + P ) \ $ )

+ (pqß\ t R PG o \ a ' ) ( a' \ T \ $) + ( p q ß \ t R P Go \ ß') ( ß ' \ T \$) + (pqß\ ( 1 + t RGo) V( 1 ) (1 + P )G o \a') (a' \T\P)

+ (pqß\ (1+ t RGo) V( 1 (1 + P )Go\ß') ( ß ' \ T \&) . (15) Here an d in th e following we sh o rten ed our n o ta tio n by ne­

glecting th e su m m a tio n sign over in term ed iate sta te s | a ') ( \ß ')) an d th e in te g ra tio n sign over th e corresponding J a ­ cobi m o m en ta p' an d q' . T herefore w henever a p ro jectio n o p e ra to r \a' )(a' \ ap p ears in in te rm ed iate sta te s m eans th a t th e following su m m atio n a n d in te g ratio n s m u st be perform ed:

\ a ' ) ( a ' \ ^ ^ ^ f p ' 2dp'q'2dq' \ p'q'a' )(p'q' a'\. (16) a

Since a 3N F is sh o rt-ran g e d its m a trix elem ents con­

ta in in g \ß) channels vanish:

(a\V( 1 (1+P )\ß) = (ß \ vRl ) (1+P )\a ) = ( ß \ v Rl ) (1+P ) \ ß ) = 0 . (17) T hus in th e \ß) channels only th e p p C oulom b force is acting an d therefore (15) reduces to

(p q ß \ T \$) = (pqß\tR P \ $ )

+ ( p q ß \ t R P G o \ a ' ) ( a ' \ T \ $ ) + ( p q ß \ t R P G o \ ß ' ) ( ß ' \ T \$) = (p q ß \ t R P \&) + ( p q ß \ t R P G o \ a ' ) ( a ' \ T \$). (18) A gain we have used th e fact th a t th e th ird te rm in (18) vanishes (see also th e rem ark afte r (7)). Also n o te th a t t R is diagonal in th e high p a rtia l waves an d consequently v R 1 does n o t co n trib u te .

In sertin g (18) in to (14) one gets (pqa\T\<P) = (pqa\tN+cP\&)

+ (pq a \ ( 1 + t N+cG 0)V4(1) ( 1 + P ) \^ )

+ (pqa\tN+cPGo\a' ) ( a ' \ T \ $ ) + (pqa\tN+cPGo \ ß ' ) x [ ( ß ' \t RP\ $) + (ß ' \ tRPGo \ a ' ) (a'\T\$)]

+ (pqa\(1 + t N + cGo)vR )(1 + P )G o \a') (a'\T\<P) + (pqa \( 1 + t N+cG o)V4 ) ( 1 + P ) G o\ß ' )

x [ ( ß ' \tRP\<P) + (ß ' \ tRPGo \ a ' ) (a'\T\<P)]. (19) Thus,

(pqa\T\&) = (pqa\tN+cP\&)

+ (p q a \(1 + t N+cG 0)V4 ) ( 1 + P ) \^ ) + (pqa\ t N+ c P G o \ ß ' ) ( ß ' \ t R P \$)

+ (pqa\(1 + t N + c G o W } 1 ( 1 + P )Go\ ß' ) ( ß' \t R P\&) + (pqa\tN+cPGo\a' ) ( a ' \ T \ $ )

+ (pqa\ t N+c PGo \ ß' )( ß ' \ t RPGo \ a' ') (a' ' \T\ $) + (pqa\(1 + t N + cGo)VR )(1 + P )G o \a') (a'\T\<P) + (p q a \(1 + t N+cG o)V4 ) ( 1 + P ) G o\ß ')

x ( ß ' \ t R P G o \ a ' ) ( a ' \ T \ P ) . (2 0) D ue to (17) th e te rm in th e fo u rth line of (20) an d th e last te rm of (20) can b e d ro p p ed . U sing th e com pleteness relatio n (3) for th e |a) an d |ß) sta te s one finally is left w ith th e coupled set of in teg ral equations in th e space of

\a) channels only:

(pqa\T\<P) = (pqa\tN+cP\&)

+ (p q a \(1 + t N+cG 0)V4 )( 1 + P ) \^ )

+ (pqa\ t N+cPGot RP \ & ) - ( p q a \ t N + c P G o \ a ' ) ( a ' \ t RP\&) + (pqa\tN+cP G o \ a ' ) ( a ' \ T \ $ )

+ ( pqa\ t N+c PGot RPGo\ a' ) (a' \T\P)

- ( p q a \ t N + c P G o \ a ' ) ( a ' \ t R P G o \a'') (a'' \T\P)

+ ( p q a \ ( 1 + t N

+

^cGq)vR1)(1 + P )G o \a') (a'\T\&) . (21) C om paring it to eq. (8) w ith 2-body forces only, th ere is one ad d itio n al c o n trib u tio n in th e leading term , (pqa\(1 + t N+cG 0)V4(1)(1 + P)\&), an d one in th e kernel, (pqa\(1 + t N + cGo)VR1') ( 1 + P ) Go\a' )(a' \T\&), b o th con­

tain in g v R 1 ( 1 + P ).

T he tra n sitio n am p litu d e for b reak u p is again given by (9) and for elastic sc a tte rin g tw o new te rm s driven by Vj (1 + P ) ap p e a r [15]:

( P \ U \ $ ) = ( P \ P G - 1 + P T + v R 1} ( 1 + P )

+ V4(1) ( 1 + P ) GqT \ $ ) . (22) Since th e s tru c tu re of th e set (21) is analogous to th e s tru c tu re of th e set (8) describing p d sc a tte rin g w hen only

pairw ise forces are acting, it follows th a t th e inclusion of th e 3N F in to th e F addeev calculations of p d elastic sc a t­

te rin g an d b re ak u p reactio n requires no new m a trix el­

em ents and num erical tools beyond th o se used in [9, 10]

an d [15].

3 The electromagnetic reactions on 3He

It was show n in [16] th a t th e basic eq u atio n s describing reactions on 3He induced by p h o to n s or electrons have th e sam e stru c tu re as th e 3N co n tin u u m Faddeev equa­

tions (1) an d (12). T he new physical ingredient is th e p h o to n ab so rp tio n o p era to r, lets call it O [16]. For a com ­ p lete b re a k u p of 3He induced by p h o to n s th e nuclear m a­

trix elem ent N , from w hich all observables can be d e te r­

m ined, is given by an au x iliary s ta te |U) w hich fulfills th e Fad d eev -ty p e equation:

\U) = tGo + x ( P + 1)V( )Gq(1 + tGo) ( 1 + P )O\&i)

+ ( t G o P + 2 ( P +1)V 4(1)G q ( 1 + tGo)P^j \U). (23)

T hen,

N = (^ o \(1 + tG o ) ( 1 + P ) O \ $ i ) + ( $o\ ( 1+ t G o ) P \ U ). (24) Here \'^i ) is th e in itial 3He b o u n d s ta te a n d \@0) is the fully an tisy m m etrized free s ta te of th re e outgoing nucle­

ons, given in te rm s of th e ir Jacobi m o m en ta and spin and isospin q u a n tu m num bers.

For th e p d b re a k u p of 3He th e nuclear m a trix elem ent is given [16] by

N Pd = ( $ q\( 1 + P )O\&i) + (<Pq\ P \ U), (25) w here th e final s ta te is d eterm in ed by th e p ro to n -d e u tero n relative m o m en tu m eig en state |qq ) an d th e d e u tero n wave function \4>d):

(&q\ = (^ d \ (q \. (26)

L et us consider first th e case w ith o u t 3NFs:

\U) = tG o(1 + P )O\&i) + t G o P \ U ). (27) P ro je c tin g eq. (27) on sta te s |a) an d |ß) (in th e |ß) sta te s only th e screened C oulom b force VcR is acting) one gets

(p q a \U ) and (pqß|U)

(p q a \t N +cG 0 ( 1 + P ) O Wi) + (p q a \t N+cG 0P \U)

= (pqß\tRGo(1 + P )O\&i) + ( p q ß \ t R Go P \U) = ( p q ß \ t R G o ( 1 + P )O\&i) + (pqß\tR G o P \ a ' ) ( a ' \ U ) + ( pq ß \ t R G o P\ ß ' ) ( ß ' \ U). (29) Since in above eq u atio n s th e p h o to n ab so rp tio n op er­

a to r O comes always w ith 3He b o u n d s ta te therefore ( ß \ O \ ^ i ) = ( ß \ P O \ ^ i ) = 0. C onsequently th e first te rm

in (29) vanishes. T h e la st te rm is p ro p o rtio n a l to (pqß|

t R G 0P \ ß ) ( ß \ t R and th e d irect calcu latio n of its isospin p a rt gives zero. T h u s eq. (29) reduces to:

(pqß\U) = ( p q ß \ t R G o P \a' )(a' \ U). (30) In sertin g (30) into (28) one gets:

(pqa \U) = (p q a \t N+cG 0(1 + P ) O \ ^ i)

+ ( p q a\ tN+c Go P \ a ' ) ( a' \ U) + (pqa\tN+cGqP \ß ' ) ( ß ' \ U ) = (p qa \ t N + cGo(1 + P )O\&i) + (pqa\tN+cGqP \ a ' ) ( a ' \ U ) + ( pqa\ t N+c GoP\ ß ' ) ( ß ' \ t R G o P \a' ) ( a ' \ U ). (31) U sing th e com pleteness re la tio n for th e |a) an d |ß) sta te s gives:

(p q a \U) = (p q a \t N +cG 0( 1 + P ) O \&i) + (pqa\tN+cGqP \ a ' ) ( a ' \ U ) + ( p q a\ t N + c G o Pt R G oP \ a ' ) ( a ' \ U )

- ( p q a \ t N + c G o P \ a ' ) ( a ' \ t R G o P \ a ' ' ) ( a ' ' \ U ).

(32) W h en com pared w ith th e set resu ltin g from (27) for a n e u tro n -n e u tro n -p ro to n system th e re are tw o new te rm s in th e kernel: (pqa\tN+cG 0P t R G 0P \ a ' ) ( a ' \ U ) and - ( p q a \ t N + cG 0P \ a ' ) ( a ' \ t R G 0P \ a ' ' ) ( a ' ' \ U ). T h ey are iden­

tical to those in (8) for th e 3N co n tin u u m an d conse­

q u en tly also th e ir evaluation is th e sam e as for p d sc a tte r­

ing. T h e vanishing of \ß)-com ponents of th e O \ ^ ) - s t a t e caused th a t, in stead of th re e leading term s as in (8), only one leading te rm appears, w hich can be calcu lated in a sta n d a rd w ay [16].

S ta rtin g from (23) an d perform ing analogous steps w hen th e 3N F is included gives

(p q a \ U ) = (p q a \t N +cG 0(1 + P ) O \ ^i ) + (pqa

+ (pqa + (pqa - ( p q a + (pqa

1 ( P + 1)V( l ) Go(1 + tN+cGo )(1 + P ) O \ $ i ) RN+c G Q

R

G 0P \ a ' ) ( a ' \ U ) tN+c G o P t R G o P \ a ' ) ( a ' \ U )

t N + c GqP \ a ' ) ( a ' \ t R G o P \ a ' ' ) (a ' ' \ U )

2 ( P + 1)V1 4(1)Gq(1 + t N + c G o ) P\ a ' ) ( a ! \ U ). (33)

Thus, adding a 3N F resu lts in one ad d itio n al lead­

ing term , (pqa\ 1 ( P + 1)V4(1) G o( 1 + t N +cG0 ) ( 1 + P ) O \'^i) : an d one ad d itio n al kernel term , (pqa\ 1 ( P + 1)VR1 G0(1 + (28) t N + c G o ) P \ a ' ) ( a ' \ U ).

T he m a trix elem ents (pqa\ U) provide tra n sitio n am pli­

tu d e s for th e two- an d th re e -b o d y b re a k u p of 3He. Namely, for th e tw o-body b reak u p of 3He th e second te rm in (25) can be calcu lated using (30) a n d th e com pleteness of the

\a) an d \ß) sta te s, re su ltin g in:

($q\P\U) = ( $ q \ P \ a ) (a \ U) + ( $ q \ P \ß)(ß\U) = ( $ q \ P\ a ) ( a \ U) + ( $ q \ P t R G o P \a' )(a' \U) - ( $ q \ P \ a ) ( a \ t R G o P \ a ' ) ( a ' \U). (34)

Page 6 of 6 T he E u ro p e an Physical Jo u rn al A

T h e first and th ird te rm s can b e o b ta in e d from th e |a) p a rtia l-w a v e-p ro je c te d m a trix elem ents using (B.2) of ref. [9]. T he second te rm m u st be calcu lated using d irectly th e 3-dim ensional screened C oulom b t-m a trix t R accord­

ing to (D.9) of ref. [9].

For th e th re e -b o d y b reak u p of 3He th e second term in (24) can be calcu lated in a sim ilar way an d is given by

( $ o \ ( 1 + tGo )P\ U) = (<Po\P\a)(a\U) + ( $ o \ P \ß)(ß\tRGo P \ a ' ) ( a ' \ U )

+ ( $ o \ a ) (a \ tN + c Go P \ U ) + ($o\ ß ) (ß\tRGo P \ U ) = ( $ o \ P\ a ) ( a \ U) + (<Po\PtRGoP\a)(a\U)

- ( $ o \ P \ a ) ( a \ t R G o P \ a ' ) ( a ' \ U ) + (<Po\a)(a\tN+cG o P \a' ) ( a \ U ) + ( $ o \ a ) ( a \ t N + c G o P t R G o P \ a ' ) ( a \ U )

- ( $ o \ a ) ( a \ t N + c G o P \ a ' ) ( a ' \ t R G o P \ a ' ' ) ( a ' ' \ U )

+ ( $o \ t R Go P \ a ) ( a \ U ) - ( $ o \ a ) ( a \ t R G o P \ a ' ) ( a ' \ U ). (35) Here again, th e second, fifth a n d seventh te rm m ust be calcu lated using d irectly th e 3-dim ensional screened C oulom b t-m a trix t R . For th e second, (<T0\ P t R G 0P \a) X ( a \ U ), a n d seventh, (@0\tR G 0P \ a ) ( a \ U ), te rm th e cal­

cu latio n follows expressions (D .6), (D.7) an d (D .8) of ref. [9]. For th e fifth m a trix elem ent, (@0\a) X

(a\ t N + cG 0P t R G 0P \ a ' ) ( a ' \ U ) , th e corresponding expres­

sions of ref. [9] are (A.19) an d (B.1). T he calcu latio n of th e rem aining, |a) p artia l-w a v e-p ro je c te d m a trix elem ents in (35) follows (B.1) of ref. [9].

4 Summary

We ex ten d ed our ap p ro ach to include th e p p C oulom b force in to th e m o m en tu m space 3N Faddeev calculations, p resen ted in refs. [9, 10] for elastic p d sc a tte rin g and b rea k u p in case w hen only pairw ise forces are acting, to include also a 3N F an d to tr e a t reactions induced by in te r­

action of electrom agnetic probes w ith th e 3He nucleus. It is based on a s ta n d a rd form ulation for sh o rt-ran g e forces an d relies on th e screening of th e long-range C oulom b in­

tera ctio n . In o rd er to avoid all u n c e rta in ties connected w ith th e ap p licatio n of th e partial-w ave expansion, u n ­ su itab le w hen w orking w ith long-range forces, we ap p ly d irectly th e 3-dim ensional p p screened C oulom b t -m atrix .

For each rea ctio n considered in th e p resen t study: elas­

tic p d sc a tte rin g an d b reakup, two- an d th re e -b o d y decay of th e 3He nucleus induced by real or v irtu a l photons, th e resu ltin g coupled set of in teg ral e q u atio n s in th e finite space of |a) channels only, in co rp o rates th e co n trib u tio n s of th e p p C oulom b in tera ctio n from all partial-w ave sta te s u p to infinity. A dding a 3N F resu lts in a set of Faddeev- ty p e e quations w ith th e sam e s tru c tu re as in th e case w hen only 2N in teractio n s an d p p C oulom b force are acting. O n to p of th a t for each rea ctio n one new co n trib u tio n in th e leading te rm an d in th e kernel ap p ears. T hese tw o a d d i­

tio n a l te rm s have th e sam e form in d ep en d e n t if th e pp C oulom b force is actin g or not.

Solutions of th e resu ltin g Faddeev eq u atio n s in th e form of p artia l-w a v e-p ro je c te d m a trix elem ents, to g eth er w ith th e ad d itio n al m a trix elem ents calcu lated d irectly w ith th e 3-dim ensional screened C oulom b t -m atrix , pro ­ vide tra n s ito n am p litu d es from w hich num erous observ­

ables can be calculated.

Since in [9,10] th e p ractical feasibility of our form ula­

tio n has been d o cu m en ted in case of p d elastic sc a tte rin g an d break u p , th e p resen ted extension of sim ilar stru c tu re will also be feasible an d will allow to ap p ly th a t approach w ith th e com plete nuclear H am ilto n ian to analyses of num erous d a ta from 3N h adronic an d electrom agnetic reactions.

T his work was su p p o rted by th e P olish 2008-2011 science funds as a research p ro ject No. N N202 077435. It was also p a r­

tially su p p o rted by th e H elm holtz A ssociation th ro u g h funds provided to th e v irtu a l in stitu te “Spin and strong Q C D ” (VH-VI-231).

Open Access T his article is d istrib u ted u nder th e term s of th e C reative Com m ons A ttrib u tio n N oncom m ercial License which perm its any noncom m ercial use, distrib u tio n , and reproduction in any m edium , provided th e original author(s) and source are credited.

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