IL I~UOVO CIMENTO VOL. 90 A, N. 3 1 Dieembre 1985

Centre-of-Mass Problem for Hadrons from the Quark Model.

A. HOSAKA, T. K A Z o o and H. TOKI

Department o] Physics, Tokyo Metropolitan University - Setagaya, Tokyo 158, Japan

(ricevuto il 20 Giugno 1985)

Summary. - - We study the use of the Peierls-Yoccoz and the Peierls- Thouless procedures to remove the centre-of-mass motion from the Hartree-Fock states in order to construct hadrons in terms of the rela- tivistic quark model. Using explicitly the nonrelativistie harmonic- oscillator wave functions, we show the consequence of the above pro- eedures and the point to take care when several Hartree-Fock states are linearly combined.

PACS. 12.40. - Models of strong interactions.

Pions p lay an essential role in nuclear physics. This is not only due to the s t rong coupling of pion with nucleons bu t also to its ex t reme small mass. Hence, it is a fundamen ta l question in nuclear physics to find out why pion has such an ex t remely small mass among all the hadrons and how pion couples wi th nucleons. The quark model is supposed to answer this question.

Recent ly , W ~ s E and his collaborators (1) have approached the above ques- t ion concerning pion in a relat ivist ic quark model in the line of l~ambu and

Jona-Las in io (Z~JL) (2). Their model s tar ts f rom a model Hami l ton ian wr i t ten

in te rms of only quarks wi th the chiral invar iance as stressed essential for the

construct ion of pion b y I~JL. The chiral invar iance requires the in teract ion

strengths of the scalar t e r m (~o) 2 and the pseudoscalar t e r m (~75 v~) 2 identical.

The scalar t e r m is used to construct the Ha r t r ee -Foek poten t ia l to b ind quarks

(1) W. WEIss: in Quarks, Mesons and Isobars in ~Vuclei, edited by R. GU~RDIOLA and ±. PoLLs (World Scientific Publ. Co., Singapore, 1983). V. B~R~ARD, R. BROCK- MA~, M. SC~AD~, W. W~IS~ and E. W ~ . ~ : Nuel. Phys. A, 412, 349 (1984). (2) Y. NAMB~Y and G. Jo~A-LAsI~IO: Phys. l~ev., 122, 345 (1961); 124, 246 (1961).

315

316 A. HOSAKA, T. KAJINO a n d If . TOKI

to form baryons. The remaining pseudoscalar t e rm is as strong and acts as the residual interact ion in the I~PA to bring down the pion mass f rom the Har t r ee -Fock energy. Through the breaking of the chiral symmetry , this model yields the Goldberger-Treiman relation as usual and hence the strong 3W~? coupling (3).

There are many other models to deal with pions and baryons f rom the quark model (4). Generally, quarks occupy single-particle states and the col- lections of these lead to baryons. At present, the removal of the centre-of- mass mot ion seems to be the source of ambiguities for their masses and the form factors (4). Hence, we shall discuss the Peierls-Yoccoz (~) and Peierls- Thouless (3) methods to remove the centre-of-mass motion from the relativistic

Har t r ee -Fock wave functions (7). We shall s tar t with a relativistic Har t ree -Fock (HF) wave function for a

n-body system

(1) q~(x~--X, x ~ - - X , ..., x ~ - - X ) ,

whose centre of the t t F potent ia l is placed at a co-ordinate X. Note tha t X is different f rom the centre-of-mass co-ordinate R = (l /n) ~ x~. This appear-

J

ance of the ad hoe co-ordinate X for the H F potent ia l centre indicates the violat ion of the translat ional invariance; the mixture of the centre-of-mass mot ion in the Har t ree-Fock wave function. PE]3~RLS and Yoccoz (5) have proposed the following procedure to remove the centre-of-mass motion:

(2) Try(x1 , ..., x ; P) = fd3X exp [iPX] O ( x 1 - x , ..., x - x ) .

Let us show first t ha t this P Y wave funct ion ~rY corresponds to the wave funct ion of Tegen et al. (3), which is constructed in a step-by-step method

b y going into momen tum space:

(3) Tr r (x l , ..., x,,; P ) =

(2~)~"" (2~/~

(s) R. TEGV.~, R. B~OeKMA~ and W. WEIS]~: Z. .~hys. A, 307, 339 (1982). (4) A.W. THOMAS : in Advances in Nuclear Physics, Vol. 13, edited by J .W. NEGELE and E. VOGT (Plenum Publ. Co., New York, N.Y., 1984). (5) R.E. P~IE~LS and J. Yoccoz: P~oe. l~hys. Soe., 70, 381 (1957). (6) R.E. PEI~LS and D. J. THOYLESS: 2~tcl. Phys., 38, 154 (1962). (7) C.W. WO~G: Phys. l~ev. D, 24, 1416 (1981).

CENTRE-OF-MASS PROBLEM FOR H A D R 0 ~ S FROM THE QUARK MOD~L 317

The last expression is the wave funct ion of Tegen et al. (3) up to the normaliza-

t ion factor. Le t us introduce now the centre-of-mass co-ordinate R and the intrinsic co-

ordinates r~, r~, ..., r . , whose summat ion ~ r~ ~ 0 and x~ = R + r~. Then t

p) =fd3X exp [iPX] q~(r~ ~- R - - X , ..., r . -~ R - - X ) ---- (4) T~(Xl, Xn; • • u ~

= exp [iPR]~d3X ' exp [iPX'] q~(r~ - - X ' , . . . , r . -- X') o

3

As can be seen in eq. (4), the desired plane wave funct ion for the centre-of-

mass appears and only the intrinsic co-ordinates are left in the rest. However ,

the intrinsic pa r t expressed in the integral fo rm contains P in general.

The successive prescr ipt ion proposed b y PE~E~T,S and THOULESS (6) is tO remove this P dependence in the intrinsic par t . This is done b y performing another project ion:

= ( d3P' (5) TeT(x~, . . . , x . ;P) ~ ( 2 ~ ) 3 / ( P ' ) e x p [ i ( P - - P ' ) R ] ~ r y ( x ~ , . . . , x , ; P ' ) ,

where ](P') is considered as a var ia t ional funct ion to be de te rmined b y energy minimizat ion (e). l~ote t h a t ] (P ' )= ( 2 z ) 3 ~ ( P ' - - P ) brings back TpT to Tpy.

In s t ead of going th rough the above minimizat ion procedure for ](P), we shall

hereaf ter discuss the special case, where ](P') is set constant , ] (P ' )= 1.

(6) T~(x~, ..., x . ; P ) =

= ~ e x p [ i ( P - - P ' ) R ] dSXexp [iP'X]q~(x~--X,. . . , x ~ - - X ) =

= exp [iPR] ~(x~ - - R , ..., x~ - - R) = exp [iPR] ~ ( r l , ..., r~) "

This double-project ion me thod brings the H F wave funct ion into the plane wave funct ion for the centre-of-muss mot ion s a d the intrinsic wave function, which corresponds to the H F wave funct ion with the centre of the H F poten- t ial being a t the centre-of-mass co-ordinate R.

We shall now app ly the P Y and the P T procedures to the pionie case

(n = 2). Since the difficulty of the centre-of-mass prob lem and the point we

wan t to br ing out can be demons t ra ted even a t the nonrelat ivist ic level, we

take the nonrelat ivis t ic Gaussian wave functions. Her% we take a l inear com-

binat ion of two s tates denoted by 2 and P. The 0s single-particle wave func- t ion is

1 (7) ~0~(x) = (/~)~ exp [ - x~12~],

where fl is the oscillator constant . The S pionic s ta te is constructed b y coupling

318 A . HOSAKA, T. K A J I N O ~ n ~ H . TOKI

two of the above wave function for a quark and an antiquark:

( 8 ) (~/y(X 1 - - X , X 2 - - X ) - -

The PY procedure yields

1 (t~)t exp [ - [(Xl --x)~ + (x~ - X)q/2t~].

(9) ~ ( x l , x2; P) = Ts(R, r; P) = exp [iPR] exp [--flP*/4] exp [-- r*/4fl],

where the separation of the centre-of-mass motion from the intrinsic motion is perfect, as well known for the Gaussian wave function. :Note that the term exp [--fiP~/4] drops by the normalization procedure of the intrinsic wave func- tion. The PT projection simply changes this P-dependent term to (2z/5) t. The 0p single-particle wave function is

(10) %~.~(x) = 9 ~ -fi exp [-- x'12fl] Y1~($).

The P pionie state is obtained by coupling the above wave functions for a quark and an antiquark into the total spin zero:

(11) ~p(xl - -X, x~--X) = [~o,(X~ --X) ® 7,o,(X~ --X)] °.

The PY wave function becomes

(12) Tp(R, r; P) =

: e x p [ i P R ] ( - - ~ ) ( 6 f l - - f 1 2 1 ~ - - r ~ ) e x p [ - - f l P ~ / 4 ] e x p [ - - r ~ / 4 f l ] .

In this case, we find an undesired P dependence in the intrinsic part of the wave function, which cannot be removed by the normalization procedure. The further PT projection removes this P dependence and provides

(13) ~p(R, r; P) = exp [iPR] fl r~ (2~fl)t fl~ exp [ - rq4~] .

We assume now that the pionie state is written in terms of the linear combination of the S and P pionie states, where the trivial centre-of-mass part shall be dropped and E and P are understood to be normalized:

(14) [~> = aDS> -{- b]P> .

As usual the coefficients a and b are to be determined by diagonalizing the Hamiltonian matrix. We should, however, be careful with the orthogonaliza-

CENTRE-OF-MASS PROBL E M FOR HADRONS FROM THE QUARK MODEL 3 1 9

tion. The wave functions IS). and [_P} are not orthogonal:

3 - - t i p s (15a) (SIP}eT '~ (fisp~ _ 6tip2 4- 15)~'

where P Y denotes the one of Peierls-Yoccoz and P T of Peierls-Thouless. Hence, even the pion wave funct ion is normalized; (~[~} ~- 1, the sum of the a s and b 2 is not 1, a S + b 2 ¢ 1. Instead,

(16a) a ~ + b 2 + 2 ( f l sp4_ 6tips + 15)~ ab = 1 for PY ,

(16b) a2+bS-~2] /~ab- - - -1 for P T .

To deal with a skew basis, it is easier to diagonalize the basis set

(17) IS} -~ x[S} + y]P} , ]P} : ~v'[S} + y']S}.

Choosing x : 1 and y----0 with the use of the orthogonalization and the normalizat ion condition, x' and y' are determined. We find in bo th cases

( i s ) 13} = l o s } - 1 (2~fl)~i,~ exp [-- r2/4],

(19) [/5) : [ l s ) - 1 ~/~ (2nfl)~ (1 - - r2/3) exp [--r~/4] .

This is the expected result, lqamely, with the constraint of the relat ively 0 angular momen tum for the pionic state, the lower Gaussian wave functions are those of s states with higher node numbers, lqote t h a t p-wave (1----1) relat ive wave ftmctions with the spin S ---- 1 to have the 0 to ta l spin are not allowed due to the par i ty . Hence, the P ¥ project ion and the PT project ion on the harmonic-oscillator Har t ree-Fock wave functions after performing diag-

onalization lea4 to the usual expansion of the intrinsic wave functions. In the

general ease of the relativistic wave functions, such simple expressions as eqs. (18) and (19) are not obtained, bu t the essence should be the same as above.

We shall discuss now how to apply the above methods to the relativistic case by taking a simple example in which the intrinsic energy of ~ two-particle system will be calculated. Although we have obtained the intrinsic wave func- t ion through the P Y or PT method, we do not know ye t the corresponding

320 x . H O S A K A , T. K A J I N O and H . T O K I

Hamil tonian. To do this, let us follow the essence of the PT method and p u t the centre-of-mass co-ordinate and momen tum equal to zero:

(20) H(p~, p2, ...; x~, x2, ...) : H(P, k~, k~, ...; R , rl , r2, ...) -+

P-o, R-o ~ H ( k l , k~, ...; rl, r~, ...) ,

where p ' s and x~s are the momen tum and the co-ordinate of each particle, P and R those of the centre of mass and k's and r ' s are for the intrinsic motion.

Fi rs t of all, we shall show tha t this procedure makes sense for the non- relativistic case. The Hamil tonian of a two-part icle system in the harmonic-

oscillator potent ia l is

p[ m ~ ~ p~ m~o ~ (21a) H(p l ,p~; x, , x2) : ~-~ -[- ~ - x ] -[- ~ ~- - ~ - x ] .

The corresponding eigenfunction is

(21b) ~o(x,, x~) ~ exp [-- x~,]2] exp [-- x]/2]

and its eigenvalue is E - - - - 2 E 0 - - - - 2 . ~ o . Following the above procedure for the t tami l tonian by pu t t ing P = R = 0 and the PT project ion method for

the wave function, we find

: P~ #w 2 (22a) H~.I ~ ~- ~2- r ~ , ~(r) ~ exp [-- r2]4t~],

where k : 1(p1--p2), rl : Xl--x~. and # = m/2. Then, we can define the in- trinsic energy as the expectat ion value of the <( intrnsie Hamil tonian )) with the

(, intrinsic wave funct ion )~

(23) ~ . , = < R , > .

Note t h a t this Hamil tonian and the wave funct ion satisfy the eigenequation

such that / / , o l ~ ( r ) = Erol~v(r). Let us come now to the relativistic case and consider the simplest case,

where two particles move freely:

(24) (0tip1 "4- a2P2)~(1)V(2) : (Eo + E0)~(1)~o(2) •

Transforming to the centre of mass and the relative co-ordinates, we find for

the expecta t ion value

(25a) ( P ({~1-~--0[2)-~-- k(~l-- ~)~ : 2Eo. /

C E N T R E - O F - M A S S P R O B L E M F O R I t A D R O N S F R O M TI-IE QWARI~ M O D E L 321

The Gordon ident i ty enables us to write this as

+ <kS> = 2Eo,

I f we mult iply 2Eo by the above equation, we find the energy-momentum relation

(250) <p2} + 4(k2> __ 4E~.

Hence, we can identify 4<k~> as the mass due to the intrinsic motion. This simple exercise suggests the procedure to set the intrinsic energy for the relativistic case

(26) M °- 2Eo<H~,~o > . R-~0

To demonstra te this method, we take the relativistic harmonic-oscillator model (3)

c U(x) = -~ x~ . (27) _,7 = a~p~ + (1 ÷ ~ ) U(x,) + a~p~ + (1 + ~ ) U(x~),

The lowest eigenfunction is the produc t of

i/xo '~ Z (28) ~ ( x ) = N o . _ _ . . _ ~ ~ exp[ - -x~ /2x~] ,

-

wherexozC-'3-~'andz=(~)or(~)withNo being the normalizat ion con- \ ] x ]

stant. The corresponding eigcnvalue is 2Eo ---- 2 ~-- 2 %/3/Xo. Following the proce- dure to pu t P - + 0 and R ~ 0, we get the intrinsic Hamil tonian and the intrinsic wave funct ion

(29a)

(29b)

H,n~ = (~, - - a~)k + (1 + fi,) U(r /2) + (~ + fi~) U( - - r / 2 ) ,

/ i/x. ( i/ o ]

(30) M2 V ~ [ 9 4 8 ] = Xo ~ + 2--3-Y <~'~> -2Eo.

The mass or the intrinsic energy is obtained by mult iplying 2E 0 by the expecta- t ion value of the intrinsic Hamil tonian

3 2 2 A. H0SAKA, T. ~ J ~ ¢ O a n d H. TOKI

This expression depends on the spin coupling. We find the mass

M _ M _ J0.747 for S--~O, E 2Eo [ 0.792 for S = 1.

Hence, the intrinsic energy is about ~ of the to ta l energy, which is different f rom the nonrelativist ie case of ½.

This value can be unders tood qual i tat ively by considering again the free two-part icle mot ion:

,,o.= 2E- = 2Eo = o.71.

Therefore, the intrinsic energy of the relativistic two-part icle system is larger t han t ha t of the nourelativistic case. In general case, we can apply the above me thod to obtain the intrinsic energy or the centre-of-mass correction energy and calculate the value numerically.

In summary, we have proposed two systematic methods to remove the centre-of-mass motion f rom the Har t ree -Fock wave functions (products of single-particle wave functions centred at an a rb i t ra ry point). The Peierls- Yoccoz me thod separates the centre-of-mass co-ordinate f rom the intrinsic pa r t bu t it includes an undesirable centre-of-mass mo me n t u m dependence. The Peierls-Thouless me thod removes this mo me n t u m dependence f rom the intrinsic state. Fur thermore , the special case of the Peierls-Thouless procedure has the a t t rac t ive feature of producing the correct centre-of-mass wave func- t ion and the intrinsic wave funct ion as the Har t ree -Foek wave funct ion cen- t red at the centre-of-mass co-ordinate. Since the above two methods are mathemat ica l ly straightforward, t hey can be applied for any type of Har t ree- Foek wave functions.

In the nex t step, we have pointed out t ha t the resulting wave functions af ter the P Y and P T methods are not orthogonal by taking simple wave functions. This nonor thogonal i ty has to be worried once one takes a linear

combinat ion of several H F states. We then apply the PT method to obtain the intrinsic energy for the non-

relativistic and the relativistic cases. We find tha t the intrinsic energy of the relativistic case is different f rom tha t of the nonrelativistie case. We plan to apply this method systematical ly to calculate the centre-of-mass correc-

tions of the relativistic models.

We are grateiul to the members of the theory group of Tokyo ~¢Ietropolitan Universi ty for constructive discussions on the centre-of-mass problem and the

quark model.

CENTRE-OF-MASS PROBLEM FOR HADRONS FROM THE QUARK MOD~L 323

• R I A S S U N T 0 (*)

Si studia l 'uso delle procedure di Peierls-Yoceoz e Peierls-Thouless per eliminare il moto del eentro di massa dagli stati di t tartree-Fock per costruire gli adroni in termini del modello relativistico dei quark. Usando esplicitamente le funzioni d 'onda non rela- tivistiehe dell'oseillatore armonico, si mostra la conseguenza delle procedure suddette e il punto da tenere in considerazione quando si combinano linearmente alcuni stati di Hartree-Fock.

(*) T r a d u z i o n e a curet del la .Redazione.

PealOMe He rro.rly,-IeSO.