Volume 227, number 3,4 PHYSICS LETTERS B 31 August 1989

SOLITONS IN THE N A M B U - J O N A - L A S I N I O MODEL

Th. MEISSNER, F. GROMMER and K. GOEKE Institut fiir Theoretische Physik II, Ruhr-L1niversitdt Bochum, D-4630 Bochum, FRG

Recieved 7 March 1989; revised manuscript received 19 June 1989

The Nambu-Jona-Lasinio model in the SU (2) sector with scalar and pseudoscalar meson fields is solved taking into account the polarization of the Dirac sea (zero-boson loop, one-quark loop). The theory is supplied with a finite proper t ime cutoff A, which is fixed to reproduce the pion decay constant f~. By a self-consistent procedure soliton solutions with baryon number B = 1 are found and investigated. The linear and non-linear versions of the model are compared and the effects of a finite current quark mass are studied.

Spontaneously broken chiral symmetry seems to be the dominant mechanism to govern the structure and the reactions of the nucleon and, may be, of low lying baryon resonances. Therefore many effective relativ- istic models [ 1-3 ] have been invented, which incor- porate this feature and are designed to simulate the main properties of non-perturbative QCD at low energies. In terms of explicit quark degrees of free- dom the most simple of those is the Nambu-Jona- Lasinio model [ 1 ], which is by various authors be- lieved to be derivable from QCD by means of some long wavelength approximation [4-8] . For a long time it has been used to describe the vacuum and to investigate some bosonic excitations or medium ef- fects [9,10]. Only very recently it was suggested to study localized solutions with baryon number B = 1. Indeed in the SU (2) sector and for classical hedge- hog boson fields these solitonic solutions were found for some special cases namely by assuming a fixed form of the meson profile [ 11-13 ] or by restricting the system to the chiral circle [ 14]. It is the aim of the present paper to show the existence and to inves- tigate the properties of solitonic solutions with B= 1 in a rather general case with the only assumption of classical a (scalar) and n (pseudoscalar) fields with hedgehog form.

The Nambu-Jona-Lasinio lagrangian in the SU (2) sector with scalar and pseudoscalar couplings reads [1]

YNJC = ¢ ( i 0 - - m o ) q / + ½G[ (q)~,)2+ (~ysxq/)Zl. ( 1 )

Here V symbolizes the Dirac spinor for the quarks with SU(2) isospin (up and down), current mass mo= md = mo and N~= 3 colors. By simple functional integral techniques [ 15-17 ] one sees that the gener- ating functional

~ N J L = f ~ @ v J . ~ e x p ( i f d 4 x o S P N J L ) (2)

is equivalent to

~ N J L = ~1~ ~ &O" ~ r exp i d 4x ' : '~NJL ,

where

(3)

Lf~jc = ~[ i~ - g( a + i~5 lrn ) ] ~/

/t 2 -- T (0"2"~ff1"2) + m°/22 0", (4)

g

with G=g2/]22 and # to be determined later. This corresponds to a = - ( g / / ~ 2 ) ~ q / and n = - ( g / /z2) ~irTs~,.

Our approximation consists in treating a and n classically. After integrating out the quarks one ob- tains the generating functional

:)"~JL =exp[ -S~rf (a , n ) ] (5)

296 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)

Volume 227, number 3,4 PHYSICS LETTERS B 31 August 1989

in terms of the effective euclidean action Self(a, rt) F =Soft(a, n) +Syfffa, tt) with the fermionic and me-

sonic parts, respectively

F S~fr = - Sp log D , (6)

f sMf= T d4XE (0"2"~ ~2) - m°/A2 d4xE a , (7) g

where D = - i ~ + g ( a + i 7 5 f n ) and Sp indicates the functional and matrix (spin, isospin and color) trace.

the fermionic action is UV divergent. It is made finite by the proper time regularization of Schwinger [ 18 ]. The non-anomalous part reads then

Re(S~ff) = ½N~ f d4XE

XTq, Tr: f dJ (XEIexp(_zI~ , I~) IXE) . (8) T

I / A 2

By now the theory contains as parameters the cur- rent quark mass too, the proper time cutoff A, the quark-meson coupling constant g, and the parameter lz. They are determined by the following prescriptions:

( 1 ) The divergence of the axial vector current of the NJL is identified with f~m 2 ~t yielding

moil2= gf~m ~. (9)

(2) The stationary phase conditions for the vac- uum values av and 7tv of the sigma and pion field, i.e. (OS~rf/Oa)v=O and (0S~fr/07t)v=0 yield n v = 0 as well as the Schwinger-Dyson equation (gap equa- tion ) for av

p2= 8Ncg212(gav , A) + m~ (10)

with

I ~ ( M , A ) - 1 i 16~2 d r r - ~ e x p ( - z M 2) . (11) I / A 2

(3) Identifying the (02~ff/0~z 2)v= - r n ] one con- cludes av =f=.

(4) One demands that the Feynman diagram of the weak pion decay in the soft pion limit (m=--,0) yields indeed f= which requires [ 12,13 ]

4N~g2I, (gf=, A) = 1, (12)

which is equivalent to demanding the kinetic energy of the mesons to exhibit the proper normalization [15,17,11].

With the prescriptions (9 ) - (12) the quark-me- son coupling constant g remains as the only free pa- rameter of the theory. In actual calculations it will be adjusted to reproduce one of the observables of the nucleon.

Using these relations between the constants of the lagrangian one can derive the self-consistent equa- tions for the various fields. For this the total energy of the system with baryon number B= 1 is needed which, following refs. [ 13,19 ] can be written as

E = r]val~val +Eo +EM -1- Ebr

with

o o A; E o = ½ N c ~ dzz -3/2

I

9

{ [('9-] × ~ e x p - z ~ - ~ e x p - r

(13)

/ t 2 f EM = ~ - d3x (02°1-7[ "2) , (14)

= - m ~ f ~ f d3x ( a - f ~ ) , (15) Eb r 2

where tx and ¢ao are the eigenvalues of the single par- ticle hamiltonians:

h = -iat. V+gfl(a+ i75 ~Tt) (16)

and

ho = - ion. V + gflf,~ ,

respectively. The sums over 2 in Eo extend over all single particle levels (positive and negative) includ- ing the localized and bound valence one (2 = val). The qval = 1 for (val > 0 and 0 for ev,i < 0.

Since there are no kinetic energy terms for the me- son fields in the lagrangian, their equation of motion are simply given by 8E/Sa= 0 and 8E/8~= 0 yielding

a ( r ) = - ~22~ So(r)+Sva~(r)rl~,~ + ~ L m ~ ,

(17)

~ r ( r ) - - g Arc /z= 47r [Po(r) + P v a , ( r ) r / ~ a l ] . (18)

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Volume 227, number 3,4 PHYSICS LETTERS B 31 August 1989

The proper time regularized scalar and axial densi- ties are composed out of the single particle contributions

&(r) = f d£2 q;;~(r, (2)On(r, g2) ,

p;(r) = f dg2 ~ ( r , 12)i75T~ (r, £2)

by

i 1 d r r - ~/2 So(r) = - ~ ,

Po(r) = - dz z-1/2

X ~ exp - z p;. •

for the actual calculations we assume the spherical hedgehog ansatz: a ( r ) = a ( r ) and n ( r ) = iJr(r). The set of eqs. ( 16 ) - (20) is treated iteratively. We start with some reasonable chose a and ~ fields and solve then the Dirac equation: h % = ~ by putting the sys- tem into a sphere with finite radius [20]. In the next step So (r) and Po (r) are evaluated which yields then new boson fields a ( r ) and n(r), etc. The procedure is repeated until self-consistency is reached. We have checked our results by using different starting meson profiles.

Actually there does not exist any solitonic (i.e. lo- calized and B = I ) solution for g < & ~ 3 . 5 (corre- sponding to a cutoff of about A ~ 660 MeV). This feature is analogous to that encountered already in the linear chiral sigma model (Gell-Mann-Levi) [ 2 ].

For g=4>g~, solitonic solutions exist and the re- suiting boson fields for m~ = 0 and m~ = 138 MeV are shown in fig. 1. Actually both solutions are rather close

1.00,

.00 .40 .80 1.20 1 . 6 0 2.00 r ( f ro)

Fig. 1. Self-consistent meson fields a and n in units off~ = 93 MeV of the linear model (with m~=0 and m~= 139 MeV, respec- tively.) as well as the non-linear model (with m~=0 and a = f~ cos 0, n = ~f~ sin 0) for a quark-meson coupling constant g = 4.

to each other and show noticeable deviations from the one, where the fields are assumed to fulfill the condition of the chiral circle, i.e. a2 + n2 = ~r~ = f 2 (with m~ = 0 ). For all these solutions the various en- ergy contributions and the quadratic baryon radii are presented in table 1.

Again the chiral circle appears as a severe restric- tion of the system. The main differences lie in the value of a ( r = 0 ) , in the height and position of the minimum of the pion field and the magnitude of the tails. There are also discrepancies in the valence quark energies in contrast to the quark-loop contributions which are rather similar.

Fig. 2. shows the baryon number density. It is given by

p(r) = ~ p~(r) +q,alp, al(r) (21)

with the single particle contribution pa ( r )= f dO ~ (r, 12) %. (r, g2). The valence part appears nor- malized to one and the quark-loop part normalized

Table 1 The various energy contributions [ due to eqs. ( 13 ) - ( 15 ) ( in MeV) ] and the quadratic baryon radius ( R 2 } (in fm 2 ) of the self-consis-

tent B= 1 solution fo rg=4 .

Eval Eo EM E~r Era, ( R 2 )

linear m~=0 483 601 122 0 1206 0.62 linear m~= 139 MeV 588 603 20 36 1247 0.58 non-linear m~ = 0 613 601 0 0 1214 0.72

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Volume 227, number 3,4 PHYSICS LETTERS B 31 August 1989

2.00-

I : 1.50- E

1.oo-

® 0,50-

E 0.00-

.,13 4.50-

.00 .25 .50 .75 1.00 5 1.50 r ( fm)

Fig. 2. Valence and one-loop distribution to the baryon density p (r) = f dO ~*q/for the linear model with m~= 139 MeV and g= 4. For clarity the values of the one-loop part are multiplied with a factor 10 in the region r>~0.4 fro.

to zero. For g = 4 the p is dominated by the valence quark contribution to 90%.

It is interesting to investigate the behavior of the solution when g is changed. Unfortunately for g> 5 the present iteration procedure provides some nu- merical problems and hence does not converge. In order to obtain a qualitative understanding we re- strict ourselves to rn~ = 0 and the chiral circle (non- linear model), where the system is much easier to solve due to the reduced number of degrees of

freedom. The solitonic energy E(g) and the quadratic bar-

yon radius R 2 (g) versus the quark-meson coupling constant g can be found in figs. 3 and 4. For 4~<g~< 8

] .~0-

1.20

~-1.00-

~0.80-

~0.60-

0.40-

0.20-

0,00 4-

]

/ i / "

/

~ Imam)

F i i i i 6 8 10 12 14 16

coup l i ng cons fan f g

wB=l - ~ - 1-Ioo

1 8 20

Fig. 3. Energy (valence, one-loop and total) (in MeV) of the self- consistent solution in dependence of the coupling constant g for the non-linear model.

1.00-

0.80-

: 0.60- E

~- 0.40

0.20.

~. total -- I-loop-contribution volence-contribution X__

. . . . . . . . . . . !

Mi~adm u l f ' ~ M t ~

clkol cir~

0 . 0 0 - - - ~ , - - . . . . - 2 8 10 12 14 16 18 20

coup l i ng cons~an~ g

Fig. 4. Quadratic baryon radius (valence, one-loop and total ) (in fm 2 ) of the self-consistent solution in dependence of the coupling constant g for the non-linear model,

the system shows a contribution of the valence quarks (qva] = 1 ), for larger g these are contained already in the regularized quark-loop expressions and terms with valence quarks do not appear explicitly (~/va~=0). Thus the one-loop-contribution to the radius shows a sudden jump at around g= 8.

The energy of the system shows a rather small de- crease with increasing g whereas the quadratic bar- yon radius exhibits noticeable changes. Energies around the nucleon mass are obtained for g~ 15. At those large g, however, the baryonic quadratic radius is only 0.25 fm 2, which is too small compared to other models, which reproduce basically all nucleon ob- servables [21-23 ]. The reason for this dilemma lies in the fact, that not the mean field energy of fig. 3 but the angular momentum and isospin projected one should be compared with the experiment. To fix the relevant values for g correlation energies of about 200-300 MeV have to be invoked yielding a lowering of the total energy [21 ]. Values of g around 4-5 are required to obtain qualitatively proper energies and radii. This supports a valence quark picture. Thus with this correlations one gets the nucleon mass at about g = 4 or 5 with a reasonable baryon radius. This fixes clearly the physical region for g to values slightly above the critical one gc.

In such a case the present Nambu-Jona-Lasinio model is dominated by the Arc valence quarks with small effects due to the polarization of the Dirac sea. This feature supports previous conclusions based on fixed meson profiles [ 12 ].

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Volume 227, number 3,4 PHYSICS LETTERS B 31 August 1989

Actual ly fig. 3 differs f rom the one o b t a i n e d in ref. [ 14 ], though bo th ca lcula t ions refer to the chiral cir- cle and should be v i r tua l ly ident ical . The reason lies

in an inapprop r i a t e scaling p rocedure of ref. [ 14] used for the curve E(g). In add i t i on the i r b a r y o n n u m b e r d i s t r i bu t ion is inaccura te by the use of a too small basis for the d iagona l i za t ion o f the Di rac h a m -

i l t on ian h. To s u m m a r i z e our points : Us ing a se l f -consis tent

p rocedure we have shown that the N a m b u - J o n a -

Las in io mode l in the zero-boson- loop a p p r o x i m a t i o n o f f the chiral circle supp l ied with a p roper t ime cu tof f exhibi ts B = 1 sol i ton so lu t ions for smal l q u a r k - m e -

son coupl ing cons tants . Hereby all o ther pa ramete r s in the lagrangian are f ixed by the S c h w i n g e r - D y s o n equa t i on and by ad jus t ing the p i o n mass as well as

the p ion decay cons t an t o f their expe r imen ta l values. The ob t a ined so lu t ions have a total m e a n field en- ergy o f abou t 1200 Me V a n d a quad ra t i c ba ryon ra- d ius o f abou t 0.6 fm 2. T h e y are no t no t iceab le inf lu- enced by a f in i te cur ren t quark mass mo.

I f the meson fields are res t r ic ted to the chiral circle ( n o n - l i n e a r m o d e l ) we have s tud ied the d e p e n d e n c e of the total energy a n d the quadra t i c b a r y o n rad ius

on the coupl ing cons t an t g. Both quan t i t i e s are de- creasing with increas ing g. Those energies have been

ca lcula ted in the l inear chiral sol i ton mo d e l [ 21 ] an d

there the projec ted nu c l eo n energy is abou t 2 0 0 - 3 0 0 MeV lower t h a n the m e a n field one.

We are thankfu l to H. R e i n h a r d t a n d R. Wt insch f rom R o s s e n d o r f ( D D R ) for useful discussions. The

presen t paper has been par t ia l ly suppor ted by the

S tud iens t i f tung des deu t schen Volkes ( B o n n ) , the B u n d e s m i n i s t e r i u m f'tir Fo r schung u n d Technologie

( I n t e r n a t i o n a l e s Biiro a n d cont rac t 06-BO-702) a n d the K F A Jtil ich (C OS Y Pro jec t ) .

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