Volume 236, number 3 PHYSICS LETTERS B 22 February 1990

T H E S O L I T O N O F T H E N A M B U - J O N A - L A S I N I O M O D E L

R. A L K O F E R 1,2

Service de Physique Thdorique 3, CEN Saclay, F-91191 Gil'-sur- Yvette Cedex, France

Received 23 November 1989

The baryon number one soliton of the Nambu-Jona-Lasinio model is investigated numerically by a self-consistent procedure for various constituent masses using proper time regularization. Stable soliton solutions exist only for constituent masses higher than 400 MeV after the cutoffwas fixed in order to reproduce the experimental value of the pion decay constant. The results for the axial coupling of the nucleon gA are almost constant over the considered range of constituent masses and are approximately fourty per cent lower than the experimental value. Additionally, an approximation is discussed which reproduces fairly accurately the properties of the fully self-consistent determined soliton profile with a greatly reduced numerical effort.

The spectra of low-lying mesons and baryons can be unders tood from the fact that one has sponta- neous and explicit breaking of the chiral symmet ry of quarks. If one is interested in hadrons made of the lightest quarks u and d the chiral l imit mu = md = 0 is obviously a good approximat ion . In this l imit QCD possesses a global U L ( 2 ) × U R ( 2 ) chiral symmetry which apparent ly is broken to the diagonal subgroup U v ( 2 ) . The axial UA( 1 ) symmetry is explicit ly bro- ken by instantons [1 ], and the remaining SUA(2) symmetry is realized in the Golds tone phase, the pions being the corresponding (would-be) Golds tone bosons.

Apart from the valence quarks present in any high energy descr ipt ion of the nucleon certain models as- sume the existence of a meson cloud surrounding the region of space where the valence quarks are located. For example, in the chiral bag model [2] a chiral symmetry preserving boundary condit ion couples the quarks inside a bag with the pseudoscalar mesons outside. However, the explicit in t roduct ion of a bag surface is not necessary for the existence of a soliton whose quant ized version is then to be ident i f ied with the nucleon and the delta. Indeed, it has been shown

Supported by a DFG postdoctoral fellowship. 2 Address after March 1, 1990: Physik-Department der TU

Miinchen, T30, D-8046 Garching, FRG. Laboratoire de l'Institut de Recherche Fondamentale du Commissariat h l'Energie Atomique.

that the most simple chirally symmetr ic model, the N a mbu- Jona -L a s in io (NJL) model, supports the existence of stable solitons [ 3-5 ]. In the present let- ter I will report numerical results of a fully self-con- sistent determinat ion of the soliton profile in the NJL model.

It can be shown by simple functional integrat ion techniques that the euclidean NJL act ion for con- stant scalar fields is in the chiral l imit given by

S ( U ) =TrA [ log(i~-- MUs) - l o g ( i ~ - M ) ] , ( 1 )

where M denotes the quark const i tuent mass, i.e. the vacuum expectat ion value of the scalar meson field, and the chiral field

Us(x ) = U Y ~ ( x ) = e x p [ i 7 5 r n ( x ) / f] (2)

describes the pion field. The symbol TrA has to be unders tood as a regularized trace over the functional space as well as a trace over Dirac, colour and f lavour indices. A convenient way of regularizing the effec- t ive action ( 1 ) in an euclidean invariant manner is given by the famil iar proper- t ime regularization method [ 6 ] which consists in replacing

T r l o g D = ½Trlog D i D

- ~ - ½ i - - d r T r e x p ( - r D t D ) (3) r

I / A 2

for D = i ~ - M U 5 and D = i ~ - M . It is easy to show

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Volume 236, number 3 PHYSICS LETTERS B 22 February 1990

that for A---,oe the above relation will become an identity. However. as the NJL model is non-renor- realizable a finite cutoff is necessary to obtain finite results ~

For the here considered time-independent chiral fields U(x) it is sufficient to minimize the energy E(U) related to the action by

S ( U ) = - f d tE(U) . (4)

As the total soliton energy E(U) can be expressed as the sum over the energies of the valence quarks and a vacuum part which can be calculated from the shifts of single particle energies when one switches on the chiral field from zero to its finite value, it is conve- nient to introduce the Dirac hamiltonians

ho = ocp + tiM, ( 5 )

h ~, = o~p + [3M Us. ( 6 )

Denoting the eigenvalues of these hamiltonians with e ° and e,, respectively, one obtains

E(U) =qEv~, +Eo, (7)

with

where e/is the energy of the lowest level in the posi- live part of the spectrum and

t l= l for el> 0,

=0 for el < 0. (10)

Note that the sum over k extends over all single par- ticle states. In deriving the expression for Eo the fact has been used that the integrals over the "proper time" r can be reexpressed for the case of the energy in terms of the complementary error function erfc (x). The field equation for U(x) is now obtained by min-

~ One might argue that in a very crude way the finite cutoff mimics the asymptotic freedom of QCD.

imization of the total soliton energy E(U). In the fol- lowing the hedgehog ansatz

U=exp l~-. 0(r) (11)

will be used. As there is no kinetic term for the me- sons the field equation for U is then simply a tran- scendental equation for the soliton profile 0(r) which reads for every given r

[~ls/(r) +So(r) ] sin(0(r) )

- [qp/(r) +Po(r) ] cos(0(r)) =0, (12)

where s~ and p~ are the following scalar and pseudo- scalar matrix elements expressed in the valence quark wave function ~4(x):

s/(r) = j- d.Q ~/(r, s~)~'/(r, .Q), (13)

f . TA" pl(r) = d~ gTl(r, ~ ) n75 T ~'l(r, ~) . (14)

The vacuum contributions So and Po can be written as a sum over all single particle matrix elements

P o ( r ) = - ½ ~ p,(r) erfc sign(eD, (16)

where s, and p, are the analogous matrix elements to (13) and (14) for arbitrary single particle states in the presence of a non-vanishing soliton profile 0(r).

Eqs. (12) - (16) are also obtained by first mini- mizing the soliton energy for arbitrary chiral fields U restricting afterwards to the hedgehog configuration. This shows that the hedgehog is a true solution to the exact field equations. For the hedgehog the hamilto- nian (6) commutes with the so-called grand spin K = 3 + ½~ and with the parity operator. Therefore the eigenfunctions ~,¢ can be classified by their grand spin K,/£3 and their parity. These eigenfunctions are ob- tained numerically by diagonalizing hc in the basis formed by the eigenfunctions of ho with good grand spin and parity. Following ref. [7] the spectrum is discretized by putting the system in a spherical box and imposing appropriate boundary conditions. The solution for the soliton profile is then found iteratively.

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Volume 236, number 3 PHYSICS LETTERS B 22 February 1990

As there are only two scales in the problem, the const i tuent mass M and the cutoffA, the solution will only depend on the ratio of these two quanti t ies. In order to relate these two quanti t ies to physical units the pion decay constant will be fixed to its experi- mental value,f= = 93 MeV.

For a finite basis j~ will depend on the posi t ion in space. For fixing the value of M one can ignore this finite size effect and use the expression of the contin- uum limit

/-2 Nc M2 F 0, (17) • ~ - - 47C2

where F(O, x) is the incomplete gamma function of order 0. The relation between the const i tuent mass and the cu to f f impl i ed b y r d = 93 MeV is shown in fig, 1. This curve looks qual i ta t ively s imilar in other re- gularization schemes like sharp euclidean invariant cutoff [8,9] or Paul i -Vi l la rs regularization [10,9] . However, the upper branch is not existing if one uses a momentum-dependent consti tuent mass falling like an inverse power for large momen ta in order to make the expression forf~ UV convergent [9 ]. For describ- ing meson proper t ies const i tuent masses a round 300 MeV have been used. This corresponds to the lower branch in fig. 1. However, for const i tuent masses M~<325 MeV I was not able to find a non-tr ivial so- lution for the soliton profile 0 ( r ) . A very s imilar re- sult was also obta ined by other groups [ 3 - 5 ] .

The total soliton energy and its different contr ibu- t ions for some values of the const i tuent mass is given

G

1000

800

-~i ¸TTT~TT , ~-'i / / !

/ /~ - !

/

6 0 0 / / /

/ 400 (

200

i I

600 ?00 800 900 1000 ,,\ (\,l~ v)

Fig. l. The relation between the constituent mass M and the cut- offA forf~= 93 MeV.

in table 1. Also given is the axial coupling of the nu- cleon gA which is related to the long distance tail of the profile function. In the chiral limit, i.e. for zero pion mass,

d 0 ( r ) _ ~, ( 1 8 )

F -

where ,4 is a constant to be found numerically [ 11,4 ]. Of course, the numerical solution is not obeying this relation strictly due to finite size effects. However, for large boxes ~2 this kind of behaviour can be seen in a double logari thmic plot very clearly for quite a large range of t". The axial coupling is then calculated from the relation

8 "2 gA =37([ ~,.t. (19)

where A is extracted from the numerical solutions. Inspired by the theoretical knowledge about the

long-distance behaviour I made the following ap- proximat ion. As the full solution shows the behav- iour (18) very clearly in the range between 1 and 3 fm for a box size D = 5 fm I only updated points for r ~ ~D matching from there on a tail obeying the the- oretical behaviour of 0( r ) . For M = 5 0 0 MeV the so- liton mass obta ined from this approximat ion is 3 MeV larger than the one obtained from a full solu- tion. The contr ibut ion from the valence quark d ropped by approximate ly 8 MeV. The last number

~2 In the actual numerical calculations 1 did lake the box size to be 5 fm. This has e.g. to be contrasted by the scale where 0(r) has dropped to half its value at the origin which varies for the different solutions from 0.40 fm to 0.43 fro. I also checked that the soliton profile does not change by increasing D fur- ther. For example, for M=400 MeV the soliton mass de- creased by 0.2 per cent for increasing D from 5 to 6 fro.

Table 1 The total soliton mass E(U) and its contributions from valence quarks Ev,l and the vacuum polarization E0 for different values of the constituent mass for the fully self-consistent solution. Given is also the axial coupling of the nucleon gA as calculated from the large r behaviour of the solilon.

i~/" Eval Eo E ( U ) gA (MeV) (MeV) (MeV) (MeV)

350 674 536 1210 0.75 400 569 642 121 I 0.79 500 389 803 I 192 0.77

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Volume 236, number 3 PHYSICS LETTERS B 22 February 1990

might seem to be not so small, however, the soliton is extremely soft against redistributing the valence en- ergy to the vacuum energy and vice versa. So chiral angles which differ only by very small amounts can give quite different Eva~ and E0 but give always al- most equal total energies.

In an additional approximation I only updated four equidistant points between r = ~D and r = ~ D then jo in ing 0 at these points and 0(0)=z~ by a rational interpolation. A comparison between the fully up- dated solution and this approximation is shown in figs. 2 and 3 for M = 350 MeV and M = 500 MeV, re- spectively. The results for this approximation for an enlarged set of const i tuent masses is shown in table 2. From the given results one can conclude that this

Z"

3K, , 7 ~i ~ ! ! , , !

/\\

, ,{,

"~? 4

F \ , !

1 ~'~

o I . . . . . . . . L . . . . . 7 Fi--; -vq

0.0 0.5 1 0 1.5 8.0 r(fm)

Fig. 2. The fully self-consistent calculated soliton profile O(r) (full line) and the one with the approximation explained in the texl (dashed line) for M=350 MeV.

3 t ~

\ L

i

J

\

d

0.0 0.5 1.0 1.5 2.0 r(fm)

Fig. 3. Same as fig. 2 for M= 500 MeV.

Table 2 Same as table 1 for the approximate solution discussed in the text.

M Eva Eo E(U') gA (MeV) (MeV) (MeV) (MeV)

350 653 561 1214 0.79 400 570 644 1214 0.80 450 479 727 1206 0.78 500 392 804 1196 0.76 600 220 949 1169 0.73 700 58 1082 1140 0.67 800 -128 1110 1110 0.65

approximation is fairly accurate, especially, if one considers the fact that the computational time needed differs by something like a factor 30-50.

From table 2 one sees that the total soliton mass is quite stable for a wide range of consti tuent masses whereas the valence quark contr ibut ion to it de- creases with increasing consti tuent mass and even becomes negative for M > 720 MeV. The soliton is only tightly bound if one chooses a large consti tuent mass as input. However, for M = 800 MeV the con- stituent mass is larger than the cutoff as can be seen from fig. 1. This obviously can be seen as an intrinsic problem of the non-renormalizable NJL model. We see also that gh does not vary much for the consid- ered consti tuent masses and is 35-45 percent smaller than its experimental value 1.23.

In conclusion, fixingf~ = 93 MeV and using proper t ime regularization, soliton solutions for the NJL

model were found numerically for constituent masses M>~ 350 MeV. No non-trivial solutions seem to exist for M~< 325 MeV. Moreover, up to M = 4 0 0 MeV the

soliton is unstable against decay into three free con- stituent quarks which is an allowed but non-physical process in the non-confining NJL model. The axial coupling of the nucleon gA comes out somewhat too low for all consti tuent masses used in the present cal- culation. The much tighter binding of the nucleon would make the use of large consti tuent masses pref- erable. That the properties of the pseudoscalar nonet can be fitted within a NJL model also with very large values of constituent masses is shown explicitly in ref. [ 12 ]. On the other hand, the use of a cutoff smaller than the consti tuent mass makes the underlying NJL model probably hard to accept as a sound model. In order to overcome this shortcomings one should look

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Volume 236, number 3 PHYSICS LETTERS B 22 February 1990

for soli tons in models with a running cons t i tuent mass

which may then even inc lude c o n f i n e m e n t in a phe-

nomeno log ica l way. Work in this d i rec t ion is in

progress.

I thank Micha! Prasza lowicz , H u g o Re inha rd t ,

Georges Ripka and Ismai l Zahed for n u m e r o u s help-

ful discussions. Par t o f this work has been car r ied out

dur ing a stay at the State U n i v e r s i t y o f N e w York at

Stony Brook, and I t hank Ger ry Brown and all m e m -

bers o f the Nuc lea r T h e o r y G r o u p for the i r hospi ta l -

ity. I am also grateful to Georges R ipka for encour -

aging me to wri te this letter, for a careful read ing o f

the manusc r ip t and helpful suggestions.

References

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see also R. Alkofer, M.A. Nowak, J.J.M. Verbaarschot and I. Zahed, Phys. Left. B 233 (1989) 205.

[2] A. Chodos and C.B. Thorn, Phys. Rev. D 12 (1975) 2733; M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Len. 51 (1983) 747.

[ 3 ] H. Reinhardt and R. Wfinsch, Phys. Lett. B 215 (1988) 577; Phys. Lett. B 230 (1989) 93.

[4] D.1. Diakonov, V.Yu. Petrov and P.V. Pobylitsa, Nucl. Phys. B 306 (1988) 809; M. Praszatowicz, Solitons in chiral quark model, BNL preprint (September 1989 ).

[5] Th. Meissner, F. Grfimmer and K. Goeke, Phys. Lett. B 227 (1989) 296; Bochum preprint (April 1989).

[6] J.S. Schwinger, Phys. Rev. 82 (1951) 664. [7] S. Kahana and G. Ripka, Nucl. Phys. A 429 (1984) 462. [ 8 ] A.H. Blin, B. Hiller and M. Schaden, Z. Phys. A 331 ( 1988 )

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[ 11 ] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552.

[12]R. Alkofer and I. Zahed, The pseudoscalar nonet for generalized NJL models, Stony Brook preprint ( 1989 ).

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