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Solution of the relativistic three quark Faddeev equation in the Nambu-Jona-Lasinio (NJL) model
Physics Letters B 318 (1993) 26-31 North-HoUand
PHYSICS LETTERS B
Solution of the relativistic three quark Faddeev equation in the Nambu-Jona-Lasinio (NJL) model Noriyoshi Ishii, Wolfgang Bentz and Koichi Yazaki Department of Physics, Universityof Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 24 June 1993; revised manuscript received 28 September 1993 Editor: C. Mahaux The relativistic Faddeev equation for quarks is solved in the NJL model including the axial-vector diquark channel together with the scalar one. It is found that the axial-vector diquark channel contributes significantly to the energy of the nucleon state. We show the region of the effective coupling constants in the scalar and axial-vector diquark channels which give about the right nucleon mass. In particular, the color current type interaction Lagrangian gives a reasonable nucleon mass.
1. Introduction Q u a n t u m chromodynamics ( Q C D ) is believed to describe the phenomena of the strong interaction in the low energy as well as the perturbative region. However, it is hard to solve Q C D in the low energy region and therefore some effective theories are needed. The N J L model [1] as a model for the effective q u a r k - q u a r k interaction is one o f them and has been used extensively to study the meson properties [2]. In recent years, it has also been used to investigate baryons [ 3-5 ]. The most accurate approach is to solve directly the relativistic F a d d e e v equation for three quarks [6]. Recently the F a d d e e v equation truncated to the scalar diquark channel has been solved numerically in the NJL model for the nucleon state [7]. (See also ref. [8] for a calculation based on a m o m e n t u m dependent interaction.) In this case the angular m o m e n t u m projection is very simple. In order to study the problem more quantitatively, we solve in this paper the relativistic F a d d e e v equation for NJL-type q u a r k - q u a r k interactions including also the axial-vector diquark channel. The angular m o m e n t u m analysis is carried out fully relativistically using the helicity formalism [9]. Instead o f sticking to a particular form of the 4-fermion interaction, we treat the effective coupling constants in the scalar diquark channel (gs) and the axialvector diquark channel (ga) as parameters in order to give a guide for constructing a Lagrangian of the NJL-type with which both mesons and baryons can be reasonably described.
2. Models for the quark-quark interaction We take the S U ( 2 ) f × SU(3)c effective quark Lagrangian o f the form £ = ~ ( i ~ - mo)g + El, where m0 is the current quark mass o f u, d quarks, and L1 is a chiral invariant 4-fermion interaction o f the NJL-type. If we. solve the F a d d e e v equation truncated to the scalar and axial-vector diquark channels, the results depend on the actual form o f El through the ratios gs rl = ~ ,
g,~
26
ga r2 = - - ,
gs
(1) 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 318, number 1
PHYSICS LETTERS B
25 November 1993
where g~ is the strength of the interaction in the pionic ( 0 - ) q~-channel and gs and ga are the ones in the scalar (0 + ) and axial-vector (1 + ) qq-channel, respectively. Their definitions are as follows: To any given El we can apply the Fierz identities to extract the relevant interaction parts as /::l,. = -½g~(~Ysv~') 2,
(2)
3
f-.t,s = -gs ~ [~( (ysC)r2flc)~ r] [~ur((C-175)r2flc) r~/],
(3)
C=I 3
l:l,a -~ -- ga ~
[~((~'/tC)
( ' r r 2 ) f l c ) ~ "T ] [ ~11T( ( C - l~,,u) (.c2,l.)flc) Tffj ],
(4)
c=l
where C = i~,2y° is the charge conjugation matrix, (fl~)ij = i~/~e~ij projects onto the color 3 channel with the normalization tr (flcflc,) = 25~c,, and {zi} are the flavour SU(2) generators with tr (zizj) = 2~ii. g~, g~, ga are related to the coupling constants appearing in the original £z. For example, for the original NJL model [ 1 ] we obtain for the ratios (1) rl = ~ , r2 = ½, while for the color current interaction type Lagrangian /~I ~--- - - g ( ~ ) / ~ " ½'~c~//)2 we get r~ = ½, r2 = ½. (,;tc, C = 1... 8, are the color Gell-Mann matrices.) Due to the chiral symmetry of Z:t, there is also an interaction term ½g~ ( ~ u ) 2 in the 0 + q~ channel and a term of the form (4) with 7u --* 7~75 in the 1- qq channel. The former one leads to the gap equation for the constituent quark mass M which is solved as in ref. [7]. The 1- qq channel is neglected here since its major component is an l = 1 state. The qq Bethe-Salpeter equation in the ladder approximation is solved as explained in ref. [7]. The T-matrix in the axial-vector diquark channel has the form
(5)
Ta(q) = [ ( 7 # C ) ( r f T 2 ) f l c ] t a ( q ) l Z u [ ( c - l y u ) ( T ; 2 T f ) f l c ] , where
ta(q) ~ = 2" 2iga gU~ + 1 -- 2 - ~ ' a H - ~"2) g~ Here
q~qV q2
(6)
Ha represents the "bubble graph",
Ha(q2) (g ~ = 6~c,c~f,f
q~q~ ) t~CC't~fft= / ~ dgk tr{[(TuC)(ZfZE)flc]iSlr(-k)T[(c-17v)(z2zf ')flc']iSF(k + q)} q2 f
d4k (--~--~)4tr[y~'iSe(k)7"iSF(k + q)],
(7)
where St(k) = (~ - M + ie) -1 is the quark Feynman propagator. The counter part of eq. (5) for the scalar diquark channel involves the quantities Ts and t,, which are given in ref. [7 ]. The diquark masses are obtained as the poles of ts (q) and ta (q). In contrast to the scalar diquark, the axial vector diquark is unbound except for abnormally large values of ga. A drawback of the present quark-quark interaction models is that they do not incorporate the confinement. It is possible to mimic confinement by including momentum dependence in the quark-quark interaction [ 10], thereby avoiding the existence of isolated colored objects like quark and diquarks. The momentum dependence of the Bethe-Salpeter and Faddeev amplitudes would also be influenced by these momentum dependent interactions. To incorporate confinement into a full Faddeev calculation is a difficult but important subject for future studies. 27
Volume 318, number 1
PHYSICS LETTERS B
25 November 1993
3. Angular momentum projection of the relativistic Faddeev equation The relativistic Faddeev equation [ 11 ] is written in operator form as
XE = Z + KEXe,
(8)
or explicitly,
X ~ b ( p , l ) = Z a b ( p , l ) + Z f ~ d4k zac ' (p,k)t c'c ( ½ q - k ) i S F ( ½ q + k ) X ~ b ( k , l ) , cc,
(9)
where X, Z are matrices in Dirac, flavour, and color space, and a, b, c, c' denote the diquark indices. We work in the center of mass frame q = (E, 0), where the total energy E should be considered as an external parameter, t c'c is the diquark t-matrix with its vertex functions stripped off (see eq. (5)). Z corresponds to the quark exchange diagram and is a product of vertex functions and the propagator of the exchanged quark [ 7]. Thus, intuitively, eq. (9) describes the scattering of a quark and a diquark. In order to describe the nucleon, we first have to project Z onto the color singlet, isospin ½ space. The color singlet projection is performed as in ref. [7], and the isospin ½ projection leads to the familiar isospin recoupling coefficient [ 11 ]. Restricting from now to the scalar and axial vector diquark channels, we thus get for the color singlet, isospin ½ part of Z:
(C-175)iSr(-P-l)r(75C) x/~(C_175)iSe(_p_l)r(Tu, C)
Za,a(p,l) = 3
x/3(C-17u)iSF(-P-l)r(75C)) _ ( C _ t 7 u ) i S F ( _ p _ l ) r ( 7 u , C)
(10)
,
while t d = diag(ts, ta~" ). For fixed momenta, Z (and therefore also KE of eq. (8)) is a matrix in Dirac spinor space and diquark channel space with the dimension (4 × 5) 2. The projection of the kernel to a total spin J is accomplished by representing it in the following basis [9]: Ipo,-ff;J,M;~.q,,~.d)
= .Afj
f
~(J) , do9L)2q+2d,M[O)
-1 ) R ( o g ) l p O ,
(11)
p;,~q,~.d).
Here D (J) is the Wigner D-function, o9 denotes the Euler angles (~b,0, g/)with do9 = s i n 0 d 0 d C d ~ , and the state on the RHS corresponds to a quark and a diquark moving along F and - F ~ = (0,0,if) with ff = [Pl) with helicities 2q and 2d, respectively. R(og) is the rotational operator which rotates this state by the angles 09. The spinor part of the state IPo,'P;2q,2d) can be chosen as a simple product state X2qe,ld with the 4 basis vectors g~q = (1,0, 0, 0) . . . . (0, 0, 0, 1 ) for the quark and the 5 basis vectors 82d = ( l , 0, 0), (0, 1,0), (0, 0, e,~d) for the diquark. Here the first component corresponds to the scalar diquark, and the others to the axial zvector diquark. (The e2d are the usual spherical unit vectors.) This simplest choice of the basis leads to the following correspondence between Dirac (diquark channel) indices a (a) and quark (diquark) helicities 2~~) (2tda) ): a = 1,2, 3,4 c°rresp°nds t°2qt~) - 2,t 2,1 2,1 ~, and a = 5,0, +, 3 , - corresponds to 2~da) = 0,0, 1,0,--1. (Here a = 5 refers to the scalar diquark, a = 0 to the time component and a = +, 3, - to the spacial spherical components of the axial vector diquark. ) Following this procedure, we obtain the following spin projected kernel: K~(sJ') .... P ) = ~1 f ,q,~'a.~q~d;~tM,(Po,P;Po,
('(q) I ~-l~(,(d)
× ~,aq,~~,
1. - l ~ r e . a a ~
dogdog'V/(2J + 1 ) ( 2 J ' + 1,~M,2q+2 d (09) ~tJ) t
t
(q)
i
(d)
t
(J~)
~'ad,a~O Sa~, (Po,P;Po, P ) S , ; ( o 9 )Sa,~,d(o9 )D~+a~,M,(o9
t-I
).
(12)
Here S.q and Sd are rotational matrices for the quark and the diquark, respectively. They have the form Sq = diag(So,Sq) where ~ is the usual rotation matrix for a Pauli spinor, and Sd = diag(1, 1 , ~ ) where Sd is the usual rotation matrix for a 3-dimensional vector. Eq. (1 2) reduces to 28
Volume 318, number 1
,~JJ,,~Mu,
(J)
dto~a~+~,~q+~ (oJ
PHYSICS LETTERS B -1
(q)
(d)
aa ~
t ~l
)S~q.~,(oJ)S~,~,,,(o~)K~o, (P0,P;P0,t'),
25 November 1993 (13)
where ~' is along the z-axis and the correspondence between helicity and matrix indices is as explained above. We remark that this is still a 20 × 20 matrix. (In the case of J = ½, the 2q + 2d = +3 components do not contribute and we get a 16 × 16 matrix. ) Using the transformation of the states (11 ) under parity [12] one can construct an unitary matrix (with respect to the helicities) which transforms the spin projected kernel into a form composed of two separate blocks corresponding to the positive and negative parities [ 13 ]. In this way one can reduce the dimension of the kernel to 1 0 x 10 ( 8 x 8 i f J = ½). We directly diagonalize the positive parity part of the kernel (13) for J = ½ after discretizing it in polar coordinates to obtain its eigenvalues 2 ( E ) . Due to eq. (8), the nucleon mass is then obtained from 2 ( m s ) = 1. To test the convergence, the dimension of the kernel (including the momentum dependence) was increased up to (8 × 100) 2. The regularization scheme has been explained already in ref. [7]: We use a sharp Euclidean cutoff after Wick rotation for the calculation of the diquark t-matrices as well as for the loop integral in the Faddeev equation. Several checks of the code and the formulation have been carried out: In the static approximation [ 5 ], where the mass of the exchanged quark is treated as infinitely heavy, the diagonalization can be carried out almost analytically. This was done including the axial vector diquark channel both with and without using the helicity formalism. These results agreed with those of the exact calculation in the limit of infinitely heavy mass of the exchanged quark. Further, in the case where the axial vector diquark is neglected, the previous results [ 7] obtained without using the helicity formalism have been reproduced.
4. Results Even if we fix the values of rl and r2, our model still has 4 parameters: the coupling constant in the pionic channel g~, the current quark mass m0, the cutoff A, and the constituent quark mass M, on which we impose the following 3 conditions: (i) the pion mass mR = 140 MeV and (ii) the pion decay constant f~ = 93 MeV are reproduced, and (iii) the gap equation is satisfied. Furthermore in this paper we confine ourselves to a fixed value of the constituent quark mass, i.e. M = 400 MeV. Therefore there are no free parameters except for rl and r2 which reflect different forms of the interaction Lagrangian. In ref. [7] the Faddeev equation was solved for r2 = 0, and for the case M = 400 MeV, it was found that rl must exceed 0.5 to form a three-body bound state. Reasonable nucleon masses were obtained with rl "" 2. In order to see the effect of axial-vector diquark channels, we show the eigenvalue curves 2 (E) varying r2 from 0 to 1 with rl fixed (rl = 1 ) by the solid lines in fig. 1. The vertical dotted line shows the threshold M + ms, where ms is the mass of the scalar diquark, and the nucleon mass for each case is obtained from 2 ( m s ) = 1. One sees that for ga > 0 the axial-vector channels contribute attractively, and the actual value of the nucleon mass is quite sensitive to ga. Thus the restriction r~ > 0.5 obtained in the calculation neglecting the axial vector diquark gets relaxed considerably. In particular, for the color current interaction type Lagrangian we have rl = ½, r2 = ½, and the nucleon mass is calculated to be 900 MeV.For the original NJL Lagrangian (rl = 2 , r2 = ½) there exists no bound state since these ratios are too small. In order to show the region of the ratios rl, r2 which give a reasonable nucleon mass (0.8 GeV < rnN < 1.1 GeV), we show in fig. 2 those parameters which lead to m s = 0.8 GeV and 1.1 GeV, and also those values which lead to the experimental value m s = 0.94 GeV. If one chooses rl < 0.5 one needs the attraction from the axial vector diquark channel in order to stay within this range of ms. We finally compare our exact results with the results of the static approximation [5 ], where the mass of the exchanged quark is treated as infinitely heavy and the Faddeev equation thus becomes separable. The comparison is shown in fig. 1 where the dashed lines refer to the static approximation. We see that for small r2 the static approximation gives a good estimate for the nucleon mass, but for larger r2 one obtains too much attraction. The 29
Volume 318, number 1 . . . .
I
. . . .
PHYSICS LETTERS B I
. . . .
I
. . . .
I?'/'t
~-,. •.q.
1.5
'
/
/ // i /
¢q, /
/
1.0
'
/
/
/
/ "/'£'?'~ ~r ~"
-
/ O.B
/
,,~" ¢~
25 November 1993
/ / 0.6
g N
0.4 0.5
-0,2
o.o
....
0.2
I .... 0.4
I .... 0.6 Total energy
I .... 0.8 E [GeV]
I,,,, 1
1.2
Fig. 1. The eigenvalues of the kernel 2(E) for r 1 = ½ and several values of r2 (r2 = 0, 0.25, 0.5, 0.75, 1.0) are shown. The solid lines are the exact results, and the dashed lines refer to the static approximation. The vertical dotted line indicates the threshold beyond which no three body bound states exist. The energy coordinates of the intersections of the curves with the horizontal dotted line are the respective nucleon masses.
0.0 0.3
0.4
0.5
0,6
0.7
0.8
rl
Fig. 2. The three lines connect those values of r I and r2 which give a nucleon mass of 800, 940 and 1100 MeV.
same holds with respect to rl, i.e., the static approximation becomes worse as rl is increased. For the case of the color current interaction type Lagrangian (rl = ½, r2 = ½), however, the static approximation works reasonably well.
5. Summary and conclusion We have solved the relativistic Faddeev equation for the nucleon state in the NJL-model truncating the q u a r k quark interactions to the scalar and axial vector diquark channels. After the projections onto isospin ½, color singlet, J~ = ]1+ were performed, we had to solve 8 coupled 2-dimensional integral equations. We treated the 2 ratios rl and r2 of eq. ( 1 ), characterizing the interactions in the scalar and axial vector channels, as parameters, restricting 0 < r2 < 1. We found that the axial-vector diquark channels contribute attractively, although the m a i n part of the attraction comes from the scalar diquark channel. In particular, if we take the color current interaction type Lagrangian and use M = 400 MeV for the constituent quark mass, the nucleon mass became 900 MeV. We also investigated the regions of rl and r2 which give a reasonable nucleon mass. For r2 < 1, rl must lie between 0.3 and 0.78, where smaller values of rt correspond to larger values of r2 and vice versa. We compared the exact results with the static approximation in order to evaluate the validity of the static approximation, and concluded that the static approximation works reasonably well for smaller coupling constants but for larger ones it tends to overestimate the attraction.
References [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1960) 345; 124 (1961) 246. [2] R. Brockmann, W. Weise and E. Werner, Phys. Lett. B 122 (1983) 201; D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188; 30
Volume 318, number 1
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25 November 1993
T. Kunihiro and T. Hatsuda, Phys. Lett. B 185 (1987) 304; T. Hatsuda and T. Kunihiro, Z. Phys. C 51 (1991 ) 49. [3] M. Kato, W. Bentz, K. Yazaki and K. Tanaka, Nucl. Phys. A 551 (1993) 541; P. Sieber, Th. Meissner, F. GriJmmer and K. Goeke, Nucl. Phys. A 547 (1992) 459. [4] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195; K. Suzuki and H. Toki, Mod. Phys. Lett. A 7 (1992) 2867. [5] A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B 286 (1992) 29. [6] R.T. Cahill, C.D. Roberts and J. Praschifka, Aust. J. Phys. 42 (1989) 129; R.T. Cahill, Nucl. Phys. A 543 (1992) 630. [7] N. Ishii, W. Bentz and K. Yazaki, Phys. Lett. B 301 (1993) 165. [8] C.J. Burden, R.T. Cahill and J. Praschifka, Aust. J. Phys. 42 (1989) 147. [9] M. Jacob and G.C. Wick, Ann. Phys. 7 (1959) 404. [10] A. H~idicke, Int. J. Mod. Phys. A 6 (1991) 3321; A.A. Bel'kov, D. Ebert and A.V. Emelyanenko, Nucl. Phys. A 552 (1993) 523; C.D. Roberts, Proc. Workshop on QCD vacuum structure, eds. H.M. Fried and B. Mfiller (World Scientific, Singapore, 1993.) [11] G. Rupp and J.A. Tjon, Phys. Rev. C 37 (1988) 1729; C 45 (1992) 2133. [12] C. Itzykson and J.-B. Zuber, Quantum field theory (McGraw-Hill, New York, 1985) p. 244. [13] N. Ishii, Faddeev approach to the nucleon in the Nambu-Jona-Lasinio (NJL) model, Master thesis, Univ. of Tokyo, 1992; N. Ishii, W. Bentz, K. Yazaki, to be published.
31
PHYSICS LETTERS B
Solution of the relativistic three quark Faddeev equation in the Nambu-Jona-Lasinio (NJL) model Noriyoshi Ishii, Wolfgang Bentz and Koichi Yazaki Department of Physics, Universityof Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 24 June 1993; revised manuscript received 28 September 1993 Editor: C. Mahaux The relativistic Faddeev equation for quarks is solved in the NJL model including the axial-vector diquark channel together with the scalar one. It is found that the axial-vector diquark channel contributes significantly to the energy of the nucleon state. We show the region of the effective coupling constants in the scalar and axial-vector diquark channels which give about the right nucleon mass. In particular, the color current type interaction Lagrangian gives a reasonable nucleon mass.
1. Introduction Q u a n t u m chromodynamics ( Q C D ) is believed to describe the phenomena of the strong interaction in the low energy as well as the perturbative region. However, it is hard to solve Q C D in the low energy region and therefore some effective theories are needed. The N J L model [1] as a model for the effective q u a r k - q u a r k interaction is one o f them and has been used extensively to study the meson properties [2]. In recent years, it has also been used to investigate baryons [ 3-5 ]. The most accurate approach is to solve directly the relativistic F a d d e e v equation for three quarks [6]. Recently the F a d d e e v equation truncated to the scalar diquark channel has been solved numerically in the NJL model for the nucleon state [7]. (See also ref. [8] for a calculation based on a m o m e n t u m dependent interaction.) In this case the angular m o m e n t u m projection is very simple. In order to study the problem more quantitatively, we solve in this paper the relativistic F a d d e e v equation for NJL-type q u a r k - q u a r k interactions including also the axial-vector diquark channel. The angular m o m e n t u m analysis is carried out fully relativistically using the helicity formalism [9]. Instead o f sticking to a particular form of the 4-fermion interaction, we treat the effective coupling constants in the scalar diquark channel (gs) and the axialvector diquark channel (ga) as parameters in order to give a guide for constructing a Lagrangian of the NJL-type with which both mesons and baryons can be reasonably described.
2. Models for the quark-quark interaction We take the S U ( 2 ) f × SU(3)c effective quark Lagrangian o f the form £ = ~ ( i ~ - mo)g + El, where m0 is the current quark mass o f u, d quarks, and L1 is a chiral invariant 4-fermion interaction o f the NJL-type. If we. solve the F a d d e e v equation truncated to the scalar and axial-vector diquark channels, the results depend on the actual form o f El through the ratios gs rl = ~ ,
g,~
26
ga r2 = - - ,
gs
(1) 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 318, number 1
PHYSICS LETTERS B
25 November 1993
where g~ is the strength of the interaction in the pionic ( 0 - ) q~-channel and gs and ga are the ones in the scalar (0 + ) and axial-vector (1 + ) qq-channel, respectively. Their definitions are as follows: To any given El we can apply the Fierz identities to extract the relevant interaction parts as /::l,. = -½g~(~Ysv~') 2,
(2)
3
f-.t,s = -gs ~ [~( (ysC)r2flc)~ r] [~ur((C-175)r2flc) r~/],
(3)
C=I 3
l:l,a -~ -- ga ~
[~((~'/tC)
( ' r r 2 ) f l c ) ~ "T ] [ ~11T( ( C - l~,,u) (.c2,l.)flc) Tffj ],
(4)
c=l
where C = i~,2y° is the charge conjugation matrix, (fl~)ij = i~/~e~ij projects onto the color 3 channel with the normalization tr (flcflc,) = 25~c,, and {zi} are the flavour SU(2) generators with tr (zizj) = 2~ii. g~, g~, ga are related to the coupling constants appearing in the original £z. For example, for the original NJL model [ 1 ] we obtain for the ratios (1) rl = ~ , r2 = ½, while for the color current interaction type Lagrangian /~I ~--- - - g ( ~ ) / ~ " ½'~c~//)2 we get r~ = ½, r2 = ½. (,;tc, C = 1... 8, are the color Gell-Mann matrices.) Due to the chiral symmetry of Z:t, there is also an interaction term ½g~ ( ~ u ) 2 in the 0 + q~ channel and a term of the form (4) with 7u --* 7~75 in the 1- qq channel. The former one leads to the gap equation for the constituent quark mass M which is solved as in ref. [7]. The 1- qq channel is neglected here since its major component is an l = 1 state. The qq Bethe-Salpeter equation in the ladder approximation is solved as explained in ref. [7]. The T-matrix in the axial-vector diquark channel has the form
(5)
Ta(q) = [ ( 7 # C ) ( r f T 2 ) f l c ] t a ( q ) l Z u [ ( c - l y u ) ( T ; 2 T f ) f l c ] , where
ta(q) ~ = 2" 2iga gU~ + 1 -- 2 - ~ ' a H - ~"2) g~ Here
q~qV q2
(6)
Ha represents the "bubble graph",
Ha(q2) (g ~ = 6~c,c~f,f
q~q~ ) t~CC't~fft= / ~ dgk tr{[(TuC)(ZfZE)flc]iSlr(-k)T[(c-17v)(z2zf ')flc']iSF(k + q)} q2 f
d4k (--~--~)4tr[y~'iSe(k)7"iSF(k + q)],
(7)
where St(k) = (~ - M + ie) -1 is the quark Feynman propagator. The counter part of eq. (5) for the scalar diquark channel involves the quantities Ts and t,, which are given in ref. [7 ]. The diquark masses are obtained as the poles of ts (q) and ta (q). In contrast to the scalar diquark, the axial vector diquark is unbound except for abnormally large values of ga. A drawback of the present quark-quark interaction models is that they do not incorporate the confinement. It is possible to mimic confinement by including momentum dependence in the quark-quark interaction [ 10], thereby avoiding the existence of isolated colored objects like quark and diquarks. The momentum dependence of the Bethe-Salpeter and Faddeev amplitudes would also be influenced by these momentum dependent interactions. To incorporate confinement into a full Faddeev calculation is a difficult but important subject for future studies. 27
Volume 318, number 1
PHYSICS LETTERS B
25 November 1993
3. Angular momentum projection of the relativistic Faddeev equation The relativistic Faddeev equation [ 11 ] is written in operator form as
XE = Z + KEXe,
(8)
or explicitly,
X ~ b ( p , l ) = Z a b ( p , l ) + Z f ~ d4k zac ' (p,k)t c'c ( ½ q - k ) i S F ( ½ q + k ) X ~ b ( k , l ) , cc,
(9)
where X, Z are matrices in Dirac, flavour, and color space, and a, b, c, c' denote the diquark indices. We work in the center of mass frame q = (E, 0), where the total energy E should be considered as an external parameter, t c'c is the diquark t-matrix with its vertex functions stripped off (see eq. (5)). Z corresponds to the quark exchange diagram and is a product of vertex functions and the propagator of the exchanged quark [ 7]. Thus, intuitively, eq. (9) describes the scattering of a quark and a diquark. In order to describe the nucleon, we first have to project Z onto the color singlet, isospin ½ space. The color singlet projection is performed as in ref. [7], and the isospin ½ projection leads to the familiar isospin recoupling coefficient [ 11 ]. Restricting from now to the scalar and axial vector diquark channels, we thus get for the color singlet, isospin ½ part of Z:
(C-175)iSr(-P-l)r(75C) x/~(C_175)iSe(_p_l)r(Tu, C)
Za,a(p,l) = 3
x/3(C-17u)iSF(-P-l)r(75C)) _ ( C _ t 7 u ) i S F ( _ p _ l ) r ( 7 u , C)
(10)
,
while t d = diag(ts, ta~" ). For fixed momenta, Z (and therefore also KE of eq. (8)) is a matrix in Dirac spinor space and diquark channel space with the dimension (4 × 5) 2. The projection of the kernel to a total spin J is accomplished by representing it in the following basis [9]: Ipo,-ff;J,M;~.q,,~.d)
= .Afj
f
~(J) , do9L)2q+2d,M[O)
-1 ) R ( o g ) l p O ,
(11)
p;,~q,~.d).
Here D (J) is the Wigner D-function, o9 denotes the Euler angles (~b,0, g/)with do9 = s i n 0 d 0 d C d ~ , and the state on the RHS corresponds to a quark and a diquark moving along F and - F ~ = (0,0,if) with ff = [Pl) with helicities 2q and 2d, respectively. R(og) is the rotational operator which rotates this state by the angles 09. The spinor part of the state IPo,'P;2q,2d) can be chosen as a simple product state X2qe,ld with the 4 basis vectors g~q = (1,0, 0, 0) . . . . (0, 0, 0, 1 ) for the quark and the 5 basis vectors 82d = ( l , 0, 0), (0, 1,0), (0, 0, e,~d) for the diquark. Here the first component corresponds to the scalar diquark, and the others to the axial zvector diquark. (The e2d are the usual spherical unit vectors.) This simplest choice of the basis leads to the following correspondence between Dirac (diquark channel) indices a (a) and quark (diquark) helicities 2~~) (2tda) ): a = 1,2, 3,4 c°rresp°nds t°2qt~) - 2,t 2,1 2,1 ~, and a = 5,0, +, 3 , - corresponds to 2~da) = 0,0, 1,0,--1. (Here a = 5 refers to the scalar diquark, a = 0 to the time component and a = +, 3, - to the spacial spherical components of the axial vector diquark. ) Following this procedure, we obtain the following spin projected kernel: K~(sJ') .... P ) = ~1 f ,q,~'a.~q~d;~tM,(Po,P;Po,
('(q) I ~-l~(,(d)
× ~,aq,~~,
1. - l ~ r e . a a ~
dogdog'V/(2J + 1 ) ( 2 J ' + 1,~M,2q+2 d (09) ~tJ) t
t
(q)
i
(d)
t
(J~)
~'ad,a~O Sa~, (Po,P;Po, P ) S , ; ( o 9 )Sa,~,d(o9 )D~+a~,M,(o9
t-I
).
(12)
Here S.q and Sd are rotational matrices for the quark and the diquark, respectively. They have the form Sq = diag(So,Sq) where ~ is the usual rotation matrix for a Pauli spinor, and Sd = diag(1, 1 , ~ ) where Sd is the usual rotation matrix for a 3-dimensional vector. Eq. (1 2) reduces to 28
Volume 318, number 1
,~JJ,,~Mu,
(J)
dto~a~+~,~q+~ (oJ
PHYSICS LETTERS B -1
(q)
(d)
aa ~
t ~l
)S~q.~,(oJ)S~,~,,,(o~)K~o, (P0,P;P0,t'),
25 November 1993 (13)
where ~' is along the z-axis and the correspondence between helicity and matrix indices is as explained above. We remark that this is still a 20 × 20 matrix. (In the case of J = ½, the 2q + 2d = +3 components do not contribute and we get a 16 × 16 matrix. ) Using the transformation of the states (11 ) under parity [12] one can construct an unitary matrix (with respect to the helicities) which transforms the spin projected kernel into a form composed of two separate blocks corresponding to the positive and negative parities [ 13 ]. In this way one can reduce the dimension of the kernel to 1 0 x 10 ( 8 x 8 i f J = ½). We directly diagonalize the positive parity part of the kernel (13) for J = ½ after discretizing it in polar coordinates to obtain its eigenvalues 2 ( E ) . Due to eq. (8), the nucleon mass is then obtained from 2 ( m s ) = 1. To test the convergence, the dimension of the kernel (including the momentum dependence) was increased up to (8 × 100) 2. The regularization scheme has been explained already in ref. [7]: We use a sharp Euclidean cutoff after Wick rotation for the calculation of the diquark t-matrices as well as for the loop integral in the Faddeev equation. Several checks of the code and the formulation have been carried out: In the static approximation [ 5 ], where the mass of the exchanged quark is treated as infinitely heavy, the diagonalization can be carried out almost analytically. This was done including the axial vector diquark channel both with and without using the helicity formalism. These results agreed with those of the exact calculation in the limit of infinitely heavy mass of the exchanged quark. Further, in the case where the axial vector diquark is neglected, the previous results [ 7] obtained without using the helicity formalism have been reproduced.
4. Results Even if we fix the values of rl and r2, our model still has 4 parameters: the coupling constant in the pionic channel g~, the current quark mass m0, the cutoff A, and the constituent quark mass M, on which we impose the following 3 conditions: (i) the pion mass mR = 140 MeV and (ii) the pion decay constant f~ = 93 MeV are reproduced, and (iii) the gap equation is satisfied. Furthermore in this paper we confine ourselves to a fixed value of the constituent quark mass, i.e. M = 400 MeV. Therefore there are no free parameters except for rl and r2 which reflect different forms of the interaction Lagrangian. In ref. [7] the Faddeev equation was solved for r2 = 0, and for the case M = 400 MeV, it was found that rl must exceed 0.5 to form a three-body bound state. Reasonable nucleon masses were obtained with rl "" 2. In order to see the effect of axial-vector diquark channels, we show the eigenvalue curves 2 (E) varying r2 from 0 to 1 with rl fixed (rl = 1 ) by the solid lines in fig. 1. The vertical dotted line shows the threshold M + ms, where ms is the mass of the scalar diquark, and the nucleon mass for each case is obtained from 2 ( m s ) = 1. One sees that for ga > 0 the axial-vector channels contribute attractively, and the actual value of the nucleon mass is quite sensitive to ga. Thus the restriction r~ > 0.5 obtained in the calculation neglecting the axial vector diquark gets relaxed considerably. In particular, for the color current interaction type Lagrangian we have rl = ½, r2 = ½, and the nucleon mass is calculated to be 900 MeV.For the original NJL Lagrangian (rl = 2 , r2 = ½) there exists no bound state since these ratios are too small. In order to show the region of the ratios rl, r2 which give a reasonable nucleon mass (0.8 GeV < rnN < 1.1 GeV), we show in fig. 2 those parameters which lead to m s = 0.8 GeV and 1.1 GeV, and also those values which lead to the experimental value m s = 0.94 GeV. If one chooses rl < 0.5 one needs the attraction from the axial vector diquark channel in order to stay within this range of ms. We finally compare our exact results with the results of the static approximation [5 ], where the mass of the exchanged quark is treated as infinitely heavy and the Faddeev equation thus becomes separable. The comparison is shown in fig. 1 where the dashed lines refer to the static approximation. We see that for small r2 the static approximation gives a good estimate for the nucleon mass, but for larger r2 one obtains too much attraction. The 29
Volume 318, number 1 . . . .
I
. . . .
PHYSICS LETTERS B I
. . . .
I
. . . .
I?'/'t
~-,. •.q.
1.5
'
/
/ // i /
¢q, /
/
1.0
'
/
/
/
/ "/'£'?'~ ~r ~"
-
/ O.B
/
,,~" ¢~
25 November 1993
/ / 0.6
g N
0.4 0.5
-0,2
o.o
....
0.2
I .... 0.4
I .... 0.6 Total energy
I .... 0.8 E [GeV]
I,,,, 1
1.2
Fig. 1. The eigenvalues of the kernel 2(E) for r 1 = ½ and several values of r2 (r2 = 0, 0.25, 0.5, 0.75, 1.0) are shown. The solid lines are the exact results, and the dashed lines refer to the static approximation. The vertical dotted line indicates the threshold beyond which no three body bound states exist. The energy coordinates of the intersections of the curves with the horizontal dotted line are the respective nucleon masses.
0.0 0.3
0.4
0.5
0,6
0.7
0.8
rl
Fig. 2. The three lines connect those values of r I and r2 which give a nucleon mass of 800, 940 and 1100 MeV.
same holds with respect to rl, i.e., the static approximation becomes worse as rl is increased. For the case of the color current interaction type Lagrangian (rl = ½, r2 = ½), however, the static approximation works reasonably well.
5. Summary and conclusion We have solved the relativistic Faddeev equation for the nucleon state in the NJL-model truncating the q u a r k quark interactions to the scalar and axial vector diquark channels. After the projections onto isospin ½, color singlet, J~ = ]1+ were performed, we had to solve 8 coupled 2-dimensional integral equations. We treated the 2 ratios rl and r2 of eq. ( 1 ), characterizing the interactions in the scalar and axial vector channels, as parameters, restricting 0 < r2 < 1. We found that the axial-vector diquark channels contribute attractively, although the m a i n part of the attraction comes from the scalar diquark channel. In particular, if we take the color current interaction type Lagrangian and use M = 400 MeV for the constituent quark mass, the nucleon mass became 900 MeV. We also investigated the regions of rl and r2 which give a reasonable nucleon mass. For r2 < 1, rl must lie between 0.3 and 0.78, where smaller values of rt correspond to larger values of r2 and vice versa. We compared the exact results with the static approximation in order to evaluate the validity of the static approximation, and concluded that the static approximation works reasonably well for smaller coupling constants but for larger ones it tends to overestimate the attraction.
References [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1960) 345; 124 (1961) 246. [2] R. Brockmann, W. Weise and E. Werner, Phys. Lett. B 122 (1983) 201; D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188; 30
Volume 318, number 1
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T. Kunihiro and T. Hatsuda, Phys. Lett. B 185 (1987) 304; T. Hatsuda and T. Kunihiro, Z. Phys. C 51 (1991 ) 49. [3] M. Kato, W. Bentz, K. Yazaki and K. Tanaka, Nucl. Phys. A 551 (1993) 541; P. Sieber, Th. Meissner, F. GriJmmer and K. Goeke, Nucl. Phys. A 547 (1992) 459. [4] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195; K. Suzuki and H. Toki, Mod. Phys. Lett. A 7 (1992) 2867. [5] A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B 286 (1992) 29. [6] R.T. Cahill, C.D. Roberts and J. Praschifka, Aust. J. Phys. 42 (1989) 129; R.T. Cahill, Nucl. Phys. A 543 (1992) 630. [7] N. Ishii, W. Bentz and K. Yazaki, Phys. Lett. B 301 (1993) 165. [8] C.J. Burden, R.T. Cahill and J. Praschifka, Aust. J. Phys. 42 (1989) 147. [9] M. Jacob and G.C. Wick, Ann. Phys. 7 (1959) 404. [10] A. H~idicke, Int. J. Mod. Phys. A 6 (1991) 3321; A.A. Bel'kov, D. Ebert and A.V. Emelyanenko, Nucl. Phys. A 552 (1993) 523; C.D. Roberts, Proc. Workshop on QCD vacuum structure, eds. H.M. Fried and B. Mfiller (World Scientific, Singapore, 1993.) [11] G. Rupp and J.A. Tjon, Phys. Rev. C 37 (1988) 1729; C 45 (1992) 2133. [12] C. Itzykson and J.-B. Zuber, Quantum field theory (McGraw-Hill, New York, 1985) p. 244. [13] N. Ishii, Faddeev approach to the nucleon in the Nambu-Jona-Lasinio (NJL) model, Master thesis, Univ. of Tokyo, 1992; N. Ishii, W. Bentz, K. Yazaki, to be published.
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