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Dynamical generation of the spectrum of fermions in non-abelian gauge field theories
DYNAMICAL
GENERATION
IN NON-ABELIAN V.A. MIRANSKY,
26 March 1981
PHYSICS LETTERS
Volume lOOB, number 2
OF THE SPECTRUM OF FERMIONS
GAUGE FIELD THEORIES V.P. GUSYNIN and Yu.A. SITENKO
Institute for Theoretical Physics, Kiev-130,
USSR
Received 1 September 1980
The dynamical mechanism of spontaneous breakdown of chiral invariance in nonabelian gauge theories is proposed. The dynamical masses of quarks in quantum chromodynamics are estimated within this framework.
In this paper the dynamical mechanism of spontaneous breakdown of chiral invariance, suggested in refs. [l-4] is used to investigate the fermion mass spectrum in non-abelian gauge field theories, in particular, the mass spectrum of quarks in massless quantum chromodynamics (QCD) (recently a similar mechanism was also considered by Mandelstam [S]). As will be shown, this mechanism supplemented with the conditions of asymptotic freedom and confinement explains naturally the main regularities of the dynamics of broken chiral symmetry of hadrons. We calculate the mass spectrum of quarks in QCD and propose a possible explanation of the existence of a finite number of quark generations. The physical phenomenon underlying this mechanism is closely connected with the phenomenon of the fall into the strong Coulomb centre V = -c~/r (cu = Ze2/4n > 1, Z > 137) which takes place when using the Dirac equation [6]. As is known, the vacuum becomes unstable with respect to the e+e--pair creation by the centre for these supercritical values of (Y.This instability manifests itself in the occurrence of solutions which correspond to the Breit-Wigner resonances [6]. In gauge theories with chiral invariant lagrangians (i.e. zero bare masses of fermions) the problem of the supercritical Coulomb centre is substituted by the problem of tightly bound fermion-antifermion states and the Breit-Wigner resonances are substituted by tachyons [2-41. The occurrence of tachyons indicates that the vacuum of the normal (massless) phase is un0 03 l--9163/8
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0 North-Holland
stable. The character of the vacuum rearrangement (phase transition) is determined by the quantum numbers of the tachyons. It has been shown in refs. [2,3] ’ that in the ladder approximation the properties of the tachyonic solutions of the Bethe-Salpeter (BS) equation coincide in general features for the case of abelian [colour group U(l)] and non-abelian [colour group SU(N)] theories. If the coupling (Y[o =g2/4n for U(1) and cr = g2(i@ - 1)/8+V for SU(N)] exceeds its critical value (Y, = n/3, there exists a K2-plet of scalar (Jpc = O++) colourless tachyons and a K2-plet of pseudoscalar (0-+) colourless tachyons for the case of the SUL(K) X SUR(K) chiral flavour group. The mass spectrum of tachyons is degenerate in parity and quantum numbers of the flavour group and takes the form [2,3] (for such values of (Ythat (Y- (11,Q 1) 2
m,ch - A2 exp [ns/h] ,
s=1,2..., (1)
h = i(3a/rr
- I)“2 ,
where A is a Lorentz invariant ultraviolet cutoff (in the following we shall pass to a local theory without cutoff). It is essential that these tachyons possess a chiral charge Q, = +2 i.e. their BS wavefunction is of the form (OlT(&$&)lP) or
157
PHYSICS LETTERS
Volume lOOB, number 2
should be emphasized that since the mass has appeared as a result of spontaneous breaking of chiral invariance, it corresponds to the dynamical part of the mass of the constituent quark (the masses of current quarks are connected with an explicit breaking of chiral invariance in QCD [9]). The spectrum of the quark dynamical masses can be determined by requiring that tachyons disappear in the stable phase and instead of them the (K2 - 1)plet of pseudoscalar Goldstone bosons appears (in this paper we shall not touch on the U(1) problem which is connected with singlet flavour current). It follows from the Ward identities for axial currents [ 10,l l] qpr&(P>
P’> = s-’
(P’)YghO + &g
s-l
(2)
(here I$ is the axial vertex and Afl are (K2 - 1) flavour matrices) that such a way of determining the mass m is equivalent to looking for nontrivial solutions of the Schwinger-Dyson equation for the mass function B(pz) [S-l(p) = $A(p2) - B(p2)]. In the euclidean domain in the ladder approximation the BS equation for pseudoscalar Goldstone bosons takes the following form (in the Landau gauge):
(s2 +m2)&)
= -$
* d4q’ J ~ x(4’) ) (4 - q’)2
(3)
where m is the fermion mass, q is the relative fermionantifermion momentum and x is the Goldstone boson wavefunction defined by
O;@ = xflr5x.
(4)
(3) over the angle variables, we get
6 +4x(x)
0
e(x’ - x) +I X
1
x’x(x’)
(5) )
*’ The physical reason of different types of the vacuum rearrangement for the case of the supercritical Coulomb centre and for the case of gauge theories with supercritical values of the coupling constant (in the former case the charge is shielded [6] while in the latter case the fermion mass is generated) is discussed in detail in ref. [4]. Note that in ref. [S] the phenomenon connected with the strong colour Coulomb field was used to describe the colour charge screening.
158
where a = m2/A2, x = q2/A2. The last equation turn is equivalent to the differential equation (d/dx){x2 d[(x + a)x]/dx} with two boundary {x2dKx {xd[(x
in its
+ (3c~/47r)xx = 0,
(6)
conditions:
+4xl/dx)JX,o= 0,
(7)
+a)xl/dx+ 6 +4xII,,1 = 0.
(8)
The general solution of eq. (6) has the form u=(x+a)X=U1
+cu2,
(9)
where
(PI 3
qcr=P;-P,¶
26 March 1981
u1 =F(r,r*,2;-y), U2 =yPrF(I, +y-‘*F(C)
(lOa)
-r*,
2r; -y-l)
-r, 2r*; -y-l),
(lob)
+ih,r* =f _ ih,y =x/u, F is a hypergeometric function. From the boundary condition (7) it follows that c = 0. For a Q 1, using asymptotic expansions for F [ 121, we get the following equation from the boundary condition (8)
I=$
[(ctgh 7rX/nA)a]l/2 sin[hlna-X(X)]
=O,
(11)
where C(h) = arg [r(2r)/@(r)], r is the Euler y-function. For the supercritical cx> (Y, = n/3 (h > 0) eq. (11) yields the mass spectrum rnz m A2exp(-ns/A),
s = 1,2 ... .
(12)
We shall be interested in (a) the transition to a local theory without cutoff and (b) the improvement of the ladder approximation. An essential point here is the transition from the coupling constant (Yto the running coupling constant a(q2). The function a(q2) determines the region of momenta responsible for the fermion mass generation (i.e. the region where a(q2) 2 cvc). In abelian theory this is the region of large momenta. The relation (12) indicates the existence of additional ultraviolet divergences as compared with ordinary perturbation theory. To remove these divergences, the additional coupling constant renormalization [4] or the extension of the symmetric operator in eq. (5) up to the self-adjoint operator [ 131 are required. In the latter case the unbounded spectrum appears as
Volume IOOB, number 2
rn: = m”2exp(--?rs/h),
PHYSICS LETTERS
s = 0, ?l , *2 ... )
(13)
where % is an arbitrary dimensional parameter. The use of such a type of relations as (13) in order to describe the quark mass spectrum was discussed in ref. [14]. In non-abelian theory the domain of strong coupling (01(q2) 2 a,_) is the region of small momenta. Transition to the local theory, i.e. inclusion of the region of large momenta where the coupling constant is small due to asymptotic freedom, should not qualitatively change the fermion spectrum. Therefore, in this case the cutoff parameter A2 can be retained in eq. (3) and can be identified with a value of q2 at which (~(4~) - a,. Thus, the asymptotic freedom results in an effective cutoff of the interaction which is resporr sible for the mass generation. We shall now take into account the mechanism of confinement. Since eq. (3) describes fermion-antifermion bound states, the confinement mechanism should effectively result in the appearance of the infrared cutoff parameter ~1in this equation. Moreover, the value of R = p-l should be of the order of the hadron size CJJ= 0.15 GeV). Thus, describing bound states we use something similar to the quark bag model [15] with additionally taken into account a strong colour field near the bag surface (from A-l to p-l) *‘. We emphasize that in the exact theory there should be no free parameters such as CLThe coupling constant has to be expressed there through the critical value oe, and the dimensional parameter A is the only free parameter of the theory (a manifestation of dimensional transmutation [ 171). Therefore eq. (5) with the infrared cutoff /.I should be regarded as a heuristical description of the complicated dynamics at large distances. In particular, the constant (Yis to be regarded as an effective coupling constant which characterises the interaction strength at distances from 1/A to l/p. The introduction of the infrared cutoff changes the boundary condition (7) only: (x2du/dx)]_
= 0,
p = n2/A2.
(14)
** Note, however, that the appearance of the infrared cutoff parameter can be due to the mechanism which is an alternative to the confinement mechanism, i.e. dynamical generation of the gluon mass (screening mechanism) [ 2,161.
26 March 1981
Hence it follows that c=---i_u; u2 x=p .
The second boundary condition (8) yields the equation determining the mass spectrum q=A1B2
- B,A,
=O,
(15)
where
Ai = (Xdui/dX + Ui)lX=l 2 (16)
i= 1,2.
Bi =(-Xdui/dX)lXZp,
Using power expansions for hypergeometric functions [12] the function cpcan be transposed to the following form (when m < p)
cp=
c
c
n=O
cDn(-Z)n+l)
z = a/p = m2/p2 ,
(17)
where
=k$o
on
PkGkHn-k
G = ;i
G,=l,
H
k
C h2+(k-;)2
“0 = 0,
A2 + (m
m=l m(4h2
A2 +(k + ;)2
= k
sin(h ln
p-l -
-:)k
c
fin-k),
k>
1,
t m2)l12’
; 1 G, ‘I2
ak = 6 (2arctgm% m=l ’
c= 2 [(X2 + ‘)( 4
ak +
- arctgz),
k>l;
2
tgh nh/n)l)p] 1/Z
X cos[Z(h) - 2 arctg 2AJ.
(18)
It follows that the introduction of the infrared parameter results in the disappearance of the oscillations in the mass variable In a [compare with eq. (1 l)] . In consequence, the function cphas only a finite number of zeros, i.e. the mass spectrum contains only a finite 159
number of levels. The number of nontrivial zeros zs of the function q(z) at given values of the parameters X and p can be easily determined if one takes into consideration that (a) the value of the mass m, (and consequently zs) increases with increasing parameter h *3 and (b) the minimal value of each mass m, is equal to zero. .Therefore, the appearance of each new level of the mass spectrum takes place when the following condition is satisfied
i.e. CD, E - sin (h In p-I
+ 2 arctg2X) = 0 .
(19)
Thus, the number of levels of the mass spectrum (n) is equal to the integral part of the value rI(h In p-I + 2 arctg 2h). As was to be expected, n increases with X and p-l. The parameter A is determined from the condition a(@) - (Ye= rr/3 and its order can be estimated with the use of the well-known renormalization group formula [for the group W(N)] (Y(A~) = (N2 - 1)g2(A2)/81mN = 6n(N2 - l)/iV(ll
N - 2kl)ln(A2/M2):
where M is the renormalization group parameter (M = 0.5 GeV for QCD) and k, counts the number of light quarks with mass m Q A. Using integral representations for hypergeometric functions [ 121 the transcendental equation (15) has been solved numerically with the computer for five valuesofp:p=2X 10-2(kl=2),p=1.7X lop2 (kl = 3), p = 1.5 X 10-Z (kl = 4), p = 1O-2 (kl 7 6) and p = 6 X 10-3 (kl = 8). The values of the masses m, corresponding to the routes zs are given in table 1. It can be noted that the form of the mass spectrum is not too much sensitive to the variations of p. This is caused by the weak (logarithmic) dependence of the number of routes (n) on p. In the sequel we restrict ourselves for practical applications to the two most interesting variants of QCD with six (K = 6) and eight (K = 8) quark flavours, respectively. The results obtained can be interpreted in the following way (details will be published elsewhere). *3 this intuitively obvious assertion can be proved by direct analysis of integral equation (5).
160
26 March 1981
PHYSICS LETTERS
Volume 1OOB,number 2
I. n = 1 (orthodox picture). In this case all fermion fields ~j (index i is related to the fundamental representation of the flavour group SU(K), K = 6 or K = 8) have a common dynamical mass m. As shown in table 1 the standard value 0.3 GeV [9] for m is obtained when X = 0.5-0.6. The “total” mass of the jth quark consists of the dynamical part m and the current part moj *4. The strength of the SU,(k) X SUR(k) (k < K) explicit breaking is determined by relations between the current masses mui (i = 1,2, ... k) and the dynamical mass m = 0.3 GeV. Since the subgroups with k < 3 include light (mui < m) quarks only, the explicit breaking of sum X SUR(2) and sum X SUR(3) SymmetrkS can be considered as relatively slight. Therefore, there are (k2 - 1) (k = 2, 3) pseudoscalar “almost” Goldstone mesons corresponding to these subgroups. It has to be emphasized that the mass generation mechanism considered here is essentially relativistic *’ . This circumstance sheds light on the profound distinction between pseudoscalar mesons consisting of light and heavy (moi > m) quarks: the latter (for example, D-mesons) are nonrelativistic systems [ 151 and so they cannot be regarded as “almost” Goldstone excitations in the phase with spontaneously broken chiral symmetry. II. n > 1. Let us consider two alternate versions: QCD with six quarks (K = 6) and three quark genera-
*4 More exactly, one should speak about the quark mass function B rather than the quark mass [ 9,111. *’ Indeed, the phenomenon of vacuum instability for the supercritical Coulomb centre (which is connected with the mechanism considered) is purely relativistic [6]. Table 1 K
n
kl
ml
6
8
1 2
6 3
3 1 2 4
A
m (Gel9 m2
m3
m4
0.3
-
-
-
0.6
4 2
1.2 2.1 2.1
0.3 0.7 0.7
0.3 0.3
-
1.4 2.2 2.4
8 4 4 2
0.3 1.2 3.0 2.8
0.3 1.2 1.1
0.6 0.6
0.3 0.3
0.5 1.35 3.05 3.3
Volume 1OOB.number 2
PHYSICS LETTERS
tions (n = 3) and QCD with eight quarks (K = 8) and four quark generations (n = 4). In both these cases the “bare” flavour symrnetry of QCD is the isospin SU(2) symmetry, and two fermion fields tij correspond to u and d quarks. The values of the masses m, (1 < s d n) corresponding to the quarks of the (n + 1 - s) generation are given in table 1 (the mass of the quarks of the first generation is assumed to be equal to 0.3 GeV.) We note that all mass ratios m,/m,+l satisfy the condition 1.8 < msfm,,l < 3. The results corresponding to n = 2 are also given in table 1. In this case the “bare” flavour symmetry of QCD is SU(3) (for K = 6) or SU(4) (for K = 8). Heavy quarks arise in these schemes as a result of quark-gluon dynamics. The important problem is to determine how much the form of the mass spectrum (m,) will be changed if explicit breaking of chiral invariance (the current masses) is taken into account. Concluding, we should like to make some observations. As it has been emphasised, the parameter h is not free in the exact theory. Therefore the choice between the alternatives I and II should be implied by the dynamics of QCD. In the approximation used here all vacua corresponding to the different masses m, are stable. The question arises: are all these vacua also stable in the exact theory? Or is only the deepest one, corresponding to the largest value (ml ), stable and are other vacua false? The existence of the region of strong coupling is essential for the proposed mechanism of dynamical mass generation. Also, the infrared region of strong coupling in QCD is usually assumed to be responsible for the confinement of colour. In our heuristic approach confinement is taken into account by introducing the infrared cutoff p (bag surface?) and the distances A-*
26 March 1981
the l/N expansion scheme for investigation of the proposed mechanism of mass generation. In particular, the use of this scheme might allow one to improve the approximation used here. In the present paper we have not touched upon the important problem of the origin of the current quark and lepton masses. It seems that this problem should be considered in the framework of unified theories with new fields and interactions (technicolour, etc.) [ 19,201. In particular, it is not excluded that only at this level the problem of the fermion generation number can be resolved. Since the mechanism considered by us not based on the specific character of QCD, it can also turn out to be useful for investigating these issues. We thank G.V. Bugrij for help with computer calculations and Professor P.I. Fomin for stimulating discussions. One of us (V.A.M.) also wishes to thank Professors V. de Alfaro, G. Furlan, A. Giovannini, N. Paver, J. Strathdee and M. Virasoro for many interesting discussions and warm hospitality.
References [I ] P.I. Fornin and V.A. Miransky, Phys. Lett. 64B (1975) 166. [ 21 V.P. Gusynin and V.A. Miransky, Phys. Lett. 76B (1978) 585. [3] PI. Fomin, V.P. Gusynin and V.A. Miransky, Phys. Lett. 78B (1978) 136. [4] V.A. Miransky, Phys. Lett. 91 B (1980) 421. [5] S. Mandelstam, Phys. Rev. D20 (1979) 3223. [6] Ya.B. Zeldovich and V.S. Popov, Usp. Fiz. Nauk 105 (1971) 403; J. Rafelski, L.P. Fulcher and A. Klein, Phys. Rep. 38C (1978) 229. [7] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [ 81 J. Rafelski, Phys. Lett. 79B (1978) 419; A. Hey, D. Horn and J. Mandula, Phys. Lett. 80B (1978) 90. [9] For a review, see S. Weinberg, Trans. N.Y. Acad. Sci., Series II, 38 (1977) 185; H. Fritzsch, CERN preprint TH-2699 (1979). [lo] K. Lane, Phys. Rev. DlO (1974) 2605. [ 111 H. Pagels, Phys. Rev. Dl9 (19791 3080. [ 121 H. Bateman and A. Erdelyi, Higher transcendental functions, Vol. 1 (McGraw-Hill, New York, 1953). [I 31 K. Case, Phys. Rev. 80 (1950) 797. [14] S. Blaha, Phys. Lett. 80B (1978) 99; L. Jenkovszky, preprint ITP-78-50E (1978) Kiev.
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[ 151 A. Chodos, R. Jaffe, K. Johnson, C. Thorn and V. Weisskopf, Phys. Rev. D9 (1974) 3471; for a recent review, see: R. Jaffe, MIT preprint CTP819 (1979). [ 161 A. de Rujula, R.C. Giles and R.L. Jaffe, Phys. Rev. D17 (1978) 285; R. Fukuda and T. Kugo, Progr. Theor. Phys. 60 (1978) 565; G. Mack, Phys. Lett. 78B (1978) 263.
162
26 March 1981
[ 171 S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [ 181 J.M. Cornwall, Phys. Rev. D22 (1980) 1452. [ 191 S. Weinberg, Phys. Rev. D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. [ 201 S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125.
GENERATION
IN NON-ABELIAN V.A. MIRANSKY,
26 March 1981
PHYSICS LETTERS
Volume lOOB, number 2
OF THE SPECTRUM OF FERMIONS
GAUGE FIELD THEORIES V.P. GUSYNIN and Yu.A. SITENKO
Institute for Theoretical Physics, Kiev-130,
USSR
Received 1 September 1980
The dynamical mechanism of spontaneous breakdown of chiral invariance in nonabelian gauge theories is proposed. The dynamical masses of quarks in quantum chromodynamics are estimated within this framework.
In this paper the dynamical mechanism of spontaneous breakdown of chiral invariance, suggested in refs. [l-4] is used to investigate the fermion mass spectrum in non-abelian gauge field theories, in particular, the mass spectrum of quarks in massless quantum chromodynamics (QCD) (recently a similar mechanism was also considered by Mandelstam [S]). As will be shown, this mechanism supplemented with the conditions of asymptotic freedom and confinement explains naturally the main regularities of the dynamics of broken chiral symmetry of hadrons. We calculate the mass spectrum of quarks in QCD and propose a possible explanation of the existence of a finite number of quark generations. The physical phenomenon underlying this mechanism is closely connected with the phenomenon of the fall into the strong Coulomb centre V = -c~/r (cu = Ze2/4n > 1, Z > 137) which takes place when using the Dirac equation [6]. As is known, the vacuum becomes unstable with respect to the e+e--pair creation by the centre for these supercritical values of (Y.This instability manifests itself in the occurrence of solutions which correspond to the Breit-Wigner resonances [6]. In gauge theories with chiral invariant lagrangians (i.e. zero bare masses of fermions) the problem of the supercritical Coulomb centre is substituted by the problem of tightly bound fermion-antifermion states and the Breit-Wigner resonances are substituted by tachyons [2-41. The occurrence of tachyons indicates that the vacuum of the normal (massless) phase is un0 03 l--9163/8
1/OOoO-0000/$02.50
0 North-Holland
stable. The character of the vacuum rearrangement (phase transition) is determined by the quantum numbers of the tachyons. It has been shown in refs. [2,3] ’ that in the ladder approximation the properties of the tachyonic solutions of the Bethe-Salpeter (BS) equation coincide in general features for the case of abelian [colour group U(l)] and non-abelian [colour group SU(N)] theories. If the coupling (Y[o =g2/4n for U(1) and cr = g2(i@ - 1)/8+V for SU(N)] exceeds its critical value (Y, = n/3, there exists a K2-plet of scalar (Jpc = O++) colourless tachyons and a K2-plet of pseudoscalar (0-+) colourless tachyons for the case of the SUL(K) X SUR(K) chiral flavour group. The mass spectrum of tachyons is degenerate in parity and quantum numbers of the flavour group and takes the form [2,3] (for such values of (Ythat (Y- (11,Q 1) 2
m,ch - A2 exp [ns/h] ,
s=1,2..., (1)
h = i(3a/rr
- I)“2 ,
where A is a Lorentz invariant ultraviolet cutoff (in the following we shall pass to a local theory without cutoff). It is essential that these tachyons possess a chiral charge Q, = +2 i.e. their BS wavefunction is of the form (OlT(&$&)lP) or
157
PHYSICS LETTERS
Volume lOOB, number 2
should be emphasized that since the mass has appeared as a result of spontaneous breaking of chiral invariance, it corresponds to the dynamical part of the mass of the constituent quark (the masses of current quarks are connected with an explicit breaking of chiral invariance in QCD [9]). The spectrum of the quark dynamical masses can be determined by requiring that tachyons disappear in the stable phase and instead of them the (K2 - 1)plet of pseudoscalar Goldstone bosons appears (in this paper we shall not touch on the U(1) problem which is connected with singlet flavour current). It follows from the Ward identities for axial currents [ 10,l l] qpr&(P>
P’> = s-’
(P’)YghO + &g
s-l
(2)
(here I$ is the axial vertex and Afl are (K2 - 1) flavour matrices) that such a way of determining the mass m is equivalent to looking for nontrivial solutions of the Schwinger-Dyson equation for the mass function B(pz) [S-l(p) = $A(p2) - B(p2)]. In the euclidean domain in the ladder approximation the BS equation for pseudoscalar Goldstone bosons takes the following form (in the Landau gauge):
(s2 +m2)&)
= -$
* d4q’ J ~ x(4’) ) (4 - q’)2
(3)
where m is the fermion mass, q is the relative fermionantifermion momentum and x is the Goldstone boson wavefunction defined by
O;@ = xflr5x.
(4)
(3) over the angle variables, we get
6 +4x(x)
0
e(x’ - x) +I X
1
x’x(x’)
(5) )
*’ The physical reason of different types of the vacuum rearrangement for the case of the supercritical Coulomb centre and for the case of gauge theories with supercritical values of the coupling constant (in the former case the charge is shielded [6] while in the latter case the fermion mass is generated) is discussed in detail in ref. [4]. Note that in ref. [S] the phenomenon connected with the strong colour Coulomb field was used to describe the colour charge screening.
158
where a = m2/A2, x = q2/A2. The last equation turn is equivalent to the differential equation (d/dx){x2 d[(x + a)x]/dx} with two boundary {x2dKx {xd[(x
in its
+ (3c~/47r)xx = 0,
(6)
conditions:
+4xl/dx)JX,o= 0,
(7)
+a)xl/dx+ 6 +4xII,,1 = 0.
(8)
The general solution of eq. (6) has the form u=(x+a)X=U1
+cu2,
(9)
where
(PI 3
qcr=P;-P,¶
26 March 1981
u1 =F(r,r*,2;-y), U2 =yPrF(I, +y-‘*F(C)
(lOa)
-r*,
2r; -y-l)
-r, 2r*; -y-l),
(lob)
+ih,r* =f _ ih,y =x/u, F is a hypergeometric function. From the boundary condition (7) it follows that c = 0. For a Q 1, using asymptotic expansions for F [ 121, we get the following equation from the boundary condition (8)
I=$
[(ctgh 7rX/nA)a]l/2 sin[hlna-X(X)]
=O,
(11)
where C(h) = arg [r(2r)/@(r)], r is the Euler y-function. For the supercritical cx> (Y, = n/3 (h > 0) eq. (11) yields the mass spectrum rnz m A2exp(-ns/A),
s = 1,2 ... .
(12)
We shall be interested in (a) the transition to a local theory without cutoff and (b) the improvement of the ladder approximation. An essential point here is the transition from the coupling constant (Yto the running coupling constant a(q2). The function a(q2) determines the region of momenta responsible for the fermion mass generation (i.e. the region where a(q2) 2 cvc). In abelian theory this is the region of large momenta. The relation (12) indicates the existence of additional ultraviolet divergences as compared with ordinary perturbation theory. To remove these divergences, the additional coupling constant renormalization [4] or the extension of the symmetric operator in eq. (5) up to the self-adjoint operator [ 131 are required. In the latter case the unbounded spectrum appears as
Volume IOOB, number 2
rn: = m”2exp(--?rs/h),
PHYSICS LETTERS
s = 0, ?l , *2 ... )
(13)
where % is an arbitrary dimensional parameter. The use of such a type of relations as (13) in order to describe the quark mass spectrum was discussed in ref. [14]. In non-abelian theory the domain of strong coupling (01(q2) 2 a,_) is the region of small momenta. Transition to the local theory, i.e. inclusion of the region of large momenta where the coupling constant is small due to asymptotic freedom, should not qualitatively change the fermion spectrum. Therefore, in this case the cutoff parameter A2 can be retained in eq. (3) and can be identified with a value of q2 at which (~(4~) - a,. Thus, the asymptotic freedom results in an effective cutoff of the interaction which is resporr sible for the mass generation. We shall now take into account the mechanism of confinement. Since eq. (3) describes fermion-antifermion bound states, the confinement mechanism should effectively result in the appearance of the infrared cutoff parameter ~1in this equation. Moreover, the value of R = p-l should be of the order of the hadron size CJJ= 0.15 GeV). Thus, describing bound states we use something similar to the quark bag model [15] with additionally taken into account a strong colour field near the bag surface (from A-l to p-l) *‘. We emphasize that in the exact theory there should be no free parameters such as CLThe coupling constant has to be expressed there through the critical value oe, and the dimensional parameter A is the only free parameter of the theory (a manifestation of dimensional transmutation [ 171). Therefore eq. (5) with the infrared cutoff /.I should be regarded as a heuristical description of the complicated dynamics at large distances. In particular, the constant (Yis to be regarded as an effective coupling constant which characterises the interaction strength at distances from 1/A to l/p. The introduction of the infrared cutoff changes the boundary condition (7) only: (x2du/dx)]_
= 0,
p = n2/A2.
(14)
** Note, however, that the appearance of the infrared cutoff parameter can be due to the mechanism which is an alternative to the confinement mechanism, i.e. dynamical generation of the gluon mass (screening mechanism) [ 2,161.
26 March 1981
Hence it follows that c=---i_u; u2 x=p .
The second boundary condition (8) yields the equation determining the mass spectrum q=A1B2
- B,A,
=O,
(15)
where
Ai = (Xdui/dX + Ui)lX=l 2 (16)
i= 1,2.
Bi =(-Xdui/dX)lXZp,
Using power expansions for hypergeometric functions [12] the function cpcan be transposed to the following form (when m < p)
cp=
c
c
n=O
cDn(-Z)n+l)
z = a/p = m2/p2 ,
(17)
where
=k$o
on
PkGkHn-k
G = ;i
G,=l,
H
k
C h2+(k-;)2
“0 = 0,
A2 + (m
m=l m(4h2
A2 +(k + ;)2
= k
sin(h ln
p-l -
-:)k
c
fin-k),
k>
1,
t m2)l12’
; 1 G, ‘I2
ak = 6 (2arctgm% m=l ’
c= 2 [(X2 + ‘)( 4
ak +
- arctgz),
k>l;
2
tgh nh/n)l)p] 1/Z
X cos[Z(h) - 2 arctg 2AJ.
(18)
It follows that the introduction of the infrared parameter results in the disappearance of the oscillations in the mass variable In a [compare with eq. (1 l)] . In consequence, the function cphas only a finite number of zeros, i.e. the mass spectrum contains only a finite 159
number of levels. The number of nontrivial zeros zs of the function q(z) at given values of the parameters X and p can be easily determined if one takes into consideration that (a) the value of the mass m, (and consequently zs) increases with increasing parameter h *3 and (b) the minimal value of each mass m, is equal to zero. .Therefore, the appearance of each new level of the mass spectrum takes place when the following condition is satisfied
i.e. CD, E - sin (h In p-I
+ 2 arctg2X) = 0 .
(19)
Thus, the number of levels of the mass spectrum (n) is equal to the integral part of the value rI(h In p-I + 2 arctg 2h). As was to be expected, n increases with X and p-l. The parameter A is determined from the condition a(@) - (Ye= rr/3 and its order can be estimated with the use of the well-known renormalization group formula [for the group W(N)] (Y(A~) = (N2 - 1)g2(A2)/81mN = 6n(N2 - l)/iV(ll
N - 2kl)ln(A2/M2):
where M is the renormalization group parameter (M = 0.5 GeV for QCD) and k, counts the number of light quarks with mass m Q A. Using integral representations for hypergeometric functions [ 121 the transcendental equation (15) has been solved numerically with the computer for five valuesofp:p=2X 10-2(kl=2),p=1.7X lop2 (kl = 3), p = 1.5 X 10-Z (kl = 4), p = 1O-2 (kl 7 6) and p = 6 X 10-3 (kl = 8). The values of the masses m, corresponding to the routes zs are given in table 1. It can be noted that the form of the mass spectrum is not too much sensitive to the variations of p. This is caused by the weak (logarithmic) dependence of the number of routes (n) on p. In the sequel we restrict ourselves for practical applications to the two most interesting variants of QCD with six (K = 6) and eight (K = 8) quark flavours, respectively. The results obtained can be interpreted in the following way (details will be published elsewhere). *3 this intuitively obvious assertion can be proved by direct analysis of integral equation (5).
160
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PHYSICS LETTERS
Volume 1OOB,number 2
I. n = 1 (orthodox picture). In this case all fermion fields ~j (index i is related to the fundamental representation of the flavour group SU(K), K = 6 or K = 8) have a common dynamical mass m. As shown in table 1 the standard value 0.3 GeV [9] for m is obtained when X = 0.5-0.6. The “total” mass of the jth quark consists of the dynamical part m and the current part moj *4. The strength of the SU,(k) X SUR(k) (k < K) explicit breaking is determined by relations between the current masses mui (i = 1,2, ... k) and the dynamical mass m = 0.3 GeV. Since the subgroups with k < 3 include light (mui < m) quarks only, the explicit breaking of sum X SUR(2) and sum X SUR(3) SymmetrkS can be considered as relatively slight. Therefore, there are (k2 - 1) (k = 2, 3) pseudoscalar “almost” Goldstone mesons corresponding to these subgroups. It has to be emphasized that the mass generation mechanism considered here is essentially relativistic *’ . This circumstance sheds light on the profound distinction between pseudoscalar mesons consisting of light and heavy (moi > m) quarks: the latter (for example, D-mesons) are nonrelativistic systems [ 151 and so they cannot be regarded as “almost” Goldstone excitations in the phase with spontaneously broken chiral symmetry. II. n > 1. Let us consider two alternate versions: QCD with six quarks (K = 6) and three quark genera-
*4 More exactly, one should speak about the quark mass function B rather than the quark mass [ 9,111. *’ Indeed, the phenomenon of vacuum instability for the supercritical Coulomb centre (which is connected with the mechanism considered) is purely relativistic [6]. Table 1 K
n
kl
ml
6
8
1 2
6 3
3 1 2 4
A
m (Gel9 m2
m3
m4
0.3
-
-
-
0.6
4 2
1.2 2.1 2.1
0.3 0.7 0.7
0.3 0.3
-
1.4 2.2 2.4
8 4 4 2
0.3 1.2 3.0 2.8
0.3 1.2 1.1
0.6 0.6
0.3 0.3
0.5 1.35 3.05 3.3
Volume 1OOB.number 2
PHYSICS LETTERS
tions (n = 3) and QCD with eight quarks (K = 8) and four quark generations (n = 4). In both these cases the “bare” flavour symrnetry of QCD is the isospin SU(2) symmetry, and two fermion fields tij correspond to u and d quarks. The values of the masses m, (1 < s d n) corresponding to the quarks of the (n + 1 - s) generation are given in table 1 (the mass of the quarks of the first generation is assumed to be equal to 0.3 GeV.) We note that all mass ratios m,/m,+l satisfy the condition 1.8 < msfm,,l < 3. The results corresponding to n = 2 are also given in table 1. In this case the “bare” flavour symmetry of QCD is SU(3) (for K = 6) or SU(4) (for K = 8). Heavy quarks arise in these schemes as a result of quark-gluon dynamics. The important problem is to determine how much the form of the mass spectrum (m,) will be changed if explicit breaking of chiral invariance (the current masses) is taken into account. Concluding, we should like to make some observations. As it has been emphasised, the parameter h is not free in the exact theory. Therefore the choice between the alternatives I and II should be implied by the dynamics of QCD. In the approximation used here all vacua corresponding to the different masses m, are stable. The question arises: are all these vacua also stable in the exact theory? Or is only the deepest one, corresponding to the largest value (ml ), stable and are other vacua false? The existence of the region of strong coupling is essential for the proposed mechanism of dynamical mass generation. Also, the infrared region of strong coupling in QCD is usually assumed to be responsible for the confinement of colour. In our heuristic approach confinement is taken into account by introducing the infrared cutoff p (bag surface?) and the distances A-*
26 March 1981
the l/N expansion scheme for investigation of the proposed mechanism of mass generation. In particular, the use of this scheme might allow one to improve the approximation used here. In the present paper we have not touched upon the important problem of the origin of the current quark and lepton masses. It seems that this problem should be considered in the framework of unified theories with new fields and interactions (technicolour, etc.) [ 19,201. In particular, it is not excluded that only at this level the problem of the fermion generation number can be resolved. Since the mechanism considered by us not based on the specific character of QCD, it can also turn out to be useful for investigating these issues. We thank G.V. Bugrij for help with computer calculations and Professor P.I. Fomin for stimulating discussions. One of us (V.A.M.) also wishes to thank Professors V. de Alfaro, G. Furlan, A. Giovannini, N. Paver, J. Strathdee and M. Virasoro for many interesting discussions and warm hospitality.
References [I ] P.I. Fornin and V.A. Miransky, Phys. Lett. 64B (1975) 166. [ 21 V.P. Gusynin and V.A. Miransky, Phys. Lett. 76B (1978) 585. [3] PI. Fomin, V.P. Gusynin and V.A. Miransky, Phys. Lett. 78B (1978) 136. [4] V.A. Miransky, Phys. Lett. 91 B (1980) 421. [5] S. Mandelstam, Phys. Rev. D20 (1979) 3223. [6] Ya.B. Zeldovich and V.S. Popov, Usp. Fiz. Nauk 105 (1971) 403; J. Rafelski, L.P. Fulcher and A. Klein, Phys. Rep. 38C (1978) 229. [7] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [ 81 J. Rafelski, Phys. Lett. 79B (1978) 419; A. Hey, D. Horn and J. Mandula, Phys. Lett. 80B (1978) 90. [9] For a review, see S. Weinberg, Trans. N.Y. Acad. Sci., Series II, 38 (1977) 185; H. Fritzsch, CERN preprint TH-2699 (1979). [lo] K. Lane, Phys. Rev. DlO (1974) 2605. [ 111 H. Pagels, Phys. Rev. Dl9 (19791 3080. [ 121 H. Bateman and A. Erdelyi, Higher transcendental functions, Vol. 1 (McGraw-Hill, New York, 1953). [I 31 K. Case, Phys. Rev. 80 (1950) 797. [14] S. Blaha, Phys. Lett. 80B (1978) 99; L. Jenkovszky, preprint ITP-78-50E (1978) Kiev.
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[ 151 A. Chodos, R. Jaffe, K. Johnson, C. Thorn and V. Weisskopf, Phys. Rev. D9 (1974) 3471; for a recent review, see: R. Jaffe, MIT preprint CTP819 (1979). [ 161 A. de Rujula, R.C. Giles and R.L. Jaffe, Phys. Rev. D17 (1978) 285; R. Fukuda and T. Kugo, Progr. Theor. Phys. 60 (1978) 565; G. Mack, Phys. Lett. 78B (1978) 263.
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[ 171 S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [ 181 J.M. Cornwall, Phys. Rev. D22 (1980) 1452. [ 191 S. Weinberg, Phys. Rev. D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. [ 201 S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125.