6
Z. Phys. C Particles and Fields 32, 585-590 (1986) F kzles [Or Phy~'k C and FL= ds Springer-Verlag 1986 Phenomenology of Dynamical Symmetry Breaking in QCD Herman J. Munczek Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA Received 20 May 1986 Abstract. A consistent treatment of the QCD quark propagator and quark-antiquark bound state equ- ations is presented which follows the Nambu and Jona-Lasinio approach to the discussion of chiral symmetry breaking. An expression is obtained for the dynamical, momentum dependent, mass. In the appro- ximation used here the dynamical mass is determined by M 0, its value at zero momentum, and by the strong coupling constant ~s and bare quark mass mo. In the limiting case of no explicit chiral symmetry breaking, i.e., mo = 0, this expression coincides in form with the one obtained by Chang and Chang in their renormalization-group analysis. In this limit chiral symmetry remains broken and we show the explicit appearance of a Nambu-Goldstone pion. A consistent calculation of the pseudoscalar, scalar and vector meson masses gives values ofmo, ~s and Mo well in step with other estimates. This makes possible a calculation off,, the pion decay constant, in reasonable agreement with experiment. The relative smallness of the pi-meson mass has been, for a long time, the main motivation for the many attempts to describe it as a would-be Nambu- Goldstone boson. The pion's quantum numbers indi- cate that the symmetry to be spontaneously broken is the chiral symmetry of the Lagrangian for the fermions of which the pion is presumably composed, i.e. the up and down quarks of quantum chromodynamics (QCD). Strict chiral symmetry of the Lagrangian forbids an explicit mass term for the fermions. The pion mass, however, even if small, is not zero and this appears to require the inclusion in the QCD Lagran- gian of a small bare, or "current-algebra," mass parameter too. A conventional perturbative treatment built on the (almost) chiral symmetric vacuum would yield a fermion propagator with a mass m(p2) propor- tional to the current algebra mass rn 0 and which does not support the notion of the pion bound state as a Nambu-Goldstone boson. Specifically, if rno vanishes so does m(p2) and the chiral symmetry is not broken. A crucial step towards a better understanding of the spontaneous breaking of chiral symmetry was taken by Nambu and Jona-Lasinio [1] (NJL). Although the interaction chosen for their discussions was nonre- normalizable they showed that, by choosing a per- turbative expansion around a new (broken) vacuum in which the fermions are assumed to have already a mass Mo (not present in the Lagrangian), they could obtain the explicit appearance of a massless pseudoscalar as well as other massive bound states of different quan- tum numbers. The dynamically generated mass Mo was determined by a condition referred to as the "gap equation", by analogy to superconductivity theory. It is essential in the NJL approach that there be consis- tency between the Schwinger-Dyson (SD) equation, used to obtain the fermion propagator, and the Bethe- Salpeter (BS) equation for the bound states. That is, the same renormalization condition (namely, the gap equation) should be used in the treatment of both equations. This has been emphasized by Lane [2] who studied the extension of the NJL approach to QCD with inclusion of renormalization group ideas. More recently, Chang and Chang [3] have studied the NJL mechanism for the dynamical generation of the mass M o in QCD. They find that M o is a renormalization-group invariant that can be expressed in terms of the invariant QCD cut-off A c, i.e., Mo = Ace 116. In further work [4] they have applied their techniques to the case in which there is in the Lagran- gian an explicit mass term too. They thus obtain an expression for M(p 2) which depends on two independ- ent renormalization-group invariants: Mo = M(0) and Ac. The analytic structure of M is the usual one. However, in the critical limit m o = 0,M vanishes identically for space-like p2 < _ Mg e -= - M~, while for real p2 between - M~ and M if, M varies smoothly between 0 and Me. In this way, dynamical symmetry breaking is maintained when making the continuous transition from a nonchiral to a chiral-symmetric Lagrangian. The purpose of the work to be discussed here is to study the QCD low energy bound states in the light of

Phenomenology of dynamical symmetry breaking in QCD

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Page 1: Phenomenology of dynamical symmetry breaking in QCD

Z. Phys. C Particles and Fields 32, 585-590 (1986) �9 F kzles [Or Phy~'k C

and FL= ds �9 Springer-Verlag 1986

Phenomenology of Dynamical Symmetry Breaking in QCD Herman J. Munczek Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA

Received 20 May 1986

Abstract. A consistent treatment of the QCD quark propagator and quark-antiquark bound state equ- ations is presented which follows the Nambu and Jona-Lasinio approach to the discussion of chiral symmetry breaking. An expression is obtained for the dynamical, momentum dependent, mass. In the appro- ximation used here the dynamical mass is determined by M 0, its value at zero momentum, and by the strong coupling constant ~s and bare quark mass mo. In the limiting case of no explicit chiral symmetry breaking, i.e., mo = 0, this expression coincides in form with the one obtained by Chang and Chang in their renormalization-group analysis. In this limit chiral symmetry remains broken and we show the explicit appearance of a Nambu-Goldstone pion. A consistent calculation of the pseudoscalar, scalar and vector meson masses gives values ofmo, ~s and Mo well in step with other estimates. This makes possible a calculation off,, the pion decay constant, in reasonable agreement with experiment.

The relative smallness of the pi-meson mass has been, for a long time, the main motivation for the many attempts to describe it as a would-be Nambu- Goldstone boson. The pion's quantum numbers indi- cate that the symmetry to be spontaneously broken is the chiral symmetry of the Lagrangian for the fermions of which the pion is presumably composed, i.e. the up and down quarks of quantum chromodynamics (QCD). Strict chiral symmetry of the Lagrangian forbids an explicit mass term for the fermions. The pion mass, however, even if small, is not zero and this appears to require the inclusion in the QCD Lagran- gian of a small bare, or "current-algebra," mass parameter too. A conventional perturbative treatment built on the (almost) chiral symmetric vacuum would yield a fermion propagator with a mass m(p 2) propor- tional to the current algebra mass rn 0 and which does not support the notion of the pion bound state as a Nambu-Goldstone boson. Specifically, if rno vanishes so does m(p 2) and the chiral symmetry is not broken.

A crucial step towards a better understanding of the spontaneous breaking of chiral symmetry was taken by Nambu and Jona-Lasinio [1] (NJL). Although the interaction chosen for their discussions was nonre- normalizable they showed that, by choosing a per- turbative expansion around a new (broken) vacuum in which the fermions are assumed to have already a mass Mo (not present in the Lagrangian), they could obtain the explicit appearance of a massless pseudoscalar as well as other massive bound states of different quan- tum numbers. The dynamically generated mass Mo was determined by a condition referred to as the "gap equation", by analogy to superconductivity theory. It is essential in the NJL approach that there be consis- tency between the Schwinger-Dyson (SD) equation, used to obtain the fermion propagator, and the Bethe- Salpeter (BS) equation for the bound states. That is, the same renormalization condition (namely, the gap equation) should be used in the treatment of both equations. This has been emphasized by Lane [2] who studied the extension of the NJL approach to QCD with inclusion of renormalization group ideas.

More recently, Chang and Chang [3] have studied the NJL mechanism for the dynamical generation of the mass M o in QCD. They find that M o is a renormalization-group invariant that can be expressed in terms of the invariant QCD cut-off A c, i.e., Mo = Ace 116. In further work [4] they have applied their techniques to the case in which there is in the Lagran- gian an explicit mass term too. They thus obtain an expression for M(p 2) which depends on two independ- ent renormalization-group invariants: Mo = M(0) and Ac. The analytic structure of M is the usual one. However, in the critical limit m o = 0,M vanishes identically for space-like p2 < _ Mg e -= - M~, while for real p2 between - M~ and M if, M varies smoothly between 0 and Me. In this way, dynamical symmetry breaking is maintained when making the continuous transition from a nonchiral to a chiral-symmetric Lagrangian.

The purpose of the work to be discussed here is to study the QCD low energy bound states in the light of

Page 2: Phenomenology of dynamical symmetry breaking in QCD

586

the NJL approach and using some results on dynam- ical mass generation which, although obtained under different approximations, bear in some cases a close connection to those of Reference 4. As mentioned above, the treatment of the Bethe Salpeter bound- state equation should be consistent with that of the Schwinger-Dyson equation for the momentum de- pendent fermion mass M. Instead of trying to extend to the BS equation the same renormalization group methods developed in [3, 4] for the calculation of M, I present in Sect. I an approximation to the SD equation which yields a mass M whose momentum dependence is functionally similar to that obtained by Chang and Chang [4]. In place of their two parameters M o and Ac, this form depends on M o and on a s, an effective strong- coupling constant. For large space-like momentum it has the asymptotic behaviour M ,,~ [ l n ( - p2)]- 1 in- stead of,~ [ln ( - p2)]-a,., where d,, is the anomalous mass dimension [-2, 5]. In the critical case mo= 0, notably, both forms for M(p 2) are identical.

In Sect. II I introduce the BS equation in the approximation compatible with the one used for the SD equation. This permits, as in NJL, the use of the gap equation in order to solve for the bound state's mass eigenvalues. Focusing on d~i quark-antiquark bound states with equal bare mass m o it is found that the pseudoscalar mass is proportional to (m o Mo) 1/2, thus vanishing when the explicit chiral symmetry breaking is turned off, i.e. when mo = 0. The scalar and vector mesons' masses instead remain finite and, in fact, depend very weakly on mo when m o << M 0. As in NJL the scalar is very weakly bound, it's mass is rno ~- 2Me - 2Mo el/2. If the scalar meson is the 6 (980), we have then M o = 297 MeV. The experimental value of the p-meson mass is obtained for ~s ~- 0.71. Finally, the parameter mo= (m, + rod)~2 is determined by the pion mass to be m o = 5.9 MeV. One can see that the range of values for M(p2), which could be called a running constituent mass, as well as the values of C~s and mo are quite in step with standard QCD phenomenology [6].

The determination of all free parameters permits one to undertake, in Sect. III, the calculation off~. This is feasible because the asymptotic form for M(p 2) makes certain long-range integrals possible. Then, depending on the approximation chosen to calculatef~, its value is found to range from 57 MeV to 79 MeV, in reasonable agreement with the experimental value of 93 MeV. Conclusions as well as an Appendix follow.

I. Schwinger-Dyson Equation and Dynamical Constituent Mass

In the Landau gauge and in the one gluon exchange, or ladder, approximation the inverse unrenormalized propagator has the form

s ; ' ( p ) = sJ - M ( f ) . (1)

M(p z) is to be obtained from the corresponding SD equation

H.J. Munczek: Phenomenology of Dynamical Symmetry Breaking in QCD

a s i . 2 1 1 d4 q, M(p 2) =mo - - ~ J M ( q )q2 _ M2(q2)(q p)~

(2)

where as = 92/4~, and g is the gauge coupling constant. If too is zero, a trivial solution to (2) is M = 0. This is also what one would obtain in a standard perturbative approximation since the usual perturbative ground state is built on a basis of massless free fermions. The NJL approach [1] was to assume a different vacuum state in which the fermions already possess a mass Mo. Since their interaction kernel was a constant instead of 1/(q -p)Z, the expression (2) for Mo became a numer- ical relationship: the gap equation.

In what follows I propose to study the properties and phenomenological consequences of the following ap- proximation in the solution of (2). Instead of using some constant value for M inside the integral, as in a perturbative treatment, one can argue that, since there is a pole at q = p in the integrand of (2), a significant contribution will come from that region and therefore it is reasonable to choose M = M(p z) inside the in- tegral. With this soft gluon dominated approximation the gap equation is now

2 ____aS i ~ _ MZ(PZ)(ql _1 p)2 d4 q" M(p 2) = mo -- M(p ) ~ - ~ J q 2

(3) Since M is a constant inside the integrand, the in- tegration can be performed, except for the fact that it has an ultraviolet divergence. To renormalize (3), as well as the BS equation to be considered later, one can use systematically the equation for M ( 0 ) - M o S0 , which can be written as

1 - m o / M o - as i 1 ~ S q2(q2 _ M2~ d4 q" (4)

The integral on the right is the only divergent one that will appear in subsequent calculations. For example, dividing (3) by M(p 2) and subtracting from (4), the resulting integral is convergent and it can be easily performed leading to the following expression for M - M(p 2)

mo~(1 - M/Mo) = M{(1 -- MZ/pZ)ln (1 - p2/M2) ~s

+ In MZ/MZ~ }, (5)

where M 2 = M2o e. Equation (5) is the renormalized version of the gap equation (when mo 4: 0) and gives M as an implicit function of p2 and of the parameters mo, M 0 and ct s. An alternative form is

{ ' (6)

Expression (6) for M is similar in form to the expression given by Chang and Chang [4]. The differences are in

Page 3: Phenomenology of dynamical symmetry breaking in QCD

H.J. Munczek: Phenomenology of Dynamical Symmetry Breaking

the value of the coefficient in front of the logarithm and in the power of the exponent of the bracket ( - 1 instead of - din). The analytic properties are essentially the same; namely, a branch point at M2(p 2) = p2 and a propagator pole at precisely that value ofp 2. For large spacelike p2 the asymptotic behavior is M ~ [ln ( _ p2)] - 1. For mo << Mo it is easy to verify numerically that there is a solution which has a branch point at the value p2 such that MZ(p 2) ---p2. The solution remains positive but steadily decreasing as p 2 ~ _ oe. As mo decreases to zero, p2 reaches M 2 and M vanishes for p2 less than ( - M2). This is the critical case discussed by Chang and Chang. Actually, direct inspection of (5) shows that: either M = 0 (no chiral symmetry brea- king), or M satisfies the gap equation

M 2 p2/M2)(1 -M2/p 2) ~22(1 - = 1, (7)

in which case there is spontaneous chiral symmetry breaking and the results of the approximation intro- duced in this paper coincide exactly with those given in [4]. Notice that there is only one free parameter in (7), namely, Mc = Mo el/2. In the present calculation it is related to the (mass) renormalization of (2), while within the context of a renormalization group analysis [3] Mo can be related to the invariant QCD cut-off.

For calculational purposes it is useful to introduce dimensionless quantities in (7) by defining

M 2 -2WM2 (V )/ c, x-pE/M . (8) Equation (7) can now be written as

/~" = (/~ - x) (u- ~), (9)

or in the parametric form

# = t (m -'), x = #(1 - t). (9a)

As t varies in the range (0, oe), # and x vary in the ranges

/_z

1 .0

l / e ,/~

d c b y / / / / /

- 2 . 0 - - 1 . 0 0 . 0 1 .0

X

Fig. 1. The dimensionless squared dynamical mass p(x) for several values of2 ~ m o rr/(% Mo) as obtained from (10). Curve a corresponds to the critical case 2 = 0. Curve b is for 2 = 0.088, obtained with the values ofmo, M o and c~ s determined here. Curves c and d correspond to 2 = 0.5 and 1, respectively

in QCD 587

(1,0) and (1, - 1) respectively, thus giving the numer- ical relationship between/~ and x. Actually one finds that, with an accuracy of about one percent, p can be expressed in those ranges as a quadratic function of x:

U = #0 "[- l x "-[- (�89 - - ]AO) X2, (9b)

with #o = e - 1. When m o -Y: 0 the implicit dimensionless equation for

/~(x) can be obtained by squaring (6) and it reads

# = e - l ) . 2 2 + l n / ~ + 1-- In 1-- , (10)

where 2 = lrmo/(c~sMo). Figure 1 illustrates the behavior of #(x) for several values of 2 including the critical case 2 = 0. To the right of the branch point line p = x there are no real solutions to (10) and all curves decrease monotonically to zero as x ~ - oe.

II. Bethe-Salpeter Equation and OQ Bound States

Let us consider a possible 6 and d quarks bound state. For the purposes of this work it is sufficient to take their bare masses to be equal. With the same conditions and assumptions on which the SD equation is based, the BS vertex function satisfies the equation

1 o~ s i~7uSr(q+ Zp(k) - 3 rc rc )zp(q)Se(q-)

�9 TvG,~(q - k)d4q, (11)

where q+ = q _+ �89 G. v(q) = (gu~ - (quq~/qE))l/q 2 and SF(q)= [ ~ - M ( q 2 ) ] -~" is the Feynman propagator with the dynamical mass M determined in the previous section. The same approximation that led to (3) is used here, i.e., we set inside the integral m(q2)-~m(k2+) and zp(q)--*Zp(k). It is not the aim here to determine Zp(k) but, instead, to obtain the eigenvalues p2 of the BS equation which give the bound states' squared masses. It is sufficient then to consider the equation for zp(O) = Z,

lc~ s i Z - 3 rc ~2~Tu(q+ + M)Z((I- + M)Tv

. I 1 1 1 qZ+ - M 2 q2_ _ M 2 G,~(q) d4q. (12)

In this equation M -- M(p2/4). The scalar vertex func- tion is the simplest one to solve and I shall start with it to exemplify some of the operations involved and also to provide an estimate of M o.

II .A Scalar Bound State

With Z proportional to the unit matrix, (12) reduces to

1 as i 2 lp2 1 - 3 ~ ~2~(q - - - +M2)

qZ+ _ M E q2_ - M E dEq �9 (13)

Page 4: Phenomenology of dynamical symmetry breaking in QCD

588

Because the term in brackets is even in q any factors odd in q would not contribute. Taking advantage of this, one can rewrite (13) in the form

1 as ~ I 1 d4q %2(M z __ lp2) n q2 _ M 2 q2 ;g

. ss 1- : (14) q2+ M 2 q2_ M 2 q2 J"

Using the gap equation (3) and with mo =0 , one obtains

2%(M2 - �88 M(p2/4)) = 0, (15) n

where Lo is the integral inside the brackets in (14) and is given by

4 {~tan-1 1 In (1 + bl~)}, (16) L~ = 7 b -

where b E = (4M2/p z) - 1. L o does not have a zero, On the other hand for small b 2 we have L o ~ lib. There- fore, the solution to (15) is b = 0 or

�88 = MZ(�88

This is the condition determining the momentum p~ at which the propagator has a pole. Because m o = 0 the corresponding mass is the critical mass, i.e., p2 = 4M z. The natural candidate for a dO scalar bound state is the a-meson with mass m a = 980 MeV. This identification yields

M~ = �89 = 490 MeV, Mo = 297 MeV. (17)

As in NJL the scalar has zero binding energy. We shall see later that m o ~- 5.9 MeV << Mo, and in this circum- stance the corrections to the determination ofrna are, as in NJL, also very small.

II.B Vector Bound State

As in the scalar case, the solution of the BS equation for spin one depends weakly on m o when mo << Mo and the calculations are performed at m o =0. The vector meson vertex function for a polarization eu can be written as

z=A~+Br ~-p-- 0. (18)

Inserting this form in (A.1) of the Appendix one obtains two coupled linear homogeneous equations for the coefficients A and B. Their form is not very illuminating so the details are presented in the Appendix. The determinant condition for the system of equations gives the mass squared eigenvalue p2 in terms of Mo and a s only. There is no extra dependence on a cutoffas in the NJL treatment. The difference is obviously due to the use, here, of a renormalizable interaction.

Adopting the critical case value Mc = 490 MeV, (10) yields 2M(m~/4)= 834MeV for mo=770MeV. This

H.J. Munczek: Phenomenology of Dynamical Symmetry Breaking in QCD

can be interpreted as a binding energy of 64 MeV. This is to be compared with a binding energy of 210 MeV, which we would obtain if we would use a constant mass in the propagator (a simpler, but still self-consistent, approach than the one we are using here). Using (A. 15) and (A. 16) as well as the integrals given in the Appendix one finds that p2 = m 2 is obtained for a s = 0.71.

II.C Pseudoscatar Bound State

The pseudoscalar vertex has the form

Z= A7~ + BTsP, (19)

which when replaced in (A.1) leads to the equations

A((mon/Mas) - (pZLo/2)) + B(p2MLo) = 0, (20)

A(M(L o + 2Lo) ) - B ( 3n + L - 4 L \ as

+ (M 2 + �88 0 + 250) ) = 0, (21) l

where L, L, L o, L o are integrals given in the Appendix. They all have finite values at p= = 0.

The appearance of mo in (20) is due to the systematic use of the gap equation (3) in its derivation. If mo= 0 there are two solutions: one is p2 = 0 which signals the presence of a Nambu-Goldstone boson. The other solution is A = 2 MB. If this is used in (21) the equation becomes

4n/a s = 3/4 + ( m 2 - lpZ)L, 0.

To satisfy this would require as-~ 4n instead of a s = 0.71. Therefore, a massless pion is the only solution.

For the case m o r 0, we can approximate by drop- ping from (20) all terms of order p4/M4 or higher and also terms of order mop2/M 3. The values of the integrals at p2 = 0 are given in the Appendix. We have then from (20)

A ( n m o l m 2 ) m 2 kasMo 2 ~ + BMo7 = 0, (22)

and from (21)

We can finally obtain from (22) and (23)

2 n / 2 n + as ) (24) m 2=moMo --

With rn== 140MeV and the values M o = 2 9 7 M e V and as = 0.71 found previously we find that the average mass of the up and down quarks is

m o = 5.9 MeV, (25)

a value well within the range of other estimates [6, 7].

Page 5: Phenomenology of dynamical symmetry breaking in QCD

H.J. Munczek: Phenomenology of Dynamical Symmetry Breaking in QCD

IlL Calculation of the Pion Decay Constant

The pion decay constantf~ has the experimental value of 93 MeV. There are several formulas which provide approximate theoretical calculations of f~ and that express it in terms of the quantities calculated in Sect. I and II. Those formulas involve long range integrals which sometimes are possible to perform only in the critical case mo = 0,

A formula whose derivation [8] is based on the Goldberger-Treiman relation at the quark level ex- presses f~ as

cO

f 2 = ~ M . ( [ z M ( - z)/(z + M2( - z))2]dz, t41~-

(26)

where Mp is the dynamical mass at the pole position. With the asymptotic behavior obtained previously for M the integral is convergent only in the critical case m o = 0. In that case we have also Mp = Mc = 490 MeV. The integral can be performed numerically yielding cO

[ z M ( - z)/(z + M 2 ( - - z))]dz " 0.345 Mc, (27) o

so we have from (26)

f , -~ 0.162 Mc = 79 MeV. (28)

Another approximate formula for f~ is based on a generalization of perturbation theory [9] which re- spects the chiral Ward identities and reads

3 cO

z d z (z + M 2 ( - z)) 2" (29)

A numerical integration using (10) gives the critical limit result

f , -~ 0.116 Mc = 57 MeV. (30)

The form of (6) for M permits one to perform the calculation in (29) even when m 0 r 0, since the as- ymptotic behavior of M ( - z ) makes the integrals convergent. Using the values of too, Mc and ct s given previously, a numerical integration yields*

f~ ~- 0.127 Mc = 63 MeV, (31)

an improvement over the critical limit value. The difference is due to the fact that M, which is positive in the integration region, increases when m o > 0.

An important factor in possible further improve- ments in the calculation of f~ is the asymptotic behavior of M, which would be altered by a re- normalization group approach. This would also alter its dependence on the other parameters, which for

* Here the value Mc = 498 MeV has been used. This is obtained from M (p~)=p~ is (6) when m o = 5.9 MeV and when the branch point 2 2 2

2 2 = (980 MeV) 2 as estimated in Sect. IIA located at 4pc = mo

589

consistency would have to be determined from a renormalization group improved BS equation. Such a project is under study.

IV. Conclusions

The dynamical constituent mass M, in the approxim- ation developed in Sect. I, appears to be a useful tool in the discussion of dynamical chiral symmetry breaking. An intriguing point is its close similarity in form to that of the M obtained by Chang and Chang in their renormalization group analysis approach.

In the form presented here, the dynamical mass is particularly suitable to verify the explicit existence of the Nambu-Golds tone boson. By using the NJL approach to solve the bound state equations, M is completely determined in terms of the 6,p and ~ meson masses. The quite acceptable values obtained for Mo, mo and a s indicate that it is worthwhile to continue the study of the applications of the present approach. One such application is the calculation of f~ in Sect. III. The reasonable values obtained there suggest a possible avenue for improvement. That would be the modification of the asymptotic behavior of M as given, for example, by the renormalization group methods of [3]. How those methods extend to the BS equation is a question, though, that has to be solved in order to provide a self-consistent phenomenological discussion.

Acknowledgements. I wish to thank John Ralston for many useful discussions and Vince Reinert for the computer graphics. This research was supported by DOE Contract No. DE-FG02- 85ER40214.

Appendix

A.I Bethe-Salpeter Equation and Related Integrals

Because the term in brackets is even in q, (12) for the vertex can be written as (M = M(p2/4))

37~ �9 Z = 7uT~ZTpTuL,~ + 7.(�89 + M)Z( - �89 + M)Lo

~s

- Z L - Y,(�89 + M)Z( - �89 + M)Tp M~#, (A.1)

where, with q+=q_+�89 L,p,M,p,L o,L a r e the integrals:

i 2 L~a = ~ [(q+ -- M2)(q 2_ - M2)q 2] -a q~qad4q

- P~PP r (A.2) - g , o L 1 - p2 ~2,

i 2 L = ~7[(q+ - MZ)(q 2_ - M2)q2] - lqZd4q , (A.3)

Page 6: Phenomenology of dynamical symmetry breaking in QCD

590

i 2 i q~ q~ M~p=~2~[(q+ - M2)(q 2- - M2)q2] - q2

P~P~ ~, =-- g~pm I + ~-1v12, (A.4)

L o = ~2~[(q2+ -- M 2 ) ( q 2 -M2)q2]-Xd4q. (A.5)

The divergent integrals L 1 and L2 can be expressed in terms of L and another integral L as

4L1 + L 2 = L,

L 1 + L 2 = L

i 2 = ~2I [(q+ -- M2)(q z- - M2)q 2] -1

(p. q)2 ,4 " )5 a q. (a.6)

M~ and M 2 are obtained in terms of L o and another integral L o as

4M1 + M2 = L o

Mx + M 2 = L o

= ; ~ [(q % - M2)(q 2_ - M2)q 2]

(p.q)2 ,4 �9 p ~ a q. (A.7)

The convergent integrals L o and Lo can be calculated by using Feynman parametrization and (in the case of Lo) partial fraction decomposition. With the notation

4M 2 b 2 --- - - 1, (A.8) p2

one obtains, for b 2 > O,

4 2 Lo ~ { ~ t a n -11 = ~ -

For p2 = 0, one has L o = 1/M~ and L o = 1/(4Mg). To eliminate the divergences in L and L one can use

the SD gap equation (3), which can be written as

~ ( m~ 1) i 1 d4 (A.11) ~skM- =~S(q2+-M2)q2 q. L and L can then be expressed in terms o fL o and L o by inserting in their integrals the identity

2 1 2 l t = [ ( q 2 - - M 2 ) + ( M - g P ) + P ' q ] q 2 - (A.12)

The term (p.q)(1/q 2) does not contribute and one obtains after some standard integrations

H.J. Munczek: Phenomenology of Dynamical Symmetry Breaking in QCD

) L = % \ M 1 +(M2-�88 (A.13)

I n ( m o ) L = ~ \ ~ - - 1 +~6+ �88 2- �88 o. (a.14)

It follows from the equations above that all the integrals can be expressed in terms of just L o and L o, given by (A.9) and (A.10).

A.2 Vector Meson Equations

With g = Ae + Bee inserted in (A.1) one can obtain

3n - - - A = A { 4 L 1 + 2L 2 - L

as

+ (M 2 + �88 + M2 - 2Lo)}

- Bp2M(2M~ + M2 - 2Lo), (A.15)

3n B = B { - L + ( M 2 +�88 2 } - A M M 2 . (a.16)

as

Neglecting terms of order mo/M, using (9b) to calculate M(m2/4) and using the expressions for the integrals given before one finds numerically that p2 = rap2 for a s=0.71. There is also a value Cts=20.5 which yields the same result. This value is rejected in view of other phenomenological estimates of a s and also because it would lead to a large (negative) m o in (24), thus violating the basic assumptions of the present analysis�9

Regarding the axial vector meson, an analysis similar to the one above shows that there is no real value of p2 that would satisfy its BS equation�9 This is similar to the result of NJL [1].

R e f e r e n c e s

1. Y. Nambu, G. Jona-Lasinio: Phys. Rev. 122, 345 (1961); 124, 246 (1961)

2. K. Lane: Phys. Rev. D10, 2605 (1974) 3. L.N. Chang, N.P. Chang: Phys. Rev. D29, 312 (1984); N.P. Chang,

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