Masses of Spin-1 Mesons and Dynamical Chiral Symmetry Breaking

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Masses of spin-1 mesons and dynamical chiral symmetry

breaking

Bing An Li

DepartmentofPhysics and Astronomy, University of Kentucky

Lexington, KY 40506, USA

Abstract

In terms of an e ectivechiral theory of mesons it is shown quantitatively that in

the limit of m = 0, the masses of , ! , K (892), m , a , f (1286), K (1400), and

q 1 1 1

f (1510) mesons originate from dynamical chiral symmetry breaking.

1 1

The origin of the masses of hadrons is always one of the most imp ortant topics in hadron

dynamics. The mass of a hadron is asso ciated with chiral symmetry breaking. There are

implicit chiral symmetry breaking from the current quark masses and dynamical chiral sym-

metry breaking. The Nambu-Jona-Lasinio mo del[1], starting from massless quark, has non-

vanishing quark condensate which means dynamical chiral symmetry breaking. On the other

hand, dynamical chiral symmetry breaking in Quantum Chromo dynamics(QC D ) has b een

studied extensively[2]. QC D has dynamical chiral symmetry breaking. It is well known that

in the chiral limit, the o ctet pseudoscalar mesons are Goldstone mesons and massless. From

the theory of chiral symmetry breaking Gell-Mann, Oakes, and Renner[3] have obtained

m = (m + m ) < 0j j0 >: (1)

u d

The smallness of the masses of the current quarks leads to the light pseudoscalar mesons.

The heavy meson is asso ciated with U (1) problem[4]. Why the mass of the meson is

much heavier than pion's mass? In this letter we try to nd the origin of the meson's mass.

In Ref.[5] wehave prop osed an e ectivechiral theory of mesons(pions, , , ! , a (1260), and

f (1286)). In the case of two avors the Lagrangian of this theory is constructed by using

U (2) U (2) chiral symmetry and the minimum coupling principle

L R

L = (x)(i @ + v + a mu(x)) (x)

m ( + ! ! + a a + f f ) (x)M (x); (2) +

i i

2 2

i i

where a = a + f , v = + ! , and u = expfi ( + )g, m is a parameter and M

i i 5 i i

is the quark mass matrix. In Eq.(2) u can b e written as

1 1

u = (1 + )U + (1 + )U ; (3)

5 5

2 2

where U = expfi( + )g. The intro duction of the couplings b etween the pseudoscalar

i i

mesons and the quarks is based on the formalism of the nonlinear mo del. The Vector

Meson Dominance(VMD) is a natural result of this theory,Weinb erg's rst sum rule is

derived from this theory analytically. A uni ed study of the pro cesses of normal parity and

abnormal parity is presented by this theory. Most theoretical results agree with the data

within ab out 10%. Besides the success in the phenomenology of mesons, this theory has

dynamical chiral symmetry breaking(Eq.(124) of Ref.[5])

3 2

); (4) < 0j (x) (x)j0 >=3m g (1 +

2 2

2 g

where g is the universal coupling constant and de ned as

2 2

f f

g = + ; (5)

2 2

m 6m

2 2 2

where f is the pion decay constant and f = 186MeV , m = m =g . Eq.(4) means that the

parameter m of the Lagrangia n(Eq.(2)) is the indication of the dynamical chiral symmetry

breaking. Use of the Eq.(4) leads to the mass formula of pion(Eq.(1))[5]. In the chiral limit,

there are three parameters: cuto , m, and m and g is determined by the ratio =m(see

Eq.(129) of Ref.[5]). The purp ose of this letter is to illustrate that in the limit of m =0,the

masses of spin-one mesons are resulted by dynamical chiral symmetry breaking and m can

b e determined theoretically and is no longer an input. The intro duction of the mass terms

of the spin-1 mesons to the Lagrangian(Eq.(2)) is necessary. Otherwise, this theory cannot

be well de ned. For example, without these mass terms the mixing b etween the axial-vector

meson and the corresp onding pseudoscalar meson cannot b e erased, the physical axial-vector

and pseudoscalar elds cannot b e de ned. On the other hand, in the e ective Lagrangian

of meson elds derived from Eq.(2), except for the kinetic terms of the spin-1 mesons, the

spin-1 elds only app ear in the covariant derivatives(see Eq.(13) of Ref.[5])

D U = @ U i[v ;U]+ifa ;Ug:

Because the a elds are in the anticommutator, a sp ontaneous symmetry breaking mecha-

nism leads to the mass di erence b etween the axial-vector and vector mesons. However, the

vector elds are in the commutator, there is no way to generate a mass term for the vector

meson in this theory. Therefore, the mass terms of spin-1 mesons must b e intro duced.

In QC D , in principle, the mass of the meson should b e determined by a dynamical

equation of b ound state. In this theory KSFR sum rule[6]

g = f f (6)

can b e taken as the equation used to determine m . This sum rule is obtained by using

PCAC and current algebra in the limit of p ! 0. g is the coupling constantof and

f is the coupling constantof!, which is de ned in the limit of p ! 0. In Ref.[5]

KSFR sum rule is satis ed numerically. On the other hand, this sum rule can b e derived

analytically from this theory.From Eq.(2) the currents are found to b e

i i

V = ; A = :

2 2

These currents observe current algebra. Without the quark mass term the Lagrangian (Eq.(2))

is global U (2) U (2) chiral symmetric and b oth the isovector axial-vector current are con-

L R

served. The mass term of the pion is generated from the quark mass term[7]

TrM d ps (x; p); (7) (x)M (x)= iT r M s (x; x)=

F F

(2 )

where the quark propagator satis es the equation[5]

f (i@ + p + v + a ) mu M gs (x; p)=1: (8)

5 F

Using the solution of Eq.(8)[5] and to the leading order in quark mass expansion, we obtain

M M = m + ; (9)

i i

where m is expressed by Eq.(1). With the pion mass term PCAC is derived from the

Lagrangian(Eq.(2))

i 2 i

@ A = m f :

2 5

Therefore, the KSFR sum rule can b e derived from this theory in the same way as in Ref.[6].

It is necessary to p oint out that due to the limit of p ! 0, the coupling constant f of

Eq.(6) is de ned in nonphysical region. However, KSFR sum rule agrees with data within

10%. This fact means that in the limit of p ! 0, f is very close to the physical one. The

e ectivechiral theory[5] provides an explanation to this prop ertyoff . The expression of

f determined in Ref.[5] (see Eq.(48) of [5]) shows a weak dep endence of f on the pion

momentum (in the reasonable range of the coupling constant g, the contribution of the terms

which are prop ortional to the pion momentum is less than ab out 10%). In Ref.[5] g =0:35

is chosen and

f = ; (10)

which is indep endentof p . g is determined to b e[5]

g = gm : (11)

Substituting Eqs.(10),(11) into the KSFR sum rule it is found

: (12) m =2

Using Eq.(5), we obtain

2 2

=6m : (13) m

Therefore, the mass of meson is no longer an input and is determined by the parameter

m of Eq.(2) completely. In the limit of m = 0, there are only two parameters in the

q 6

Lagrangian(Eq.(2)), which are cuto and m. Using Eq.(4), we obtain

2 3 2

2 2

3 3

m =6<0j j0 > (3g + : (14) )

In the limit of m = 0, the mass of meson originates from dynamical chiral symmetry

breaking. Combining Eqs.(5),(13), we obtain

2 2 3 3

m = =0:094GeV ; < 0j j0 >= (0:247) GeV : (15)

The mass of meson is determined to b e

m =0:751GeV : (16)

It is only 2% away from the exp erimental value of 0.77GeV. The numerical value of m is

slightly di erent from the one presented in Ref.[5]. The reason is that in Ref.[5] the physical

value of m is taken as input. In the limit of m =0(q =u; d; s), we should have

m = m = m = m : (17)

K (892)

Therefore, in the limit of m = 0, the masses of the four low lying vector mesons originate

from dynamical chiral symmetry breaking. From Eqs.(4),(12),(13) it is found that

2 2

2 2 2 2 2

3 3

) =3g m =3g <0j j0 > (3g + : (18) f

The pion decay constant is the result of dynamical chiral symmetry breaking to o. Therefore,

in the limit of m ! 0, the decay constants of the o ctet pseudoscalar mesons originate from

q 7

dynamical chiral symmetry breaking. Using Eqs.(1),(14),and (18) we obtain

(g + )

m m + m

u d

= : (19)

2 1

3 3

3 g < 0j j0 >

In Ref.[5] the mass relations have b een found

1 1

2 2 2 2 2 2

(1 )m =6m +m ; (1 )m == 6m + m ; (20)

a f !

2 2 2 2

2 g 2 g

where m is the mass of the meson a (1260) and m is the mass of the meson f (1286). In this

a 1 f 1

theory m = m , the theory predicts m = m . Using Eq.(13), the mass relations(Eq.(20))

! f a

are rewritten as

2 2 2

m = m =2m ; (21)

f a

1 1

2 2

where g = m g (1 ) determined in Ref.[5]. This relation is di erent from Wein-

a 2 2

2 2 g

b erg's second sum rule[8], as p ointed in Ref.[9], it is the result of Weinb erg's rst sum rule[8]

and KSFR sum rule. On the other hand, the relation b etween m and the quark condensate

is established by using Eq.(14)

3 2 g 2

2 2 2

3 3

(3g + ) : (22) m = m =12 <0j j0 >

f a

2 2

g 2

In the limit of m = 0, the masses of a (1260) and f (1286) are resulted in dynamical chiral

q 1 1

symmetry breaking. Using g =0:35, we obtain m = m = 1388MeV and the exp erimental

f a

data are m = 1230 40MeV and m = 1282 2MeV . The deviations are ab out 10%.

a f 8

The mass relations have b een obtained in Ref.[10] in which the theory is generalized to

include the strange quark

1 1

2 2 2 2 2 2

(1 )m =6m +m ; (1 )m =6m +m : (23)

K (1400) K (892) f (1510)

1 1

2 2 2 2

2 g 2 g

Use of Eq.(13) leads to

2 2

g g

2 a 2 2 2 a 2 2

m = (m + m ); m = (m + m ): (24)

K (1400) K (892) f (1510)

1 1

2 2

g g

Substituting Eq.(21) into Eq.(24) we obtain

2 2

m m

K (892)

a a

2 2

m = (1 + (1 + ); m = ): (25)

K (1400) f (1510)

1 1

2 2

2 m 2 m

The deviations of these mass relations from the exp erimental values are ab out 3%. By using

2 2

Eq.(14), the dep endences of m and m on the quark condensate are obtained

K (1400) f (1510)

1 1

2 2

2 g 2 3 g

2 a 2 a 2 2

3 3

) m =12 <0j j0 > (3g + + (m m )

K (1400) K (892)

2 2 2

g 2 g

2 2

2 2 g 3 g

a a

2 2 2 2

3 3

<0j j0 > ) (m m ): (26) m =12 (3g + +

f (1510)

2 2 2

g 2 g

2 2 2 2

It is well known that the mass di erences of m m and m m are prop ortional

K (892)

to the masses of quarks. From Eqs.(22),(26) we conclude that in the limit of m = 0, the

masses of the four axial-vector mesons originate from dynamical chiral symmetry breaking.

To conclude, in the limit of m ! 0, in the e ectivechiral theory the masses of the eight

spin-one mesons and the decay constants of the o ctet pseudoscalar mesons originate from

dynamical chiral symmetry breaking. Three new mass relations(Eqs.(21), (24)) are found. 9

This research is partially supp orted by DOE grant DE-91ER75661.

References

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J.E.Mondula, Nucl. Phys. B199, 168(1982).

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[5] Bing An Li, U (2) U (2) Chiral Theory of Mesons, to app ear in 52D No.9(1995).

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[6] K.Kawarabayashi and M.Suzuki, Phys.Rev.Lett., 16 255,(1966); Riazudin and

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[7] Bing An LI, Phys.Rev., D50,2243(1994).

[8] S.Weinb erg, Phys.Rev.Lett., 17, 616(1966).

[9] Bing An Li, Talk presented at the International Europhysics Conference on High Energy

Physics(HEP95), July 27-Aug.2, Brussels, Belgium.

[10] Bing An Li, U (3) U (3) Chiral Theory of Mesons, to app ear in 52D, No.9(1995).

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