Asymptotic chiral symmetry, meson masses and decay constants

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Pramana, Vol. 16, No. 5, May 1981, pp. 417-423 © Printed in India.

Asymptotic chiral symmetry, meson masses and decay constants

A L A H I R I and V P G A U T A M

Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India

MS received 13 December 1980; revised 7 March 1981

Abstract. Weinberg's spectral function sum rules are examined to study the axial vector mass spectrum at the level of SU(4). New mass relations and general mass constraints are derived to predict the masses of the charmed axial vector mesons and the I=0 pseudoscalar decay constants.

Keywords. Chiral symmetrs; spectral function; sum rules; pseudoscalar mesons;

axial vector mesons.

1. Introduction

Over the years, Weinberg's two spectral function sum rules (Weinberg 1967) have been successfully employed to ascertain the extent o f symmetry violations o f hadrons.

While the success of these sum rules at the level o f SU(3) is well-known (Das e t al 1967), at the level o f SU(4) or even SU(5), where the breaking o f symmetry is ex- pected to be considerably larger, reasonable results have been obtained (Dicus 1975;

Ueda 1976; Pham et al 1976; Gautam et al 1979) in predicting the masses o f new vector mesons (belonging to the qJ and ~ multiple0 by modifying the 2nd sum rule to take into account higher order symmetry breaking effects.

In this paper, we examine the SU(4) axial vector mass spectrum along with their vector counterparts and derive new mass constraints between D(1285), E(1420) and V A (the o5 axial vector analog of , ) mesons; in addition we also obtain estimates o f the decay constants o f I = 0 pseudoscalar mesons. Throughout this paper, we shall assume, owing to the fact that the SU(6) mass relation

rn ~ A~ ~- m~ = 2 m ~

is fairly well satisfied by A 1, E and Q1 (thus suggesting that these are members o f 1 ÷÷

family) (for a discussion on this point see Caffarelli and Kang 1976) that Az, Qz, D(1285), E(1420), D**, F** along with ~F A belong to the (15) -[- (1) representation o f SU(4), (D** and F * * being the charmed and charmed-strange axial vector mesons).

In § 2, we present the formalism where we write down Weinberg's 1st and the modified second spectral function sum rules. Section 3 deals with the mass relations obtained from these sum rules while numerical results are given in § 4. Finally a brief summary o f our work is given in the concluding § 5.

P.--4

417

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418 A Lahiri and V P Gautam 2. Formalism

Assuming in analogy with the SU(3) theory, the validity of asymptotic symmetry, Weinberg's first sum rule can be written as

o 0 o 0

m 2 - - f d m 2 [ --~ + p o ' ~ ( m 2) =C[3uq-(k--1)3,oSjo ], t. m

o o (1)

where p~'J is the spin 1 spectral function of the SU(4) vector currents V i p~J and Jo' j are the spin 1 and spirt 0 analog of SU(4) axial vector currents A i and C and k are constants independent of the SU(4) index i. The second sum rule in the exact SU(4) limit

oo t~o

dm 2 piv, J (m s) = f dm s p~i (m s) = constant

0 0

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has however to be modified to take into account the effects of symmetry-breaking and provide consistency with the first sum rule (Das et al 1967; Sakurai 1967). We parametrise this breaking at the SU(4) level by writing

c o c o

f am s p~,j (m ~) = f dm s pi~j (m 2) = C [M3 o q- 1~, o ~Jo -'~ N(dsu-k-adls,j)], (3)

o o

where SU(4) invariant parameters M, N, and a are to be determined from experimental inputs.

Writing the matrix elements of SU(4) currents V~ and A i as _{/ z}

x/2-~o (01V~ Iw(q)) = g~o ~,(q); ~¢/2~o

(OIA~lO(q))

^{= g~D}

%@

V'2qo (01

V~lKS(q) > = gk

e ( q ) ; V'~qo <0{A~ [zr(q)> --- if~ qt*' etc. (4) and restricting ourselves to ground state mesons only, we obtain from equation (1) the set of relations

2 2 ~ 2

= = gF*/mF* 1

C g~/mp g R , / m K , = g~o,/m~, = 2 2 _~ g~/m~A _~_ f ~ ,

~. 2 __ z 2 2 gF**/mF** d- f ~ ,

= gQJmQ~ -~ f ~ -- gD**/mD** q- f ~ = 2 2

(5)

where Q~, D**, F** are the axial-vector counterparts of the vector mesons K*, D*

and F* and

(g~)~

(g~,)2

X--"

1

c = 2., v m~ =~V

^m~,

='X~V (g~)s _m~'

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Chiral symmetry, meson masses and decay constants 419

Z~'~A (gsA )2 m~ -q- ~ P (fs)~ = ~ A - ~ A -b ~ p (f~)2, (gX~)z

^,

l f g , (g~)~

= X'L/_,A ~'A + ~z' (fD"], (6)

0 8 t r l 5 ~ t715 oO 0 0

~"

^{_ _} ^_

N~ 6V6V

^{~ "}

^gvgv

and 0 = ~

^{6 v} ^{6 v} ^{- -} ^{Z T} ^,

V m~ V Z-..ev m2 V ~ v m v

= ,,,- ~ASA /_.,A m~ + e "f" f .

~ =

/_,,4 m] Jr/_,e " ~ '

X ^{a m•} ⁺ ^{pro s} ⁽⁷⁾

where Z V denotes summation over o~, ~ and ~ mesons, 2: A over D, E and ~A mesons and Ze over ~, ~' and ~¢ mesons, ~ being the c~ pseudosealar analog of ~.

The modified second sum rule given by (3) leads to

and

gp~ =g~A, = C[M+ U(~7+ ~)]

2~/3 ~-6 (8)

~ . = g ~ . . = C M - F -- ,

2 ~

[ ( '

g~, =g~-- = C M + N ,V/~**

Z V (gsV)2 = ,A(g~)2 = C M + N [ ( ' - - ~ +o)] ~-~ = g s ~,

~ v(g~, )2

= / _ , ~ , ~ A

~ ,..as)~

=

C [ M __ J ~2 a N ] = g]~,

~,~g>~. : ~ ~.~;-

: ~ r M + ,j --- co,

V gsV g~ = ~ A gSA g~ = CN V-~ (

⁼

fl)' (9)

• ^{v g15} ^~ ^CN

=CN(

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420 A Lahiri and V P Gautam

In the following section, we shall use the set of relations (5) - - (9) to obtain

mass

constraint equations.

3. Mass rules

Before deriving the mass rules, let us note that the parameter C appearing in (1) and (5) can be expressed as

2

C -- gPz -- 2 ~ z,

m p

using the

2 = 2 m~ f~. Thus equations (8) which take the form 1966) relation gp

c [

A~ c - f ~ M + N v'5 - ~ '

51]

mbl = C.Z f + N -- _{2 V~} -}- ;

- - N 1 a

m D * * C----f~

m ~ , , - c C f ~ [ M -+- N ( - - - ~ - - ~ )3;

(1o) KSRF (Kawarabayashi and Suzuki 1966; Riazuddin and Fayyazuddin

by using (5), become

[ (' 5)]

(2 *--fB/f~) rob** -- 2 M + N 2~/3*

2 ~ [ ( 1

. ~ ) ]

( 2 - - f ~ / f ~ ) m ~ * * ---2 M + N - - ~ - - .

Equations (6) and (12) then imply

m~ 1 = 2

m 2

_p'

2 2 Qt

(2 --f~/f~) m s --- 2 m~,,

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(12)

(13a)

(13b)

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Chiral symmetry, meson masses and decay constants 421

2 2 = m ~ , (13c)

(2 - - f ~ / f~) rob** 2

(2 - - f ~ / f ~ ) m~F,, = 2 m~F ,, (13d)

where (I 3a) is the well-known Weinberg (1967) mass relation.

It is interesting to note that (13b) to (13d) simplify to the following relations mb~ = 2 m~¢,; m~9** = 2 m b , ; m~v** = 2 m~,, (14) if we assume that all the decay constants f,,, fK' fD, f F are equal. Equation (14) may be looked upon as a generalized version of the Weinberg relation (13a).

Equation (14) also leads to an equal spacing sum rule

- - m z = m ~ * * 2 ( 1 5 )

m.41 ^{Q 1} ^{- -} mF**'

if we make use of the analogous mass rule for the vector mesons

- - m ~ ¢ , = r o b , - - m ~ - , , ( 1 6 )

mp

whioh follows from (3).

In addition, we also obtain for axial vector mesons D, E and ~ a , the following mass constraint equation by eliminating the coupling constants occurring in (6), (7) and (9)

tmZ r z _~ ~,'~fl)2 (m2yf215 - - ~n) -[- (m2Yfg,~n + fl)z (m2yf]~ _ g2) Y JO, 8 _~. (m Yf15, o + ~ 2 x/'~aS)2(m~yj~8 _{- -} g~) = ( m y 3 2 f15 g15 ) 2

(m~y f 2 ° _ g2o) (m y f s ~ ~ - - g~) _{- -}2 (m y f o , s 2 2 -]- V' 35) ( m y f8,15 2 -}- fl)

( m y fls, O -{-

for Y = D, E and TA respectively, where s : = c - ( s ¢ ) ' ; s ; = c -

p p

(17a, b, c)

f g = c x - ( f o),,

p

p p p

Numerical solutions of (14) and (17) are now taken up in the following section.

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422 A Lahiri and V P Gautam 4. Numerical calculations

Using the experimental masses for the vector mesons p, K*, D* and F* as inputs, (14) gives

MA---1.1GeV; m Q = l . 2 6 G e V ; m D * * = 2 . 8 4 G e V ; m F * * = 3 . 0 3 G e V . (19) While the experimental numbers for mo** and mE** are awaited, o u r prediction f o r the masses o f A 1 and Qx mesons are in good agreement with their experimental values o f 1.1 GeV* and 1.28 GeV respectively.

In o r d e r to solve the t h r e e equations given by (17), we first o f all note that these relations involve five unknown parameters viz. k, l, fs, f15 a n d f o . However, o u r experience with the vector mesons shows (Ueda 1976) that it is reasonable to assume the validity o f U(4) symmetry: k = l , l = 0 . Also ~ may be t a k e n as a pure octet state with f,~ = f , (Gell-Mann et al 1968). This allows us to solve f o r f~5 and f0 from (17) by taking the experimental masses f r o m any one o f the ( I = 0 , J P = I +) set o f mesons viz. (D, ~FA) or (E, TA) a.s inputs. ** By expressing fs, fl~ and f0 in terms o f the decay constants o f the physical states ~7, ~7' and ~ (keeping in view t h a t % is an almost pure ck pseudoscalar state) as

A = A ; ix5 = ½(f ' - f o = (20)

o u r results are

f * t ' / f ~ --- 1.6; f * t c / f ~ --- 0.8,

corresponding to the set m D = 1.285 GeV, r a t a --- 3.51 G e V a n d f*t'/f~ = 1.1; f ~ c / f i r = 0 " 4 ,

for the other set m E = 1.42 GeV; m~F A = 3.51 GeV.

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5. Conclusions

We have recorded in this paper the results obtained f o r axial vector mesons f r o m Weinberg's spectral function sum rules when extended to SU(4). In particular, we have obtained generalized mass relations o f the f o r m (mA) " = 2(m°) 2, f o r I = 1 and 1/2 axial vector and vector mesons respectively. T h e masses obtained o f the c h a r m e d and charmed-strange axial vector mesons will provide a useful guide to the search for this class o f mesons. F o r the isoscalars, we have obtained generalized mass constraint relations which relate the pseudoscalar decay constants to the masses o f

*It may be remarked that the new experimental data indicate that the Al-resonance is wider and may be at a higher mass than previously thought (Particle Data Table 1980).

**The mass of the WA meson is around 3.51 GeV, see Particle Data Table (1980).

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Chiral symmetry, meson masses and decay constants 423

I = 0 a x i a l v e c t o r m e s o n s . T a k i n g (m D, m T a ) o r (m E, raTA ) as i n p u t s , we h a v e a l s o o b t a i n e d e s t i m a t e s o f t h e d e c a y c o n s t a n t s f,7' a n d f , lc, w h i c h a r e in t u n e w i t h o t h e r t h e o r e t i c a l e x p e c t a t i o n s .

References

Caffarelli R V and Kang K 1976 Phys. Lett. B65 386

Das T, Mathur V S and Okubo S 1967 Phys. Rev. Lett. 19 470 Dicus D A 1975 Phys. Rev. Lett. 34 846

Gautam V P, Bagchi B and Nandy A 1979 J. Phys. G5 885 Gell-Mann M, Oakes R J and Renner B 1968 Phys. Rev. 175 2185 Kawarabayashi K and Suzuki M 1966 Phys. Rev. Lett. 16 255 Particle Data Table 1980 Rev. Mod. Phys. 52 No. 2 Pham X Y, Gauron P and Kang K 1976 Phys. Rev. D14 794 Riazuddin and Fayyazuddin 1966 Phys. Rev. 147 1071 Sakurai J J 1967 Phys. Rev. Lett. 19 803

Ueda Y 1976 Phys. Rev. D14 788

Weinberg S 1967 Phys. Rev. Lett. 18 507

Asymptotic chiral symmetry, meson masses and decay constants