Masses of light quarks in quantum chromodynamics

Nuclear Physics B193 (1981) 368-380 © North-Holland Publishing Company

MASSES OF LIGHT QUARKS IN QUANTUM CHROMODYNAMICS* W. HUBSCHMID and S. MALLIK t Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

Received 2 January 1981 We try to determine light quark masses by considering sum rules for the vacuum expectation value of the time-ordered correlation function of two divergences of the axial vector current. The evaluation is carried out at momenta high enough for the non-perturbative contributions to be negligible. We find that the average mass of the up and down quark at a momentum of 1 GeV lies between 3.3 and 7.9 MeV while that for the strange quark lies between 84 and 212 MeV. The ranges of values reflect predominantly the uncertainty in the absorptive part in the low energy region (~<1.7GeV).

1. Introduction T h e r e n o r m a l i s e d ( s c a l e - d e p e n d e n t ) masses of q u a r k s are essential p a r a m e t e r s in describing the h a d r o n s p e c t r u m a n d weak i n t e r a c t i o n t h e o r y a n d serve to c o n s t r a i n the g r a n d unified models. W h i l e the study of heavy q u a r k - a n t i q u a r k systems provides a m e a n s of d e t e r m i n i n g the masses of these quarks, a d e t e r m i n a t i o n of the light q u a r k masses m u s t be indirect, since they are, in general, n o t m a n i f e s t e d in o b s e r v a b l e d y n a m i c a l effects. T h e first d e t e r m i n a t i o n of the i n d i v i d u a l light q u a r k masses, a s s u m i n g SU(6) s y m m e t r y for the m e s o n wave functions, was carried out by L e u t w y l e r [1]. His result for the average mass of the up a n d d o w n q u a r k is 5.4 M e V , while the mass for the s t r a n g e q u a r k lies b e t w e e n 125 a n d 150 MeV. T h e p o i n t of r e n o r m a l i s a t i o n a s s u m e d implicitly in this d e t e r m i n a t i o n is given by a typical m e s o n mass like that of rho. S u b s e q u e n t l y , m a n y o t h e r a t t e m p t s [2] have b e e n m a d e to d e t e r m i n e these masses u n d e r different sets of a s s u m p t i o n s . R e c e n t l y there have b e e n i n t e r e s t i n g d y n a m i c a l calculations to d e t e r m i n e [3] or to p u t lower b o u n d s o n [4] light q u a r k masses. T h e s e works are b a s e d o n s u m rules for the v a c u u m e x p e c t a t i o n value of the t w o - p o i n t f u n c t i o n of the d i v e r g e n c e of axial vector currents. A t short distances this q u a n t i t y can b e calculated from Q C D p e r t u r b a t i o n t h e o r y a n d s u m rules are o b t a i n e d by e q u a t i n g it to its dispersion r e l a t i o n in t e r m s of physical states. A r e m a r k a b l e idea p r o p o s e d by Shifman, V a i n s h t e i n a n d Z a k h a r o v (SVZ) [5] in the context of such s u m rules is that the s h o r t - d i s t a n c e a p p r o a c h b a s e d o n a few * Work in part supported by Schweizerischer Nationalfonds. 1 Now at the Institut for Theoretische Kernphysik, Universit~itKarlsruhe, Kaiserstrasse 12, D-7500 Karlsruhe, West Germany. 368

IV. Hubschmid and S. Mallik / Masses of light quarks in QCD

369

terms of perturbation theory may be extended to larger distances if one adds to it some small non-perturbative effects, which may be viewed as representing the confining mechanism. The result obtained for single resonances on the basis of these sum rules is indeed impressive, but the theoretical basis for such a strong form of duality in the low energy region still remains to be investigated [6]. In our work we deal with two types of similar sum rules to estimate the masses of light quarks. We, however, evaluate the sum rules at momenta higti enough for the non-perturbative contributions to be entirely negligible. This necessitates the consideration of physical states in the low energy region (~<1.7 GeV), which we assume to consist of 7r(K) and ~'(K'), the first radially excited states of ~-(K). Although the parameters of 7r'(K') are not available experimentally at present, we have estimated them in various ways. That perturbation theory should apply at as low a m o m e n t u m as 1.7 G e V has been amply confirmed by SVZ. In sect. 2 we collect the results for second-order perturbation theory as well as the non-perturbative terms for the vacuum expectation value of the correlation of two axial vector divergences. In sect. 3 we write down two types of sum rules and discuss their ranges of validity as well as their relationship with one another. In sect. 4 we first estimate the parameters of our model for the low energy region and then evaluate the quark masses with these parameters. Finally, in sect. 5 we discuss our result and compare it with those of others; in particular, we find it hard to justify an assumption implicit in a recent derivation [4] of lower bounds for the quark masses.

2. The pseudoscalar density correlation function To be able to work with quark masses in the leading order of chiral symmetry breaking, one considers the two-point correlation function of the divergence of the axial vector currents, P ( s ) = i f d x e iqx (OIT(Ot~A . t( x ) O A~,v ( O ) ) I O ) ,

s=q 2 ,

(2.1)

where O " A , (x ) = O" ( (t(x ) y , y s d (x )m = 2rh~t(x )iysd (x ) ,

(2.2)

rfi being the average mass of up and down quark. The case for the strangeness changing current is entirely analogous. The function P ( s ) can be calculated in perturbation theory for s large enough. (fig. 1) the bare Up to second order in the strong coupling constant (c~s= g~/47r) 2

-2×

• -<512>-

Fig. 1. Diagrams to calculate Pp(s) in perturbation theory up to second order. Dashed lines represent gluons.

370

W. Hubschmid and S. Mallik / Masses of light quarks in OCD

quantity Po(s) in the dimensional regularisation scheme (D = 4 - e ) is [4] (A = - s ) , - e + l ° g ~-~ + c l + × {4E 2

365

log2 4' +

~.

(2 log / 47r_ y)x+ ~) log 4/z+

c21] .J

(2.3)

Here ~ is the unit scale of mass. cl and c2 are two constants. The renormalised second-order result can be improved by the renormalisation group equation, which in this case has an inhomogeneous term determined by the singular part of P0(A) [7]. In terms of the running coupling constant as(A) and mass m (A) (with 3 flavours), a~(A) ~-

4 9 log ( A / A 2 ) '

re(A) (O/s(A) ~ 4/9 ' m(/z2) -- \O~s(/Z2)1

(2.4)

the solution of this equation is given by (in the MS scheme of renormalisation) pp(A) = R~2 4rh2(A ) (9 log ~A- ~ - ~17- ) ,

(2.5)

apart from an arbitrary linear monomial in A. Here A fixes the strength of strong interaction. In our calculation we take A = 200+ 100 MeV. As s decreases, the confining forces play an increasingly important role. This is signalled, according to SVZ, first by the appearance of the non-perturbative (power) corrections rather than by the breakdown of the perturbation series (the coupling constant growing large). The specific method suggested by them to get the nonperturbative terms is based on operator product expansion. In our case we have i I d x eiqXT{3~,A "* (x)O~A " (0)} = CI(A)I + C 2 ( A ) m q q + C 3 ( A ) G , ~. .G. .

+ C 4 ( A ) q F q q F q +" • •,

(2.6)

f ' being the appropriate matrices in the spin and colour space. Calculating the vacuum-to-vacuum matrix element of eq. (2.6) in perturbation theory, only the unit operator survives and CI(A) is identified with pp(A). To summarise the method of calculating the coefficient of other operators, as described by SVZ, one has to calculate perturbatively the matrix elements of both sides of eq. (2.6) between one-quark, one-gluon and two-stark states respectively. Only the one-gluon matrix element involves a closed fermion loop. The contribution to C3(A) of the first two diagrams in fig. 2 is ~ f7a+3 3(3a+l)(a-1)log~a+l 1 48rrA ~ a 5 - - + 2 a2~/a ~---~-lJ'

37l

W. H u b s c h m i d a n d S. M a l l i k / M a s s e s o f light q u a r k s in Q C D

I

X .. I

a

I

Fig. 2. Diagrams contributing to the coefficmnt"of G..G~ "~ in the operator product expansion. Dashed lines represent gluons.

where a = 1 +4r~2/A. To leading order in l / A , C 3 ( A ) is then 50~ s

C3~A)-24~~

O/s

Ofs

127rA

8~-A"

(2.7)

The subtracted term (the third diagram in fig. 2) is the contribution of mqq to the gluon matrix element. Alternatively, and much more simply, one can calculate this coefficient directly in configuration space by using quark propagators in the external field of gluons*. The non-perturbative pieces (renormalization group improved) are [5] p.p(A) = 4rh 2(A){ 8 ~ (0] °~s G ~a G ~ , 10)- 31

+

(Ol,~ (,~,,, + dd)10)

~'as(A) a2 ( o l - - u c r ~ y s t a ddcr ~ yst o u

+~(uy~t u + dyM ad)

E q = u,d.s

qY"taql0) ]

(2.8)

where t ° are the colour matrices (Tr tat b = 26ab). TO evaluate the 4-quark vacuum expectation values, S V Z advocate saturating by the vacuum intermediate state. Then using the P C A C relation

(OJ~(au+dd)lO)=- ~1m 2=f=, e2

(2.9)

the matrix elements of quark operators can be determined. H e r e f,, is the pion decay constant, f~ = 0.95 m=. Finally we add the perturbative and non-perturbative pieces to get {

(

'~

3 4th2(Zl) zi 9 l o g p,2 Pp+np(A) = 8~r~

17]

"B'2

,

2 2 }

6 / +~-~(M~ +2m=f,~)

28 rrol~(Zl) 4 +27 A ~ rn~ff~,

(2.10)

up to an arbitrary monomial linear in A. H e r e M1 is related to the vacuum a ¢~va expectation value of G , ~ G , a p.va M 4 = (0] -a~ G~G ]0) = 0.012 G e V 4 , 71"

* We are grateful to H. Leutwyler for showing us this m e t h o d of calculation.

(2.11)

372

W. Hubschmid

and S. Mallik / Masses of light quarks in QCD

obtained by SVZ. Somewhat shall see, the non-perturbative sum rules.

larger values are preferred by others [6] but, as we terms play a negligible role in our evaluation of the

3. Sum rules We shall consider two well-known types of sum rules relating the high and low s regions. The first is the finite-energy sum rule (FESR) [8] obtained by considering the contour of fig. 3 (k =O, 1,2,. . .)

I0

so sk Im P(s) ds = (-l)ksok+t

If so is sufficiently equation to get

I0

77 d0 Re {e -i(k+l)eP(s

large, one can use (2.10) to evaluate

= _sg eei”)]

.

(3.1)

side of this

the right-hand

(3.2) where 1: (so) = (-)” lord0 ( log2 $+ and fi = is the invariant

B2) a2 cos {k8 +(Y arc tan (o/log

$)]

,

(3.3)

mass

Note that the integrals (3.3) depend only logarithmically on so. In practice, k should be restricted to low integer values, because higher k’s emphasize the upper end of the integral, where, in general, a model for the absorptive part is expected to be less accurate. P(s) satisfies a dispersion relation in terms of sigularities due to hadronic physical intermediate states. At a space-like point s = -A,

P(A) =

(3.5)

apart from two subtractions. The second type of sum rules, the dispersion sum rules (DSR), are obtained by equations the nth derivative (n 3 2, to avoid unknown

W. Hubschmid and S. Mallik / Masses of lightquarks in QCD

373

subtraction constants) of the dispersion relation to the same quantity given by the perturbative and non-perturbative terms for A sufficiently large, 8 3rr24tfiZ(A) { l + 2 a s ( A ) ( ~ + A " ) + r r n ( n~1-~1 ) 2 3 (M4+2m2f2~) }

+(n+l)n(n

28 m4f4,~ - 1)~71rad/t) A3 4

2

laA,_l[ _rn•f= =n(n- , [(mZ+A),+l l

co

+1 f9 Im P(s) ds l ~- -,~ ( - ~ - - + ~ 7 ~ j ,

1

(3.6)

where An = l + ~ + g + . • • + 1 / ( n - 2 ) ( n ~>3). The high energy part of the integral (s/> so) may again be evaluated with Pp+np(S) given by eq. (2.10). The two types of sum rules are, in fact, not different and it is easy to obtain the F E S R ' s (3.1) from D S R ' s (3.6). The left-hand side of (3.6) represents essentially the result of differentiating Po+np(A) n times. Instead, we could have assumed a dispersion representation for Pp+np(A) also and performed the same operation on it. Assuming I m P p + . p ( S ) = I m P ( s ) for s ~>s0, the integrals from So to ~ would cancel out on both sides of the equation. The dispersion denominators may now be expanded in powers of s/A and coefficients of equal powers of A may be equated on both sides. The n dependence disappears in the relations so obtained and relating the m o m e n t s of Im Pp+,p(S) to that over the circular contour (fig. 3) we get the FESR's. The two sets of sum rules may, however, be very different in practice. Recall that the non-perturbative terms appearing in the sum rules are, in fact, the first two terms of an infinite series, which must at least converge. In the FESR, it is a power series in M~/so, where M2 is a mass scale set by the non-perturbative series, perhaps related to a typical hadronic mass like that of p meson. This series converges, once So is large enough. On the other hand, for the D S R ' s the non-perturbative

Fig. 3. Contour in the s-plane for the finite-energy sum rules.

374

W. Hubschmid and S. Mallik / Masses of light quarks in QCD

t e r m s c o n s t i t u t e a series in n M Z / a . Clearly, even for a v a l u e of A for which t h e s e c o n d - o r d e r Q C D p e r t u r b a t i o n t h e o r y is a g o o d a p p r o x i m a t i o n , the result o b t a i n e d f r o m such a s u m rule can be m e a n i n g l e s s if n is t a k e n large e n o u g h . In a d d i t i o n , n is also r e s t r i c t e d by t h e fact t h a t t h e p e r t u r b a t i v e c o n t r i b u t i o n on the left of t h e e q u a t i o n grows like log n for large n. ( F o r t h e F E S R ' s , t h e r e is also a r e s t r i c t i o n on k, b u t t h a t is for a different p r a c t i c a l r e a s o n , as a l r e a d y m e n t i o n e d . ) In o u r e v a l u a t i o n we t h e r e f o r e c o n s i d e r the D S R for n = 2 only. In the F E S R the p e r t u r b a t i v e (and n o n - p e r t u r b a t i v e ) result for P ( s ) has b e e n a v e r a g e d o v e r a circle in t h e c o m p l e x s - p l a n e , while in D S R it has b e e n u s e d at a s p a c e like p o i n t A w h e r e t h e a m p l i t u d e is s u p p o s e d to b e r a t h e r s m o o t h . This m a y not c o n s t i t u t e a d e f e c t of F E S R o v e r D S R , p r o v i d e d So is a w a y f r o m a n y t h r e s h o l d . B o t h t y p e s of s u m rules can, of course, b e p u t in the e x p o n e n t i a l (or B o r e l i m p r o v e d ) f o r m [5], w h e r e a n o n - p e r t u r b a t i v e t e r m b e h a v i n g like A " is s u p p r e s s e d b y a f a c t o r 1 I n I. This w o u l d b e a significant i m p r o v e m e n t if o n e is w o r k i n g at low So o r ,:1. W e , h o w e v e r , p l a n to w o r k at values of So o r d large e n o u g h for the c o n t r i b u t i o n of t h e n o n - p e r t u r b a t i v e t e r m s to b e negligible a n d h e n c e such i m p r o v e m e n t s a r e n o t essential for us.

4. E v a l u a t i o n of the s u m rules

O u r m o d e l for t h e a b s o r p t i v e p a r t in t h e low e n e r g y r e g i o n * ( u p to s = So, to be d e t e r m i n e d p r e s e n t l y ) is given by ~" a n d ~r', lying on the d a u g h t e r t r a j e c t o r y to ~-, c o n s i d e r e d in the n a r r o w w i d t h a p p r o x i m a t i o n : Im P ( s ) = rrrn~f ~ {~(s - m ~ ) + r6(s - m 2 , ) } ,

(4.1)

w h e r e r is the s q u a r e of the r a t i o of ~r' c o u p l i n g to ~r c o u p l i n g with t h e d i v e r g e n c e of t h e axial v e c t o r c u r r e n t : r =

(m~,f~,lrn~f,,)

2 .

(4.2)

E x p e r i m e n t a l l y t h e r e is as y e t no clear e v i d e n c e for ~ ' . B u t t h e o r e t i c a l l y the p r e s e n c e of such d a u g h t e r t r a j e c t o r i e s is i n d i c a t e d in all m o d e l s we k n o w of; V e n e z i a n o a m p l i t u d e [10], n o n - r e l a t i v i s t i c q u a r k m o d e l [11] a n d relativistic w a v e e q u a t i o n s [12] a r e t h e l e a d i n g e x a m p l e s . B e f o r e we p r o c e e d to e s t i m a t e the p a r a m e t e r s of t h e m o d e l , it is useful to n o t e d o w n t h e f o r m u l a for the q u a r k m a s s o b t a i n e d f r o m t h e s u m rules b y i g n o r i n g all c o r r e c t i o n s , p e r t u r b a t i v e a n d n o n - p e r t u r b a t i v e . This will give us a fairly a c c u r a t e i d e a of h o w the q u a r k m a s s d e p e n d s on t h e v a r i o u s p a r a m e t e r s i n v o l v e d in the * Non-resonant 31r or resonant p~r contributions are expected to be negligible. It is not difficult to see that the situation is analogous to the evaluation of the dispersion integral for the nucleon matrix element of the axial vector divergence to determine the correction to the Goldberger-Treiman relation. See, e.g., ref. [9].

W. Hubschmid and S. Mallik / Masses of light quarks in OCD

375

sum rules. To leading o r d e r in log (so/A 2) we have

I~

log" l ( s o / A 2 ) ,

k#O,

I~ (So)--* -

(4.3)

Lit log" (so~A2),

k =0,

T h e F E S R (k = 0) then yields rfi (so)

2~r m,,f,, 2 ~/1 + r, ~/3 So

(4.4)

a formula similar to that given in [3]. U n d e r the same approximation, the D S R reads 4 2

1

r

a

3

A" 1 I, ~ 4 m 2 ( s ) s ds]

+ 8~I r

,>

~2+--i

j .

(4.5)

If we ignore logarithmic variations in s in the integrand, we get 8 24rh2(a)1

_ l+nso/A ] (l+so/Zl),j=n(n-1)~

4 2 l+(l+m],/Ay+,

.

(4.6)

N o w taking A large enough, we regain the a b o v e formula. This derivation also shows h o w the strong n and A d e p e n d e n c e on the right-hand side of eq. (3.6) (ignoring the n o n - p e r t u r b a t i v e contributions) is only a p p a r e n t w h e n we consider the pole term jointly with the integral over the continuum. W e n o w try to estimate the p a r a m e t e r s of our m o d e l by several simple considerations. T h e mass of ~r' is d e t e r m i n e d f r o m an assumed linear R e g g e trajectory with a universal slope: 2

m,.s = 1.12(2n + j ) + m ~ ,

(4.7)

w h e r e n and j are the radial and spin q u a n t u m numbers, respectively. Thus 2 m,,, = 2.24 G e V 2, which we do not try to d e t e r m i n e any further. For the u p p e r end so of the low e n e r g y region we then have So ~> 2.6 G e V z. T h e r e are several ways to d e t e r m i n e the coupling p a r a m e t e r r. W e note that

r = (Olai3,sdlrr')/(Olai3,sdlrr)} 2

(4.8)

,

=

where ~b,~(,,,)(O) is the wave function of ~(~-') at the origin. If we assume this ratio is equal to that for O' to ~Oin the ~c family, we get

Fio,m~,

r -

~

F,~m ~

-

0.6,

(4.9)

on putting experimental n u m b e r s for the masses rno(o,) are electronic widths F~(~,).

376

IV. Hubschmid and S. Mallik / Masses of light quarks in Q C D

A n idea of the m a g n i t u d e of r also follows if we assume a non-relativistic h a r m o n i c oscillator m o d e l for the (gd) system:

05,~(x)-e x~,

&,,(x)~(3_4x2)e-X

2'

giving r = 3. Finally we m a y try to d e t e r m i n e the p a r a m e t e r s of the m o d e l by considering the three lowest m o m e n t F E S R ' s themselves, h o p i n g that they will not be too bad for o u r model. T h e three sum rules m a y be written as (k = 0, 1, 2) ( So ~k {9Ilkf92 17rk+2 ~.2 [ M 4 + 2 m 2 f 2 ] 1k-8/9 } Z \~w'' -- 6 1-8/9 q'--3- ~ S-~ "

=r

/ m ~ \ k 112 z f___~Ik_; 1 , + ---T- + - ~m,~,) 243 So

(4.10)

2 3 m=f=) 4 2 and Iak --=i ak (So). w h e r e z = 3 m_A2 So/(2~" With S o - 3 G e V 2, the n o n - p e r t u r b a t i v e terms [the last terms on both sides of eq. (4.10)] are negligibly small and by taking quotients of these sum rules, we get So 2

J3(so)

m=, = J4(so)

,

r+ 1 r

[m2,~ J2(so) [ / = \--~-o/ J3(s0)'

(4.11)

where

Jk (So) = 9 I k/9 (So) -- ~-tk-s/9 (So).

(4.12)

Such quotients are totally insensitive to the value of A, at least within the range 100 < A < 300 M e V . F r o m (4.11) we get So = 3.0 G e V 2. C o m b i n i n g with our earlier inequality for So, we then choose So to be So = 2.8 + 0.2 G e V z .

(4.13)

U n f o r t u n a t e l y the e q u a t i o n determining r d e p e n d s sensitively on the value 2

m,~,/So:

1 +r=r 1.54 x--~o / '

(4.14)

and thus it is difficult to rely on the value of r o b t a i n e d f r o m this equation. Nevertheless, we do not take it as an indication that r m a y be rather large (it gives r = 4.3 for So = 2.8 G e V 2) c o m p a r e d to that o b t a i n e d f r o m eq. (4.9) and we think the uncertainty in its values is well taken into account by assuming r = 2.5 + 2 . 0 .

(4.15)

H a v i n g d e t e r m i n e d the p a r a m e t e r s of our m o d e l in the low e n e r g y region given by eqs. (4.7), (4.13), (4.15) we insert t h e m in the f o r m u l a for q u a r k mass given by F E S R (k = 0): 2 2 zr m ~f,, ~/1 + r f(so) , (4.16) rh(1GeV) ~/~ So

377

W. Hubschmid and S. Mallik / Masses of light quarks in QCD

where

f(so) =

log

- 17.(2) 2(9I]~9 (So)-W~-8/9 (So)

'

(4.17)

is equal to unity within 5% for 2.8 < So< 3.2 G e V 2 and 100 < A < 300 M e V . Thus the F E S R gives the average of the mass of up and d o w n q u a r k as rh(1 G e V ) = 5 . 7 + 2 . 2 M e V ,

(4.18)

where the estimated error comes mainly f r o m the uncertainty in the value of r. W e have also evaluated the D S R for n -- 2 at different values of d. T h e renormalisation point invariant q u a r k mass o b t a i n e d in this way shows a d e p e n d e n c e on zl for A ~< 3 G e V 2, p r e s u m a b l y because of i n a d e q u a c y of the model at low energy, and then tends to settle to a constant value. W e pick up this near constant value to get n~ (1 G e V ) = 5.6+212 M e V .

(4.19)

A n a l o g o u s l y we can also derive sum rules for the divergence of the strangeness changing axial vector current, O , A ~ = o~, ( gyu ysS) = (mu + m d ) u i T s s ,

(4.20)

and evaluate the strange q u a r k mass f r o m the F E S R , which is again given by a f o r m u l a similar to (4.16): (mu + ms)(1 G e V ) = 4 ~ m ~ f K ~/1 + r' f(s'o) ,

(4.21)

s; where, of course, the p a r a m e t e r s of the m o d e l for the low e n e r g y region have to be d e t e r m i n e d once again. H e r e fK-~ 1.1 m~. T h e mass of K' is d e t e r m i n e d f r o m eq. (4.7) with rn~ replaced by rn 2, giving m ~:, = 2.5 G e V 2 and therefore we have So ~ 2.9 G e V 2. This value t o g e t h e r with the one d e t e r m i n e d f r o m the three F E S R ' s eq. (4.10), viz. s~ = 3.3 G e V 2, provide us with the range of s~ to be considered: Sot = 3.1 ± 0.2 G e V 2 .

(4.22)

T h e range of value of r' remains the same as for the n o n - s t r a n g e case. W e then get f r o m F E S R (mu+ms)(1 G e V ) = 161+_57 M e V .

(4.23)

T h e D S R (n = 2) for the same quantity gives (mu + ms)(1 G e V ) = 146+_52 M e V .

(4.24)

378

W. Hubschmid and S. Mallik / Masses of light quarks in QCD

5. Discussion

The evaluation of the quark masses we presented above is based on a model for the low energy region (So< 3 G e V 2) which is given by ~r(K) and 7r'(K'). The high energy region is approximated by the second-order perturbative result together with the first two non-perturbative (power) correction terms. We have tried to determine the parameters of the model by several methods. On the basis of our evaluation of the two sum rules, we find the average mass of the up and down quark at a m o m e n t u m of 1 G e V to lie between 3.3 and 7.9 MeV, while that for the strange quark lies between 84 and 212 MeV. These ranges include the values found by Leutwyler [1]. A specific feature of our evaluation of the sum rules is that the contributions from the non-perturbative terms are entirely negligible. However, this is obtained only at the price of a large uncertainty in the value of the residue of ~"(K'). Clearly this uncertainty can be eliminated by choosing So (or a similar parameter in the Borel improved sum rules of SVZ) much lower, when the non-perturbative terms gain importance. It is then necessary to ascertain somehow, for example, 'by evaluating the next one or two higher dimensional non-perturbative terms, that the remainder of the non-perturbative series is sufficiently small. This is all the more important with respect to D S R ' s with high values of n. Otherwise, by emphasizing the non-perturbative contributions and higher values of n in D S R we may push the sum rules to a corner where they may no longer be valid. We believe exactly this criticism applies to a recent work by Becchi et al. [4] who obtain a lower bound on the non-strange quark mass which is higher than the value found by us by at least a factor of 1.5. The inequality on which their lower bound is based may be obtained by ignoring the (positive definite) spectral function in eq. (3.6) (and also neglecting all logarithmic dependences on A):

3 {

8 24m2

,2

1+57r n ( n - 1 )

A E j ~>

.,01, A2

42

m~f~.

(5.1)

Now the variable M 2 = A / n may be introduced to write the above inequality as 8_~24m2(1 +Tr 2 M4'~>_ 1 4 2 ~-- M4 ] ~ ~-~ m ~f,~. Here the bound M = 490 MeV to non-perturbative the zeroth-order

(5.2)

depends crucially on the value of M 2. Becchi et al. choose* get 2rfi (1 GeV)/> 24 MeV (A = 200 MeV). This choice makes the (power) corrections contribute - 7 0 % of the principal term, i.e., perturbation result. Lacking a basic understanding role of the

* Becchi et al. rewrite the inequality for 2m, transfering all M 2 dependence on the right and retaining only the first two terms in the expansion of the square root. This generates a fictitious maximum of the right-hand side of the inequality as a function of M E, which corresponds to their choice of M 2.

w. Hubschmid and S. Mallik / Masses of light quarks in QCD

379

p o w e r c o r r e c t i o n s at p r e s e n t , in p a r t i c u l a r , w i t h o u t a r o u g h i d e a of the m a g n i t u d e of t h e r e m a i n i n g n o n - p e r t u r b a t i v e t e r m s , which, in fact, is a series in 1 / M 2, such a small v a l u e of M 2 resulting in a h u g e " c o r r e c t i o n " m u s t b e v i e w e d with suspicion. E s s e n t i a l l y similar s u m rules w e r e c o n s i d e r e d by S V Z in t h e i r original w o r k on Q C D s u m rules [5], w h e r e t h e y restrict t h e l o w e r limit of the a c c e p t a b l e v a l u e of M 2 b y r e q u i r i n g that t h e c o r r e c t i o n t e r m s s h o u l d n o t c o n t r i b u t e m o r e t h a n = 3 0 % of the p r i n c i p a l t e r m , which s e e m s to w o r k well with the sum rules c o n s i d e r e d by t h e m . Such a r e q u i r e m e n t for (5.2) l e a d s to M = 600 M e V , giving 2rh (1 G e V ) / > 18 M e V (A = 200 M e V ) , which is a b o u t the u p p e r e n d of t h e r a n g e of values for t h e s a m e q u a n t i t y f o u n d b y us. A significantly large v a l u e for t h e q u a r k mass can, of course, be o b t a i n e d f r o m t h e s u m rules b y a s s u m i n g t h e p r e s e n c e of singularities which c o n t r i b u t e significantly in the low e n e r g y region. H o w e v e r , t h e y are likely to d i s t u r b the validity of the low e n e r g y t h e o r e m s b a s e d on P C A C a n d c u r r e n t a l g e b r a . R e g a r d i n g the w o r k of ref. [3], o n e c a n n o t justify the choice of n a n d A in the D S R within the c o n t e x t of the p r o b l e m itself; the r e a s o n i n g is o n l y indirect, b a s e d on e x p e r i e n c e s with similar sum rules, w h e r e t h e results can be c h e c k e d with experiment. Finally, we h o p e to h a v e s h o w n in d e t a i l that the a c c u r a c y in d e t e r m i n i n g (or p u t t i n g a n o n - t r i v i a l l o w e r b o u n d on) the light q u a r k m a s s e s b a s e d on Q C D s u m rules is l i m i t e d b y the u n c e r t a i n t y in t h e s p e c t r a l f u n c t i o n in the low e n e r g y r e g i o n (0.5 G e V 2 ~< s ~< 2.5 GeV2), which o n e m a y try to d e t e r m i n e in t e r m s of e i t h e r the singularities of p h y s i c a l states o r t h e p e r t u r b a t i v e c o n t r i b u t i o n c o r r e c t e d b y n o n p e r t u r b a t i v e terms. A n y p r e c i s e d y n a m i c a l d e t e r m i n a t i o n of the q u a r k m a s s e s will n e c e s s a r i l y d e m a n d b e t t e r k n o w l e d g e of at least o n e of t h e two m e t h o d s of d e t e r m i n i n g the s p e c t r a l f u n c t i o n in the low e n e r g y region. W e a r e v e r y g r a t e f u l to H. L e u t w y l e r for suggesting this i n v e s t i g a t i o n a n d for m u c h h e l p a n d suggestions d u r i n g the w o r k . W e also t h a n k R. C r e w t h e r , J. G a s s e r , P. M i n k o w s k i a n d H. G e n z for useful discussions.

References [1] H. Leutwyler, Nucl. Phys. B76 (1974) 413 [2] M. Testa, Phys. Lett. B56 (1975) 53; J. Gasser and H. Leutwyler, Nucl. Phys. B94 (1975) 269; G. Furlan, N. Paver and C. Verzegnassi, Nuovo Cim. 32A (1976) 75; S. Weinberg, in Festehrift for I.I. Rabi, ed. L. Motz (NY Academy of Sciences, NY, 1977); S. Mallik, Helv. Phys. Acta 50 (1977) 825; P. Minkowski, The charged weak current and quark masses, Bern University preprint (1980); P. Langacker and H. Pagels, Phys. Rev. D19 (1979) 2070; J. Gasser and A. Zepada, Nucl. Phys. B174 (1980) 445; J. Gasser, Hadron masses and sigma commutator in the light of chiral perturbation theory, Bern University preprint (1980);

380

[3] [4] [5] [6] [7] [8]

[9] [10] [11] [12]

W. Hubschmid and S. Mallik / Masses of light quarks in QCD H. Pagels and S. Stoker, Magnitude of the light current quark masses, Rockefeller University preprint COO-2232B-194 (1980 A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, V.A. Novikov, L.B. Okun and M.A. Shifman, Sov. J. Nucl. Phys. 27 (1978) 274 C. Becchi, S. Narison, E. de Rafael and F.J. Yndurain, CERN preprint TH.2920 (1980) M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448,519 J.S. Bell and R. Bertlmann, Nucl. Phys. B177 (1981) 218 D.J. Gross, in Methods in field theory, Les Houches Session XXVIII 1975, ed. R. Balian and J. Zinn-Justin (North Holland, Amsterdam 1976) R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166 (1968) 1768; R. Shankar, Phys. Rev. D15 (1977) 755; E.G. Floratos, S. Narison and E. de Rafael, Nucl. Phys. B155 (1979) 115 H.F. Jones and M.D. Scadron, Phys. Rev. D l l (1975) 174 G. Veneziano, Nuovo Cim. 57A (1968) 190 R.H. Dalitz, 18th Int. Conf. on High-energy physics, Berkeley, 1966 (University of California Press, 1967) H. Leutwyler and J. Stern, Phys. Lett. 73B (1978) 75; Nucl. Phys. B133 (1978) 115