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Model independent determination of QCD quark masses
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
MODEL INDEPENDENT
DETERMINATION
16 August 1984
OF QCD QUARK MASSES
M. K R E M E R 1, N.A. P A P A D O P O U L O S a n d K. S C H I L C H E R
lnstitut far Physik, Johannes Gutenberg Universiti~t, 6500 Main& West Germany Received 5 January 1984 Revised manuscript received 25 April 1984 The method of approximate analytic continuation by duality (ACD), which permits to extrapolate a two-point function from its asymptotic QCD form down to Q2 = 0, is used to determine the value of the light quark masses in QCD. For the value of the four-quark coridensate advocated by Shifman, Vainshtein and Zakharov we obtain (m u + m d) = (21 _+2) MeV a t Q 2 = 1 G e V 2.
A s Q C D b e c a m e the accepted theory of strong interactions, a t t e n t i o n focussed on the d e t e r m i n a t i o n of the f u n d a m e n t a l p a r a m e t e r s of the lagrangian, the strong coupling c o n s t a n t a s a n d the q u a r k masses m i. A d e t e r m i n a t i o n of the q u a r k masses [1] is m u c h m o r e difficult than that of the coupling constant, as they m a n i f e s t themselves m a i n l y in low energy region where p e r t u r b a t i v e Q C D b e c o m e s i n a d e q u a t e . Previous calculations b a s e d on Q C D sum rules or finite energy sum rules therefore either y i e l d e d o n l y lower b o u n d s [2,3] or h a d to m a k e m o r e or less ad hoc assumptionS on the c o n t i n u u m in the p s e u d o s c a l a r channel [4]. In this n o t e we will present a d e t e r m i n a t i o n of the q u a r k masses which is a l m o s t i n d e p e n d e n t of the c o n t i n u u m . It is b a s e d on a m e t h o d of a p p r o x i m a t e analytic c o n t i n u a t i o n b y d u a l i t y ( A C D ) p r o p o s e d p r e v i o u s l y [5] which allows the r e c o n s t r u c t i o n of an a n a l y t i c a m p l i t u d e at small space-like q2 given o n l y its a s y m p t o t i c form for large q2. W e follow the p i o n e e r i n g w o r k of ref. [2] a n d consider the two p o i n t function of the divergence of two axial currents II(q 2) =
ifdx eiqx,
(1)
which has in Q C D the o p e r a t o r p r o d u c t e x p a n s i o n [2] YIQCD(t) = ( m , + r o d ) 2 ( - ( 3 / 8 ~ ' 2 ) [ ( 1 + ~ a J r r ) t
l o g ( - - t / / , 2 ) - - ( a s / r r ) t 10g2( -- t / t t 2 ) ]
+ ( 3 / 4 ~ r 2 ) ( m 2 + m 2 _ m . m d ) l o g ( - - t / i t 2) + ( m u + m d ) ( g t q ) / 2 t -- ~<(as/~r)GZ>/t
+ (l12~r/27)as2/t2},
(2)
or, after r e n o r m a l i z a t i o n group i m p r o v e m e n t I I ( t ) = (3/8~r 2 ) ( r h . + rh d )228/9( - t ) ( 91108( - t / A 2 )] t/9 _ ~10 [log( -- t / A 2 )] + 2 ' 7 / 9 [ ( r h 2 + th 2 - F n , t h d ) / t ] 9 [ l o g ( - t / A 2 ) ] - 7 / 9 +
8/9
~l[lOg(-t/A2)] -8/9 l o g l o g ( - t / A 2 ) }
_ ( th u + th d )228/9 t - 1 [log ( _ t / A 2 )] - 8/9 X (~<(aJ~r)G2>
- ½ ( m . + m d ) ( q q > --
~721ras(qq>2/t
+ higher o r d e r s + p o l y n o m i a l ,
} (3)
I Supported by Bundesministerium fiir Forschung und Technologie. 476
0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
where A is the Q C D scale parameter for which we take A = (150 + 50) MeV, (7qq) the quark condensate, ((o~s/¢r)G~,~G~ ) the gluon condensate and rh i the invariant quark masses which are related to the running quark masses in the MS scheme by ( r n i ( _ Q 2 ) =rhi(½ log(Q2/A2))
4/9
1
28 l o g l o g ( Q 2 / A 2 ) 290 1 ) 36 log(QZ/A2 ) + 3---g--log(Q2/A2 ) .
(4)
In the following we will use ((as/~)Gt~,,Gt'") = 0.012 GeV 4, which is the value given by Shifman et al. (SVZ) [6]. For the value of the dimension six operator Ots{~lq)2 we use two alternative values: (a) The value advocated by SVZ [6] Ots~qq)2 = 1.8 x 10 -4 GeV 6. This is a large value which is, to our opinion, mainly justified by its phenomenological success [5,6]. SVZ justify it, by assuming factorization for the four-quark operator at very low Q2. (b) A "self-consistent" value, which is obtained by assuming factorization at Q 2 = 1 GeV 2 and using the PCAC relation (m~ + rnd)(qq) = -f~m2,,
(5)
in an iterative calculation. We shall now present in some detail the principle involved in our extraction of the quark masses from pseudoscalar Q C D sum rules. It follows from (2) that the second derivative of H ( q 2) satisfies an unsubtracted dispersion relation 2 4
1 f ~ dt
4ff, m~. )3 + - n"(-Q2) (Q2+m 2 "17""9m~
2 (t +
QZ)3
Im II(t).
As Im H ( t ) is a positive definite function one obtains for sufficiently large Q2, where H " ( - Q 2 ) approximated by QCD, the inequality 2 3. H " ( - Q2) > 4fZm4/(Q2 + rn~.)
(6)
is well
(7)
As I I " is proportional to the square of the quark masses, (7) directly translates into a lower bound of these. At Q2 = 1 GeV2; where the nonperturbative higher order term may still be controllable, this would lead to (rn u + md)hOev > 18 MeV. In the past attempts were made to improve the bound of (7) using optimal inequalities [2], Borel transforms or to phenomenologically approximate the continuum in (6) [4]. It is however clear from (6) that the only way to obtain a model independent constraint on II~ct) ( - Q2) or equivalently a prediction of the quark masses is to analytically continue I I ~ c D ( - Q 2 ) down to Q2 = 0. For Q2 = 0 the pion-hole dominates the r.h.s, of (6) and the inequality of (7) becomes an inequality [to the accuracy to which our input parameters are determined see (5)]. The method of ACD, whose effectiveness has been tested previously [5], allows just such an extrapolation of F[~ci~ ( - Q2) to low Q2. In the following we will demonstrate how A C D is applied here. By Cauchy's theorem H satisfies the relation
2 £ r I " ( - Q 2 ) = 2~r~-i dt
n(t) , (t + Q2) 3
(8)
where the contour C of integration consists of a circle of radius R and a cut along the real axis from m 2 to R. 477
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
On the phenomenological side l'I contains a pion pole contribution and a continuum part I I ( - Q2) = 2 f ~ m 4 / ( m ] + Q2) + continuum,
(9)
with f . = 93 MeV. It is clear that the continuum contribution to I I becomes negligible compared to the pion pole when Q2 approaches zero. Taking the pion pole explicitly into account, (6) yields 2
4
2
4f~2m4/(m]+Q2)3+continuum=4f~m=/(m.+Q2)3
+2fg
~'t9mZ~
dtlmH(t) (t+Q2)
3
+
~
2 ~c
II(t)dt
R(t2+Q2)3,
(10)
where C R is the circle of radius R. In the integral along the real axis we use the approximation N
1
- ~ a.t",
(,+Q2) '
fort~[9m2,R].
(11)
.=0
The coefficients a , depend on Q2 and R and must be determined by a least square fit in the given interval. N is finite, its maximal value being limited by the number of terms known in the asymptotic expansion of I I ( t ) . With this approximation one can rewrite the integral
~2g'fm 2n . d(tt l m+ Q2) I I ( t3) -
R n 2 /'9rn ~ x---~ n 2"rrfo ~_,a,t I m I I ( t ) d t - - J o l r 2.,a.t Im I I ( t ) d t
-2 f 2 4 2n = 2~ri a¢ E a , t " Y I ( t ) d t -- 4f; m ~ _ , a , m ~ . R
Eq. (6) then reads
4f,;2rn,~ 4
a , m ,2, + continuum --~--~ 2
n=0
:c(
~ ( t + Q12 ) 3
a,t ~ 1-I(t)dt
(12)
and this is the central result for the following analysis. It is valid for all Q2 <: R. The r.h.s, simulates the pion pole and grows, as expected, rapidly as Q2 _+ 0. Typically it increases relative to the continuum as Q2 is lowered from Q2 = 1 GeV 2 to Q2 = 0 by a factor of 200. Therefore, at Q2 = 0 we obtain in a very good approximation, instead of the inequality of (7) the result 2~ri q~ dt t S -
2. a,t ~ I I ( t ) = 4f~2m=2 ~ a,m,~, n=0
(13)
n=O
which allows the determination of quark masses. Some remarks are in order at this point. First in the integral the function H must be known only on the circle C n, hence one can use the perturbatively calculated expression (3). Furthermore it should be noted that the polynomial is an approximation to 1 / ( t + Q2)3 on the real axis only, so the integral along the circle is by no means zero. As the pion pole has disappeared during the calculation, it is important to check that the pion contribution is still dominant at the 1.h.s. of (12). This can be done once the coefficients a~ are known by comparing it to the value of the actual pion-pole. The separate treatment of the pion-pole is essential for the approximation method used: The approximation of the function 1 / ( t + iQ2) 3 by a polynomial introduces an error which depends on the shape of the imaginary part of the function YI. This error is large for rapidly varying functions, in particular, the 478
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
8-function c o r r e s p o n d i n g to the p i o n - p o l e , a n d it is small for s m o o t h functions. In particular, the error is zero, if the i m a g i n a r y p a r t is a p o l y n o m i a l of a smaller degree than the fit p o l y n o m i a l . T h e r e f o r e the s e p a r a t e t r e a t m e n t of the p i o n - p o l e reduces s u b s t a n t i a l l y the error which is due to the fit. T h e final answers for the q u a r k masses d e p e n d sensitively on the value of the d i m e n s i o n six o p e r a t o r . F o r the SVZ value we o b t a i n (rn u + m a ) , o e v = (21 + 2) MeV, (rnu + m s ) , c e v = (245 _+ 10) MeV.
(14)
U s i n g the f a c t o r i z a t i o n hypothesis, for the four q u a r k c o n d e n s a t e we get ( m u + m a ) = (35 -t- 3) MeV, ( m u + ms) = (270 ___ 10) MeV.
(15)
T h e q u o t e d errors take into a c c o u n t the effects of neglected higher d i m e n s i o n a l condensates, the d e p e n d e n c e on the degree N of the fit p o l y n o m i a l a n d the R - d e p e n d e n c e for small QZ, as well as the very small d e p e n d e n c e on A, the Q C D scale. T h e y do not account, however, for the uncertainties in the values of the i n p u t condensates. This is d e m o n s t r a t e d in p a r t i c u l a r b y the very different values o b t a i n e d for the q u a r k masses in (14) a n d (15). T h e d e p e n d e n c e on the gluon c o n d e n s a t e is less d r a m a t i c . T h e analysis was d o n e for N = 2 . . . . . 5, R = 1.5 to 3 G e V 2 a n d Q 2 = 0.0 to 0.9 G e V 2 ,1. F o r small Q2 the R a n d N d e p e n d e n c e is small, yielding the q u o t e d errors. W e p r e s e n t typical results in fig. 1. It is w o r t h n o t i n g that for the SVZ value of as(Y:tq>2 the p r e d i c t e d q u a r k masses flatten off near Q2 = 0 as a m a n i f e s t a t i o n of the fact that the c o n t i n u u m b e c o m e s i n d e e d negligible. This is in c o n t r a s t to the case of a small f o u r - q u a r k condensate, so that in this case the conclusions are m o r e doubtful. W e have therefore p e r f o r m e d a similar investigation using the s u m rule I I " ( Q2)
"i
f d t I I " t ~t ~ )
I n this case one looses the positive definitiveness s u b s t a n t i a l l y greater accuracy. It turns out that for very stable a n d agrees in this case with (14) a n d stability could be o b t a i n e d . The ratios of the q u a r k masses can b e o b t a i n e d has
of the spectral function, b u t the p o l y n o m i a l fit has a the SVZ value of the four q u a r k c o n d e n s a t e the result is (15), whereas in the case of the small c o n d e n s a t e no i n d e p e n d e n t l y from chiral p e r t u r b a t i o n theory [1]. One
2 m s ( r n . + rod) = 25 -t- 2.5.
(16)
U s i n g m u / Y n = 0.72 from ref. [1] we o b t a i n 2 r n J ( r n u + rod) = 22.9 + 3.2 f r o m (14) a n d 2 m s / ( m . + rod) = 14.9 + 2.0 from (15). T h e result (14) is in a g r e e m e n t with relation (16), the result (15) is not. This c o u l d be t a k e n as an indication, in a d d i t i o n to successful p h e n o m e n o l o g y [5,6], that the four q u a r k c o n d e n s a t e does in fact have the large value a d v o c a t e d b y SVZ. In that case, however, one has to give up factorilization of the v a c u u m e x p e c t a t i o n value of the four q u a r k o p e r a t o r . The a s s u m p t i o n that the ratio of four q u a r k o p e r a t o r m a t r i x elements a p p e a r i n g in different current correlation functions can be d e t e r m i n e d b y the f a c t o r i z a t i o n h y p o t h e s i s m a y p o s s i b l y be m a i n t a i n e d . O u r values for (rn u + rod) are in g o o d a g r e e m e n t with results from b a r y o n i c sum rules in ref. [7], where ( m u + m d ) a O e v = 30--+6 M e V was o b t a i n e d a s s u m i n g f a c t o r i z a t i o n of the four q u a r k condensate, a n d ( m u + md)lOCV = 20 M e V if the four q u a r k c o n d e n s a t e was k e p t as a free p a r a m e t e r . T h e analysis of b a r y o n i c s u m rules also gives some i n d i c a t i o n s that the four q u a r k c o n d e n s a t e might b e s u b s t a n t i a l l y larger ,1 F o r the s t r a n g e q u a r k m a s s we used Q2 = _ 0.2 to 0.9 G e V 2 in o r d e r to get a b e t t e r suppression of the c o n t i n u u m .
479
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
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30
÷
E
w
oll o.'2 o:3 0.'~ a's o'6 o.'7 018 02 I-G v 2]
Fig. 1. Lower bounds of quark masses as function of Q2, the argument of the pseudoscalar two-point function. At 0 2 = 0 the bound becomes an equality. The lower curve corresponds to the SVZ value of the four-quark condensate, the upper curve to the one obtained by factorization.
t h a n i m p l i e d b y t h e f a c t o r i z a t i o n h y p o t h e s i s . O u r v a l u e ( m u + m d ) l GeV = 21 + 2 M e V is h i g h e r t h a n t h e o n e g i v e n i n ref. [1], b u t c o n s i d e r e d t h e e r r o r b a r s a n d t h e u n c e r t a i n t y i n t h e q u a r k c o n d e n s a t e , w e d o n o t r e g a r d t h i s d i f f e r e n c e as a r e a l d i s c r e p a n c y . W e t h a n k F. S c h e c k f o r h i s i n t e r e s t i n t h i s w o r k a n d f o r d i s c u s s i o n s .
Note added: T h e i n c l u s i o n o f t h e O(et 2) c o r r e c t i o n t o t h e p s e u d o s c a l a r t w o - p o i n t f u n c t i o n s c a l c u l a t e d r e c e n t l y [8], l e a v e s o u r r e s u l t s a l m o s t u n c h a n g e d .
References [1] [2] [3] [4]
[5] [6] [7] [8] 480
For a review and references see: J. Gasser and H. Leutwyler, Phys. Rep. 87C (1982) 77. C. Becchi et al., Z. Phys. C8 (1981) 335. A.L. Kataev, N.N. Krasnikov and A.A. Pivovarov, Phys. Lett. 123B (1983) 93. P.N. Truong, Phys. Lett. l17B (1982) 109; S. Narison and E. de Rafael, Phys. Lett. 103B (1981) 57; S. Narison, N. Paver, E. de Rafael and D. Trelani, Nucl. Phys. B212 (1983) 365; W. Hubschmid and S. Mallik, Nucl. Phys. B193 (1981) 368. N.F. Nasrallah, N.A. Papadopoulos and K. Schilcher, Phys. Lett. l13B (1982) 61; 126B (1983) 379; Z. Phys. C16 (1983) 323. M.A. Shifrnan, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385, 448. Y. Chung, H.G. Dosch, M. Kxemer and D. Schall, Heidelberg preprint HD-THEP-84-1, to be published in Z. Phys. C. S.G. Gorishny, A.L. Kataev and S.A. Larin, Phys. Lett. 135B (1984) 457.
PHYSICS LETTERS
MODEL INDEPENDENT
DETERMINATION
16 August 1984
OF QCD QUARK MASSES
M. K R E M E R 1, N.A. P A P A D O P O U L O S a n d K. S C H I L C H E R
lnstitut far Physik, Johannes Gutenberg Universiti~t, 6500 Main& West Germany Received 5 January 1984 Revised manuscript received 25 April 1984 The method of approximate analytic continuation by duality (ACD), which permits to extrapolate a two-point function from its asymptotic QCD form down to Q2 = 0, is used to determine the value of the light quark masses in QCD. For the value of the four-quark coridensate advocated by Shifman, Vainshtein and Zakharov we obtain (m u + m d) = (21 _+2) MeV a t Q 2 = 1 G e V 2.
A s Q C D b e c a m e the accepted theory of strong interactions, a t t e n t i o n focussed on the d e t e r m i n a t i o n of the f u n d a m e n t a l p a r a m e t e r s of the lagrangian, the strong coupling c o n s t a n t a s a n d the q u a r k masses m i. A d e t e r m i n a t i o n of the q u a r k masses [1] is m u c h m o r e difficult than that of the coupling constant, as they m a n i f e s t themselves m a i n l y in low energy region where p e r t u r b a t i v e Q C D b e c o m e s i n a d e q u a t e . Previous calculations b a s e d on Q C D sum rules or finite energy sum rules therefore either y i e l d e d o n l y lower b o u n d s [2,3] or h a d to m a k e m o r e or less ad hoc assumptionS on the c o n t i n u u m in the p s e u d o s c a l a r channel [4]. In this n o t e we will present a d e t e r m i n a t i o n of the q u a r k masses which is a l m o s t i n d e p e n d e n t of the c o n t i n u u m . It is b a s e d on a m e t h o d of a p p r o x i m a t e analytic c o n t i n u a t i o n b y d u a l i t y ( A C D ) p r o p o s e d p r e v i o u s l y [5] which allows the r e c o n s t r u c t i o n of an a n a l y t i c a m p l i t u d e at small space-like q2 given o n l y its a s y m p t o t i c form for large q2. W e follow the p i o n e e r i n g w o r k of ref. [2] a n d consider the two p o i n t function of the divergence of two axial currents II(q 2) =
ifdx eiqx
(1)
which has in Q C D the o p e r a t o r p r o d u c t e x p a n s i o n [2] YIQCD(t) = ( m , + r o d ) 2 ( - ( 3 / 8 ~ ' 2 ) [ ( 1 + ~ a J r r ) t
l o g ( - - t / / , 2 ) - - ( a s / r r ) t 10g2( -- t / t t 2 ) ]
+ ( 3 / 4 ~ r 2 ) ( m 2 + m 2 _ m . m d ) l o g ( - - t / i t 2) + ( m u + m d ) ( g t q ) / 2 t -- ~<(as/~r)GZ>/t
+ (l12~r/27)as
(2)
or, after r e n o r m a l i z a t i o n group i m p r o v e m e n t I I ( t ) = (3/8~r 2 ) ( r h . + rh d )228/9( - t ) ( 91108( - t / A 2 )] t/9 _ ~10 [log( -- t / A 2 )] + 2 ' 7 / 9 [ ( r h 2 + th 2 - F n , t h d ) / t ] 9 [ l o g ( - t / A 2 ) ] - 7 / 9 +
8/9
~l[lOg(-t/A2)] -8/9 l o g l o g ( - t / A 2 ) }
_ ( th u + th d )228/9 t - 1 [log ( _ t / A 2 )] - 8/9 X (~<(aJ~r)G2>
- ½ ( m . + m d ) ( q q > --
~721ras(qq>2/t
+ higher o r d e r s + p o l y n o m i a l ,
} (3)
I Supported by Bundesministerium fiir Forschung und Technologie. 476
0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
where A is the Q C D scale parameter for which we take A = (150 + 50) MeV, (7qq) the quark condensate, ((o~s/¢r)G~,~G~ ) the gluon condensate and rh i the invariant quark masses which are related to the running quark masses in the MS scheme by ( r n i ( _ Q 2 ) =rhi(½ log(Q2/A2))
4/9
1
28 l o g l o g ( Q 2 / A 2 ) 290 1 ) 36 log(QZ/A2 ) + 3---g--log(Q2/A2 ) .
(4)
In the following we will use ((as/~)Gt~,,Gt'") = 0.012 GeV 4, which is the value given by Shifman et al. (SVZ) [6]. For the value of the dimension six operator Ots{~lq)2 we use two alternative values: (a) The value advocated by SVZ [6] Ots~qq)2 = 1.8 x 10 -4 GeV 6. This is a large value which is, to our opinion, mainly justified by its phenomenological success [5,6]. SVZ justify it, by assuming factorization for the four-quark operator at very low Q2. (b) A "self-consistent" value, which is obtained by assuming factorization at Q 2 = 1 GeV 2 and using the PCAC relation (m~ + rnd)(qq) = -f~m2,,
(5)
in an iterative calculation. We shall now present in some detail the principle involved in our extraction of the quark masses from pseudoscalar Q C D sum rules. It follows from (2) that the second derivative of H ( q 2) satisfies an unsubtracted dispersion relation 2 4
1 f ~ dt
4ff, m~. )3 + - n"(-Q2) (Q2+m 2 "17""9m~
2 (t +
QZ)3
Im II(t).
As Im H ( t ) is a positive definite function one obtains for sufficiently large Q2, where H " ( - Q 2 ) approximated by QCD, the inequality 2 3. H " ( - Q2) > 4fZm4/(Q2 + rn~.)
(6)
is well
(7)
As I I " is proportional to the square of the quark masses, (7) directly translates into a lower bound of these. At Q2 = 1 GeV2; where the nonperturbative higher order term may still be controllable, this would lead to (rn u + md)hOev > 18 MeV. In the past attempts were made to improve the bound of (7) using optimal inequalities [2], Borel transforms or to phenomenologically approximate the continuum in (6) [4]. It is however clear from (6) that the only way to obtain a model independent constraint on II~ct) ( - Q2) or equivalently a prediction of the quark masses is to analytically continue I I ~ c D ( - Q 2 ) down to Q2 = 0. For Q2 = 0 the pion-hole dominates the r.h.s, of (6) and the inequality of (7) becomes an inequality [to the accuracy to which our input parameters are determined see (5)]. The method of ACD, whose effectiveness has been tested previously [5], allows just such an extrapolation of F[~ci~ ( - Q2) to low Q2. In the following we will demonstrate how A C D is applied here. By Cauchy's theorem H satisfies the relation
2 £ r I " ( - Q 2 ) = 2~r~-i dt
n(t) , (t + Q2) 3
(8)
where the contour C of integration consists of a circle of radius R and a cut along the real axis from m 2 to R. 477
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
On the phenomenological side l'I contains a pion pole contribution and a continuum part I I ( - Q2) = 2 f ~ m 4 / ( m ] + Q2) + continuum,
(9)
with f . = 93 MeV. It is clear that the continuum contribution to I I becomes negligible compared to the pion pole when Q2 approaches zero. Taking the pion pole explicitly into account, (6) yields 2
4
2
4f~2m4/(m]+Q2)3+continuum=4f~m=/(m.+Q2)3
+2fg
~'t9mZ~
dtlmH(t) (t+Q2)
3
+
~
2 ~c
II(t)dt
R(t2+Q2)3,
(10)
where C R is the circle of radius R. In the integral along the real axis we use the approximation N
1
- ~ a.t",
(,+Q2) '
fort~[9m2,R].
(11)
.=0
The coefficients a , depend on Q2 and R and must be determined by a least square fit in the given interval. N is finite, its maximal value being limited by the number of terms known in the asymptotic expansion of I I ( t ) . With this approximation one can rewrite the integral
~2g'fm 2n . d(tt l m+ Q2) I I ( t3) -
R n 2 /'9rn ~ x---~ n 2"rrfo ~_,a,t I m I I ( t ) d t - - J o l r 2.,a.t Im I I ( t ) d t
-2 f 2 4 2n = 2~ri a¢ E a , t " Y I ( t ) d t -- 4f; m ~ _ , a , m ~ . R
Eq. (6) then reads
4f,;2rn,~ 4
a , m ,2, + continuum --~--~ 2
n=0
:c(
~ ( t + Q12 ) 3
a,t ~ 1-I(t)dt
(12)
and this is the central result for the following analysis. It is valid for all Q2 <: R. The r.h.s, simulates the pion pole and grows, as expected, rapidly as Q2 _+ 0. Typically it increases relative to the continuum as Q2 is lowered from Q2 = 1 GeV 2 to Q2 = 0 by a factor of 200. Therefore, at Q2 = 0 we obtain in a very good approximation, instead of the inequality of (7) the result 2~ri q~ dt t S -
2. a,t ~ I I ( t ) = 4f~2m=2 ~ a,m,~, n=0
(13)
n=O
which allows the determination of quark masses. Some remarks are in order at this point. First in the integral the function H must be known only on the circle C n, hence one can use the perturbatively calculated expression (3). Furthermore it should be noted that the polynomial is an approximation to 1 / ( t + Q2)3 on the real axis only, so the integral along the circle is by no means zero. As the pion pole has disappeared during the calculation, it is important to check that the pion contribution is still dominant at the 1.h.s. of (12). This can be done once the coefficients a~ are known by comparing it to the value of the actual pion-pole. The separate treatment of the pion-pole is essential for the approximation method used: The approximation of the function 1 / ( t + iQ2) 3 by a polynomial introduces an error which depends on the shape of the imaginary part of the function YI. This error is large for rapidly varying functions, in particular, the 478
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
8-function c o r r e s p o n d i n g to the p i o n - p o l e , a n d it is small for s m o o t h functions. In particular, the error is zero, if the i m a g i n a r y p a r t is a p o l y n o m i a l of a smaller degree than the fit p o l y n o m i a l . T h e r e f o r e the s e p a r a t e t r e a t m e n t of the p i o n - p o l e reduces s u b s t a n t i a l l y the error which is due to the fit. T h e final answers for the q u a r k masses d e p e n d sensitively on the value of the d i m e n s i o n six o p e r a t o r . F o r the SVZ value we o b t a i n (rn u + m a ) , o e v = (21 + 2) MeV, (rnu + m s ) , c e v = (245 _+ 10) MeV.
(14)
U s i n g the f a c t o r i z a t i o n hypothesis, for the four q u a r k c o n d e n s a t e we get ( m u + m a ) = (35 -t- 3) MeV, ( m u + ms) = (270 ___ 10) MeV.
(15)
T h e q u o t e d errors take into a c c o u n t the effects of neglected higher d i m e n s i o n a l condensates, the d e p e n d e n c e on the degree N of the fit p o l y n o m i a l a n d the R - d e p e n d e n c e for small QZ, as well as the very small d e p e n d e n c e on A, the Q C D scale. T h e y do not account, however, for the uncertainties in the values of the i n p u t condensates. This is d e m o n s t r a t e d in p a r t i c u l a r b y the very different values o b t a i n e d for the q u a r k masses in (14) a n d (15). T h e d e p e n d e n c e on the gluon c o n d e n s a t e is less d r a m a t i c . T h e analysis was d o n e for N = 2 . . . . . 5, R = 1.5 to 3 G e V 2 a n d Q 2 = 0.0 to 0.9 G e V 2 ,1. F o r small Q2 the R a n d N d e p e n d e n c e is small, yielding the q u o t e d errors. W e p r e s e n t typical results in fig. 1. It is w o r t h n o t i n g that for the SVZ value of as(Y:tq>2 the p r e d i c t e d q u a r k masses flatten off near Q2 = 0 as a m a n i f e s t a t i o n of the fact that the c o n t i n u u m b e c o m e s i n d e e d negligible. This is in c o n t r a s t to the case of a small f o u r - q u a r k condensate, so that in this case the conclusions are m o r e doubtful. W e have therefore p e r f o r m e d a similar investigation using the s u m rule I I " ( Q2)
"i
f d t I I " t ~t ~ )
I n this case one looses the positive definitiveness s u b s t a n t i a l l y greater accuracy. It turns out that for very stable a n d agrees in this case with (14) a n d stability could be o b t a i n e d . The ratios of the q u a r k masses can b e o b t a i n e d has
of the spectral function, b u t the p o l y n o m i a l fit has a the SVZ value of the four q u a r k c o n d e n s a t e the result is (15), whereas in the case of the small c o n d e n s a t e no i n d e p e n d e n t l y from chiral p e r t u r b a t i o n theory [1]. One
2 m s ( r n . + rod) = 25 -t- 2.5.
(16)
U s i n g m u / Y n = 0.72 from ref. [1] we o b t a i n 2 r n J ( r n u + rod) = 22.9 + 3.2 f r o m (14) a n d 2 m s / ( m . + rod) = 14.9 + 2.0 from (15). T h e result (14) is in a g r e e m e n t with relation (16), the result (15) is not. This c o u l d be t a k e n as an indication, in a d d i t i o n to successful p h e n o m e n o l o g y [5,6], that the four q u a r k c o n d e n s a t e does in fact have the large value a d v o c a t e d b y SVZ. In that case, however, one has to give up factorilization of the v a c u u m e x p e c t a t i o n value of the four q u a r k o p e r a t o r . The a s s u m p t i o n that the ratio of four q u a r k o p e r a t o r m a t r i x elements a p p e a r i n g in different current correlation functions can be d e t e r m i n e d b y the f a c t o r i z a t i o n h y p o t h e s i s m a y p o s s i b l y be m a i n t a i n e d . O u r values for (rn u + rod) are in g o o d a g r e e m e n t with results from b a r y o n i c sum rules in ref. [7], where ( m u + m d ) a O e v = 30--+6 M e V was o b t a i n e d a s s u m i n g f a c t o r i z a t i o n of the four q u a r k condensate, a n d ( m u + md)lOCV = 20 M e V if the four q u a r k c o n d e n s a t e was k e p t as a free p a r a m e t e r . T h e analysis of b a r y o n i c s u m rules also gives some i n d i c a t i o n s that the four q u a r k c o n d e n s a t e might b e s u b s t a n t i a l l y larger ,1 F o r the s t r a n g e q u a r k m a s s we used Q2 = _ 0.2 to 0.9 G e V 2 in o r d e r to get a b e t t e r suppression of the c o n t i n u u m .
479
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
30
÷
E
w
oll o.'2 o:3 0.'~ a's o'6 o.'7 018 02 I-G v 2]
Fig. 1. Lower bounds of quark masses as function of Q2, the argument of the pseudoscalar two-point function. At 0 2 = 0 the bound becomes an equality. The lower curve corresponds to the SVZ value of the four-quark condensate, the upper curve to the one obtained by factorization.
t h a n i m p l i e d b y t h e f a c t o r i z a t i o n h y p o t h e s i s . O u r v a l u e ( m u + m d ) l GeV = 21 + 2 M e V is h i g h e r t h a n t h e o n e g i v e n i n ref. [1], b u t c o n s i d e r e d t h e e r r o r b a r s a n d t h e u n c e r t a i n t y i n t h e q u a r k c o n d e n s a t e , w e d o n o t r e g a r d t h i s d i f f e r e n c e as a r e a l d i s c r e p a n c y . W e t h a n k F. S c h e c k f o r h i s i n t e r e s t i n t h i s w o r k a n d f o r d i s c u s s i o n s .
Note added: T h e i n c l u s i o n o f t h e O(et 2) c o r r e c t i o n t o t h e p s e u d o s c a l a r t w o - p o i n t f u n c t i o n s c a l c u l a t e d r e c e n t l y [8], l e a v e s o u r r e s u l t s a l m o s t u n c h a n g e d .
References [1] [2] [3] [4]
[5] [6] [7] [8] 480
For a review and references see: J. Gasser and H. Leutwyler, Phys. Rep. 87C (1982) 77. C. Becchi et al., Z. Phys. C8 (1981) 335. A.L. Kataev, N.N. Krasnikov and A.A. Pivovarov, Phys. Lett. 123B (1983) 93. P.N. Truong, Phys. Lett. l17B (1982) 109; S. Narison and E. de Rafael, Phys. Lett. 103B (1981) 57; S. Narison, N. Paver, E. de Rafael and D. Trelani, Nucl. Phys. B212 (1983) 365; W. Hubschmid and S. Mallik, Nucl. Phys. B193 (1981) 368. N.F. Nasrallah, N.A. Papadopoulos and K. Schilcher, Phys. Lett. l13B (1982) 61; 126B (1983) 379; Z. Phys. C16 (1983) 323. M.A. Shifrnan, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385, 448. Y. Chung, H.G. Dosch, M. Kxemer and D. Schall, Heidelberg preprint HD-THEP-84-1, to be published in Z. Phys. C. S.G. Gorishny, A.L. Kataev and S.A. Larin, Phys. Lett. 135B (1984) 457.