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Volume 253, number 1,2 PHYSICS LETTERS B 3 January 1991
QCD determination of the strange-quark mass
C.A. Dominguez, C. van G e n d Institute of Theoretical Physics and Astrophysics, University of Cape Town, Rondebosch 7700, Cape, RSA
and
N. Paver Dipartimento di Fisica Teorica, Universitd di Trieste, and Istituto Nazionale di Fisica Nucleare, sezione di Trieste, 1-34100 Trieste, Italy
Received 1 October 1990
The strange-quark mass is determined from finite energy and Laplace transform QCD sum rules for the two-point function involving strangeness changing vector currents and their divergences. Improved QCD input and experimental data in the I = ½, S- wave Kn amplitude are used to obtain: ths = 266 + 29 MeV, or ms ( 1 GeV) -- 194 + 4 MeV, where rhs and r~ are the invariant and the running quark masses, respectively.
The strange-quark mass that enters the QCD lagrangian is an important parameter which measures the size of chiral SU (3) × SU (3) as well as flavour SU (3) symmetry breaking. It also impacts on weak hadronic physics involving strangeness. Hence, an accurate knowledge of its value is of the utmost importance. Since the values of the up- and down-quark masses are now known with reasonable precision [ 1 ], from a QCD sum rule analysis combined with experimental data in the pionic sector, it is possible to determine ms from the current algebra ratio [ 2 ]
m~ - 1 2 . 9 + 1 . 3 . (1) mu + m d
In this way one finds [ 1 ]
~ = 2 8 8 + 4 8 MeV, (2)
where rhs is the invariant quark mass, related to the running mass rhs (Q2) through
rhs 2a~(Q ) 1 (3) r~(Q2) = ( ~ - - ~ 2 ) ~ ( + a a ~ ( Q 2 ) )
at the two-loop level, and where
rtd -~2 Q 2 c~(Q2 ) - In +blnlnA--5, (4)
with d= 4, b=~d-~°-{-~d 2, and 5 23 a = g + ~ d - b , for three colours and three flavours. This determination of m~ is model independent, but unfortunately the error is rather large.
Also supported by MPI (Italian Ministry of Education ).
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 241
Volume 253, number 1,2 PHYSICS LETTERS B 3 January 1991
The authors of ref. [ 3 ] obtained bounds on ms by combining QCD information on the two-point function involving the strangeness-changing vector current divergence together with experimental data on the J = 0, I = ½ Kn system. Recent progress in the theoretical [ 4 ], as well as in the experimental [ 5] understanding of this channel calls for a reanalysis of this problem. In fact, the authors of ref. [4 ] have computed two-loop quark mass corrections of order O(m2/Q 2) and O(m4/Q 4) to the correlators of light-quark currents, and shown that logarithmic quark-mass singularities can be removed. This impacts on the order O ( m 4 / a 4 ) perturbative term and on the non-perturbative quark condensate (mqqq) term in the operator product expansion. Thanks to this development we are now in possession of an improved and more accurate theoretical expression to be used in the QCD sum rules. On the experimental side, a high statistics study of the reaction K - p ~ K - n + n [ 5 ] provides valuable information on the magnitudes and phases of the I= ½, S-wave Kit amplitude from threshold up to 2.6 GeV (earlier analyses reached only 1.6 GeV ). This is extremely important because it will guarantee a very good saturation of the hadronic dispersive integrals in the QCD sum rules, making the determination of ms model independent.
We begin by defining the two-point function
f quq~ ~2~ Hu~ =i d4x exp (iqx) (01 T[ Vu(x) V*~(0) ] [0) = ( -guvq2+quqv)Hv(q 2) + --~-Hs(q ~, (5)
where Vu(x ) + :g(x)yuu(x):, and consider the function
0 Hs(q z) ~(q2) = Oq 2 q2 (6)
From the results of ref. [4] one can easily calculate ~(Q2), Q2_ _ q2, to two loops in perturbative QCD, includ- ing mass corrections of order O (rn~/Qa), and then add non-perturbative power corrections. In this way we find
{ 24 t~2(Q2, ( zt 31) 12 r~a(Q2, ( 2zr 59) 3 rhdQ:) 17 as(Q2) + Q2 ) ~ + 0 4 + ] ~ ~(Q2)_ 8 n 2 02 1 + 3 rC 21 as(--Q 2 21 ad--Q 2)
2re ( a ' G 2 ) 8~2(msgs)[ 14adQZ) +2 (au)(1+ 17 adQ2))]} (7) + 3 Q4 3 Q4 1 + 3 ~ (gs) 3 - '
defined up to the subtraction constant
~u(O) = - ( m s ( g s - a u ) ) ,
where the two-point function ~u(Q2) is
~,=i f d4xexp(iqx) (OlT[OuVu(x)O~V~(O)] IO) •
(8)
(9)
In eqs. (7), (8) we have neglected the up-quark mass; this is justified a posteriori from the accuracy with which ms can be determined. The subtraction constant 9,(0) measures SU ( 3 ) flavour symmetry breaking in the QCD non-perturbative vacuum, and hence one expects ( ~ s ) / ( ~ u ) to be close to unity. Nevertheless, even allowing for sizeable departures from unity, the numerical value of ~,( 0 ) is about one order of magnitude smaller than the rest of the terms in ~(Q 2), and hence we shall neglect it.
The function ~(Q2) satisfies the following finite energy sum rule (FESR) [ 1 ]:
i d S l l m ~ , l R E s = 3 r~2(So)So[l+Rl(so)+24r~2(So)( rr 31)1 s ~ ~ 21 So as~o) ~ ' (10)
0
where the radiative correction R, (So) is given by
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Volume 253, number 1,2 PHYSICS LETTERS B 3 January 1991
R,(so)=a,(So)I~+2_ 2( f l 2 ] + ~ l n l n S O ] ---Z- ~, r~-;",8,) ~ ' ( l l )
and Im ~u(s) IRES is the hadronic spectral function. This FESR fixes the strange-quark mass which is obviously a function of So, the asymptotic freedom threshold, i.e. the energy squared at which the resonance contribution to the spectral function merges into the perturbative QCD expression. Predictions for rh~ from eq. (1) will be meaningful provided a wide range of values of So exists such that rh~ does not change appreciably (duality region).
We have made a fit to the experimental data on Im g(s) from threshold up to s= 6.8 GeV 2, in terms of two resonant Breit-Wigner forms for K~(1430) and K~(1950) plus non-resonant background. After substituting this fit in eq. ( 10 ) one obtains the values of rh~ shown in fig. 1 for A = 100 MeV [ curve (a) ] and A = 200 MeV [ curve (b) ]. As seen from fig. 1 there is a wide region of stability in So extending from So- 3 GeV 2 up to So = 6 GeV 2. In this region one reads
~ = 2 8 4 - 2 8 8 M e V ( A = I 0 0 M e V ) , (12) rh~=238-245MeV (A=200MeV) . (13)
These values translate into rhs( 1 GeV) = 190-193 MeV for A= 100 MeV, and n~( 1 GeV) = 192-198 MeV for A=200 MeV, where ra~( 1 GeV) is the running mass at 1 GeV [see eq. (3) ].
In order to gauge the systematic uncertainties of the above determination we perform a Laplace transform analysis of the second derivative of the function q/( Q2 ) defined in eq. ( 9 ). From the results of ref. [ 4 ] we easily obtain
3 m 2 [ 11 a s _ 2 ~'QcD(Q2) = 8rt2 QZ 1+ -~,----~- Q4 s
2n (oqG2> <msqq)( ~ ) ] + 3 Q4 + 8n2 Q4 1 + (14)
The Laplace transform oo
1 f (-~22) llmq/(s)_ £ [q / , (Q2) ]= M-~ dsexp rt 0
( 1 5 )
can be calculated in the standard fashion, with the result
350
310
f
190
150 2.0
270
~ 230
I I I 1
I I I I 2.8 3.6 4.4 5.2
So (Ge V z)
6.0 ig. 1. Results of the FESR, eq. (10), f o r A = 100 MeV [curve (a) ], and A = 2 0 0 MeV [curve (b ) ] .
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Volume 253, number 1,2 PHYSICS LETTERS B 3 January ! 991
3 rh 2 1 / ~ [ e / Q c D ( Q 2 ) ] = 8 1 r 2 M 2 [1 In ( M 2 / A z ) ] - - 2 ? l / i l l
X(,-l- d~ [3 ~ M 2 4 ( Y, ,82"~I - - --71e/(1)+ lnln~ ~171 72-- .al/J
-2~ [½ In (M~/A~)]-:~'/P' I+ --n -- -2y,~,(2)+ 8 13)'a~ In In A~
4 ~: 1 [149 fl, ,n ~ fl' ( 6 y , ) 7M"[½1n(MVA:)]-'~'/P'I_24 2 2 \,a, +1 ~(3)
~ 2 + ~ ) I n In AM----~ 2 12 (Y2--Y, '82'~I +(12 b'-]- b', ~,, .a , / j
zr<a, G2> <m, qq> { a ~ [ 2 2 ~ M 2 -t- 3 M4 +47r 2 M4 1-1- n 1 3 - 7 , ~ u ( 3 ) + 4 ln ln~-~
~6~I 71
~2 4 (Y2--Y'~-I) ]})"
/~, 7, (16)
where fl, = - 9, f12 = - 8, Y l = 2, 72 = 7.5 8 3 3 3..., and ~,(n ) is the digamma function. In eq. ( 15 ) < au > = < gs > has been assumed; a posteriori, departures from the SU(3 ) limit have no impact on the results. Concerning the hadronic spectral function, it can be written as usual as
! Im ~u(s) I HAD = __1 Im ~u(s) I RES O(So --S) + 1 im ~¢(s) IQCD O(S--So) , ( 17)
/£
where the hadronic cont inuum is identified with the asymptotic freedom expression
11m ~,(s) IocD_ r ~ s 1 + -~---~- (18) 7t
and Im ~v(s) IRES is given by the same fit to the data as in the FESR analysis. The onset of the continuum, as measured to So, is not fixed in the Laplace transform sum rule. However, it is to be expected that So be near the upper limit of the experimental data, i.e. So= 6-7 GeV z. At the same time, in order to have consistency with the FESR results So should be closer to the lower end of that range. In fig. 2 we show the result of the Laplace
350 300 I I I I
250
2OO 1.0
300 ~ ~ fb)
I i i I
2.6 4.2 5.8 7.4 9.0
M 2 (Ge v 2)
Fig. 2. Results of the Laplace transform sum rule, eq. (! 6 ), for A= 100 MeV and so=6 OeV 2 [curve (a) l, so=6.5 OeV 2 [curve (b)],andso=7 GeV 2 [curve (c)].
.£
250
200
I I l I
(a) (b) (¢)
1
150 i J ~. J 1.0 2.6 4.2 5 8 7.4 9.0
M 2(Ge V 2)
Fig. 3. Results of the Laplace transform sum rule, eq. ( 16 ), for A=200 MeV and So=6 GeV 2 [curve (a)] , so=6.5 GeV 2 [curve(b) ], and So= 7 GeV 2 [curve (c) I.
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Volume 253, number 1,2 PHYSICS LETTERS B 3 January ! 991
determination ofrhs as a function o f M 2 forA = 100 MeV and So= 6.0 GeV 2 [curve (a) ], 6.5 GeV 2 [curve (b) ], and 7.0 GeV 2 [curve (c) ]. Fig. 3 corresponds to A = 200 MeV, and So = 6.0 GeV 2 [curve (a) ], 6.5 GeV 2 [curve (b) ], and 7.0 GeV 2 [curve (c) ]. As may be appreciated from these figures, rhs depends very mildly on M 2 and exhibits a slight dependence on So. The values one reads from figs. 2, 3 are
r h s = 2 8 9 - 2 9 5 M e V ( A = 1 0 0 M e V ) , (19)
r h s = 2 3 7 - 2 4 4 MeV ( A = 2 0 0 M e V ) , (20)
corresponding to So = 6-6.5 GeV 2. Combining the FESR results (12), (13 ) with the above, we arrive at the combined prediction for rh~ of
rh~ = 2 6 6 + 29 M e V , (21)
and the running mass at 1 GeV
n~(1 G e V ) = 194+4 M e V . (22)
The result o f the present determination o f rh~ is consistent with the bounds o f ref. [ 3 ], and agrees with the estimate (2). However, in our determination the uncertainty is appreciably reduced, owing to the improved QCD expansion and the more complete phenomenological input (experimental data) used in the sum rules. The present result should be relevant in order to reduce the uncertainties o f theoretical calculations of kaon transition matrix elements, in particular o f the analysis o f CP violation in K--. 2n [ 6 ].
One of us (C.A.D.) wishes to acknowledge support f rom the Istituto Nazionale di Fisica Nucleare (Italy), and the Foundat ion for Research Development (South Africa), during the course of this research.
References
[ 1 ] C.A. Dominguez and E. de Rafael, Ann. Phys. (NY) 174 (1987) 372. [ 2 ] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 ( 1985 ) 465. [3] S. Narison, N. Paver, E. de Rafael and D. Treleani, Nucl. Phys. B 212 (1983) 365;
see also: C.A. Dominguez and M. Loewe, Phys. Rev. D 31 ( 1985 ) 2930. [4 ] D.J. Broadhurst and S.C. Generalis, Open University Report No. OUT-4102-22 (1988 ). [ 5 ] D. Aston et al., Nucl. Phys. B 296 ( 1988 ) 493. [6] G. Buchalla, A.J. Buras and M.K. Harlander, Nucl. Phys. B 337 (1990) 313.
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