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The QCD sum rule for the light-heavy quark systems
NUCLEAR PHYSICS A
Nuclear Physics A570 (1994) 167c-172~ North-Holland, Amsterdam
The QCD Sum Rule For The Light-Heavy Leonard
S. Kisslinger
Quark Systems
and Zhenping Li
Physics Department, Carnegie-Mellon Pittsburgh, PA. 15213-3890’
University
The method of QCD sum rules is applied to light-heavy quark systems, with detail treatment of the D and B mesons. We calculate the masses for the pseudo-scalar and vector states and their leptonic
decay constants
in a unified approach.
The applications of the QCD sum rule method[l] to the light (q) and heavy (Q) quark systems have been done more than ten years ago[2,3]. The heavy-light quark systems have been given considerable attention recently due to the finding of the heavy quark symmetry[4], and the effective field theory[ti]. The consequence of the heavy quark symmetry is that the pseudoscalar and the vector states of a light-heavy quark system should have the same masses and leptonic decay constants, and thus the differences of their masses and leptonic decay constants, f, come from the symmetry breaking effects that are of the order of l/M*. They have been calculated for the B systems using the static quark approximation and the l/Mg expansion[6-81. However, application of the static quark approximation to the L? system in which the heavy quark is a charm quark would be more problematic, since it may need the symmetry breaking terms that are proportional to l/M6. The focus in this paper is the application of the QCD sum rules from the standard QCD field theory to the light-heavy system. Since the effective field theory and the full theory differ at tree ievel[fi], th e connection between the QCD sum rule in the full field theory and the static quark theory is by no means trivial. Thus, investigation in the full field theory provides an important test of the QCD sum rules in the effective theory[8]. B ecause no static quark approximation is used, one could also apply the QCD sum rule in the full theory to the D system which can not be easily done in the effective theory. The starting point of the QCD sum rules is the evaluation of the two point function I&(Q) = i 1 $4zei9r (Ol~(~~(~)~~(O))l6) for the pseudoscalar n,&)
current
Jo
=: $(s)iy~&(r)
=
i J ~~e’q”(61~(V,(~)V,(0))10)
=
(%&V - g*~~z)~(1)(~2)
(11 :, and
+ ~~q”~(“)(~z)
*This work is supported by the U.S. National Science Foundation grant PHY-9023586 0375-~74~4/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved. SSDI 0375-9474(~)~88-5
(2)
L.S. Kisslinger, Z. Li I The QCD sum rule
168c
is evaluated in for the vector current V,(z) =: ~(~)~~Q(~) :. The correlator II($) two ways; it is calculated starting from QCD by the operator product expansion, and it is treated phenomenologically by a dispersion relation. This gives a microscopic nonperturbative QCD evaluation of physical properties. The QCD operator expansion can be written as CrI + us
%12) = -t
+ Cs{@(~ - GM
G(oS2)
+ Cs (g&~))2
(3)
+ G(o~G2)((rQj
where the coefficient Cr comes from the short distance correlation of the pseudo-scalar or vector currents, and has been calculated to the order CY.in the literature[9,lO], and C, are the Wilson coefficients, which represents the nonperturbative effects. Up to order o, the coefficient C&l] is given by c3
=
$!f&f
(4
(lf2)
where the positive sign corresponds to the pseudo scalar current and the negative sign to the vector current, the coefficient Cr is Gs =
M2 2(q2 - M,$)2
1
MS
(5)
q2 - M$ + K
for the pseudo scalar currents, and
c5 =
M;;
2k2 -
M6)3
for the vector current. The coefficient for the gluonic condensate Cd is given by
c_g=zk
l
(7)
127r(q2 -M$)
where the pseudoscalar (vector) efficient C’s has the expression
current gives the positive (negative)
2
sign, and the co-
MQ
3Mg+(q’-4) for the pseudo scalar current and 2M; G = -27($2 - M$)s
M; (*2 - Mi)"
(9) +
9(q2
for the vector current. The coefficient Cr is
~~c-1
MQ
24~~ ( g2 - M$ )”
_llM$
q2 - M$
for the pseudoscalar current, and
1
(10)
L.S. Kisslinger, 2. Li I The QCD sum rule
c,=_L
2M;
MO
24n3 (q2 - Mi)3
q2 - Mz,
+A+ 3
2M4 (q2 - M$2
1
169~
(11)
for the vector current. To minimize the excited states and the continuum contribution to the spectral function, the Bore1 transformation[l] shouid be used. It is convenient to define the quantity u=-
M;-q2 .-%l.c
(12)
which measures the vertuality
of heavy quarks, so that the Bore1 transformation
is
defined as
[q wln=ugifixed
(-&)“I f(w)
(13)
Thus, the sum rules for the pseudoscalar and vector current are
IIP”(WB) =
(14)
and lY(wa)
=
(15)
_
bG2)(+d 2732s3w4 B
The physical part of the pseudo scalar and the vector current is
IIp*
phys =
f2Xe-+
(16)
p 2M;
and (17) AP-MC
where A = quark system:‘: M2 = M$ +
~MQw;
and M is the mass of the meson states. The mass of the light-heavy obtained by
d bl-+B)] dwB
08)
L.S. Kisslinger, Z. Li I The QCD sum rule
17oc
-.a.
-.*.t.,
.,
I.
I.*
1.4
(cev) Figure 1. The pseudoscalar vector mass difference of the pseudoscalar state for the B systems(right). WB
and the leptonic
f, =
decay constant
(left) and the mass
is give by
(~,%ny,,) f
(19)
and
Therefore given b>
the mass difference between the pseudo scalar and the vector meson states is
(21)
The mass difference between pseudoscalar and vector mesons are obtained by adjust,ing the parameters wP and w, to insure that t.he mass difference is independent of Bore1
..b
..
.a
GB
(Ce;‘)
I.
‘.“.*I
..
.. WB
Figure 2. The pseudoscalar rector mass difference(left) of the pseudoscalar state for the D systems(right,).
.I
1.
(GeV)
and the mass
1.1
L.S. Kisslinger, Z. Li I The QCD sum rule
parameter
171c
wg for a finite WE, in the same time, the masses for the pseudoscalar
and
vector states should also be independent of WB. In Fig. 1, we show the result for the B systems, Figure l-a shows the mass difference between the pseudoscalar and vector mesons, and Figure l-b shows the result for the mass of the vector B meson states, the agreement with experiments is very good. We show the result for the C systems in Figure 2. The parameter wp and wv in this analysis are wp - 0.68 - 0.72 GeV and w, = 0.76 - 0.80 GeV for the B system and wp - 0.60 - 0.66 GeV, w, - 0.74 - 0.8 GeV. The mass difference between the pseudoscalar and vector current states is generated the different Wilson coefficients in Eq. 3 for the pseudoscalar and vector states at order (Y, and l/M*, in particular the contributions from the coefficients Cl and C’s are most important,and because they are of order (Y, and independent of MQ the quantity Mi - M$ in Eq. 21 decreases as the heavy quark mass MQ decreases, which is the opposite to the trend of the data. The renormalization group effects[8] might be the source for such discrepancy. The leptonic decay constants are obtained automatically from this analysis, we find fB = 95 MeV, f~* = 103 MeV, and fo = 130 MeV, foe = 150 MeV, which are consistent with other analysis. In conclusion, we present a unified approach to the light-heavy quark systems. We find the pseudoscalar-vector mass difference M,’ - Mi = 0.47 GeV2 for the B system which agrees with the data, and Mz - Mj = 0.40 GeV’ for the D systems. The same formalism can also be used to calculate the isospin violations for the mass difference using the methods recently applied to the neutron-proton mass difference[l2], which is in progress and will be presented elsewhere[ll].
REFERENCES 1.
M. A. Shifman, A.I. Vainshtein and V.I. Zskharov, Nucl. Phys. Bl47, 385, 448(1979). 2. E.V. Shuryak, Nucl. Phys. B198,83(1982). 3. T. M. Aliev and V. L. Eleskii Yad. Fiz. 38, 1537(1983) [Sov. J. Nucl. Phys. 38, 936( 1983). 4. N. Isgur and M.B. Wise, Phys. Lett. B232 113(1989). 5. E. Eichten and B. Hill, Phys. Lett. B234 511(1990), D. J. Broadhurst and A. G. Grozin, Phys. Lett. B267, 105(1991). 6. 7.
C. A. Dominguez and N. Paver, Phys. Lett. B276,179(1992). M. Neubert, Phys. Rev. D46, 1076(1992).
8. 9.
E. Bagan, P. Ball, V. M. Braun and H. G, Dosch, Phys. Lett. B278,457(1992). D. Broadhurst, Phys. Lett. BlOl, 423(1981), S. C. Genera& Ph.D. Thesis, Open University report No. OUT-4102-13(1984).
10. L. J. Reinders, S. Yazaki, and R. Rubinstien, 257( 1980). 11. L. Kisslinger and Zhenping Li to be published. 12. K-C. Yang, et al to appear in Phys. Rev. D.
Phys. Lett. B103,
63(1981),
B97,
Nuclear Physics A570 (1994) 167c-172~ North-Holland, Amsterdam
The QCD Sum Rule For The Light-Heavy Leonard
S. Kisslinger
Quark Systems
and Zhenping Li
Physics Department, Carnegie-Mellon Pittsburgh, PA. 15213-3890’
University
The method of QCD sum rules is applied to light-heavy quark systems, with detail treatment of the D and B mesons. We calculate the masses for the pseudo-scalar and vector states and their leptonic
decay constants
in a unified approach.
The applications of the QCD sum rule method[l] to the light (q) and heavy (Q) quark systems have been done more than ten years ago[2,3]. The heavy-light quark systems have been given considerable attention recently due to the finding of the heavy quark symmetry[4], and the effective field theory[ti]. The consequence of the heavy quark symmetry is that the pseudoscalar and the vector states of a light-heavy quark system should have the same masses and leptonic decay constants, and thus the differences of their masses and leptonic decay constants, f, come from the symmetry breaking effects that are of the order of l/M*. They have been calculated for the B systems using the static quark approximation and the l/Mg expansion[6-81. However, application of the static quark approximation to the L? system in which the heavy quark is a charm quark would be more problematic, since it may need the symmetry breaking terms that are proportional to l/M6. The focus in this paper is the application of the QCD sum rules from the standard QCD field theory to the light-heavy system. Since the effective field theory and the full theory differ at tree ievel[fi], th e connection between the QCD sum rule in the full field theory and the static quark theory is by no means trivial. Thus, investigation in the full field theory provides an important test of the QCD sum rules in the effective theory[8]. B ecause no static quark approximation is used, one could also apply the QCD sum rule in the full theory to the D system which can not be easily done in the effective theory. The starting point of the QCD sum rules is the evaluation of the two point function I&(Q) = i 1 $4zei9r (Ol~(~~(~)~~(O))l6) for the pseudoscalar n,&)
current
Jo
=: $(s)iy~&(r)
=
i J ~~e’q”(61~(V,(~)V,(0))10)
=
(%&V - g*~~z)~(1)(~2)
(11 :, and
+ ~~q”~(“)(~z)
*This work is supported by the U.S. National Science Foundation grant PHY-9023586 0375-~74~4/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved. SSDI 0375-9474(~)~88-5
(2)
L.S. Kisslinger, Z. Li I The QCD sum rule
168c
is evaluated in for the vector current V,(z) =: ~(~)~~Q(~) :. The correlator II($) two ways; it is calculated starting from QCD by the operator product expansion, and it is treated phenomenologically by a dispersion relation. This gives a microscopic nonperturbative QCD evaluation of physical properties. The QCD operator expansion can be written as CrI + us
%12) = -t
+ Cs{@(~ - GM
G(oS2)
+ Cs (g&~))2
(3)
+ G(o~G2)((rQj
where the coefficient Cr comes from the short distance correlation of the pseudo-scalar or vector currents, and has been calculated to the order CY.in the literature[9,lO], and C, are the Wilson coefficients, which represents the nonperturbative effects. Up to order o, the coefficient C&l] is given by c3
=
$!f&f
(4
(lf2)
where the positive sign corresponds to the pseudo scalar current and the negative sign to the vector current, the coefficient Cr is Gs =
M2 2(q2 - M,$)2
1
MS
(5)
q2 - M$ + K
for the pseudo scalar currents, and
c5 =
M;;
2k2 -
M6)3
for the vector current. The coefficient for the gluonic condensate Cd is given by
c_g=zk
l
(7)
127r(q2 -M$)
where the pseudoscalar (vector) efficient C’s has the expression
current gives the positive (negative)
2
sign, and the co-
MQ
3Mg+(q’-4) for the pseudo scalar current and 2M; G = -27($2 - M$)s
M; (*2 - Mi)"
(9) +
9(q2
for the vector current. The coefficient Cr is
~~c-1
MQ
24~~ ( g2 - M$ )”
_llM$
q2 - M$
for the pseudoscalar current, and
1
(10)
L.S. Kisslinger, 2. Li I The QCD sum rule
c,=_L
2M;
MO
24n3 (q2 - Mi)3
q2 - Mz,
+A+ 3
2M4 (q2 - M$2
1
169~
(11)
for the vector current. To minimize the excited states and the continuum contribution to the spectral function, the Bore1 transformation[l] shouid be used. It is convenient to define the quantity u=-
M;-q2 .-%l.c
(12)
which measures the vertuality
of heavy quarks, so that the Bore1 transformation
is
defined as
[q wln=ugifixed
(-&)“I f(w)
(13)
Thus, the sum rules for the pseudoscalar and vector current are
IIP”(WB) =
(14)
and lY(wa)
=
(15)
_
bG2)(+d 2732s3w4 B
The physical part of the pseudo scalar and the vector current is
IIp*
phys =
f2Xe-+
(16)
p 2M;
and (17) AP-MC
where A = quark system:‘: M2 = M$ +
~MQw;
and M is the mass of the meson states. The mass of the light-heavy obtained by
d bl-+B)] dwB
08)
L.S. Kisslinger, Z. Li I The QCD sum rule
17oc
-.a.
-.*.t.,
.,
I.
I.*
1.4
(cev) Figure 1. The pseudoscalar vector mass difference of the pseudoscalar state for the B systems(right). WB
and the leptonic
f, =
decay constant
(left) and the mass
is give by
(~,%ny,,) f
(19)
and
Therefore given b>
the mass difference between the pseudo scalar and the vector meson states is
(21)
The mass difference between pseudoscalar and vector mesons are obtained by adjust,ing the parameters wP and w, to insure that t.he mass difference is independent of Bore1
..b
..
.a
GB
(Ce;‘)
I.
‘.“.*I
..
.. WB
Figure 2. The pseudoscalar rector mass difference(left) of the pseudoscalar state for the D systems(right,).
.I
1.
(GeV)
and the mass
1.1
L.S. Kisslinger, Z. Li I The QCD sum rule
parameter
171c
wg for a finite WE, in the same time, the masses for the pseudoscalar
and
vector states should also be independent of WB. In Fig. 1, we show the result for the B systems, Figure l-a shows the mass difference between the pseudoscalar and vector mesons, and Figure l-b shows the result for the mass of the vector B meson states, the agreement with experiments is very good. We show the result for the C systems in Figure 2. The parameter wp and wv in this analysis are wp - 0.68 - 0.72 GeV and w, = 0.76 - 0.80 GeV for the B system and wp - 0.60 - 0.66 GeV, w, - 0.74 - 0.8 GeV. The mass difference between the pseudoscalar and vector current states is generated the different Wilson coefficients in Eq. 3 for the pseudoscalar and vector states at order (Y, and l/M*, in particular the contributions from the coefficients Cl and C’s are most important,and because they are of order (Y, and independent of MQ the quantity Mi - M$ in Eq. 21 decreases as the heavy quark mass MQ decreases, which is the opposite to the trend of the data. The renormalization group effects[8] might be the source for such discrepancy. The leptonic decay constants are obtained automatically from this analysis, we find fB = 95 MeV, f~* = 103 MeV, and fo = 130 MeV, foe = 150 MeV, which are consistent with other analysis. In conclusion, we present a unified approach to the light-heavy quark systems. We find the pseudoscalar-vector mass difference M,’ - Mi = 0.47 GeV2 for the B system which agrees with the data, and Mz - Mj = 0.40 GeV’ for the D systems. The same formalism can also be used to calculate the isospin violations for the mass difference using the methods recently applied to the neutron-proton mass difference[l2], which is in progress and will be presented elsewhere[ll].
REFERENCES 1.
M. A. Shifman, A.I. Vainshtein and V.I. Zskharov, Nucl. Phys. Bl47, 385, 448(1979). 2. E.V. Shuryak, Nucl. Phys. B198,83(1982). 3. T. M. Aliev and V. L. Eleskii Yad. Fiz. 38, 1537(1983) [Sov. J. Nucl. Phys. 38, 936( 1983). 4. N. Isgur and M.B. Wise, Phys. Lett. B232 113(1989). 5. E. Eichten and B. Hill, Phys. Lett. B234 511(1990), D. J. Broadhurst and A. G. Grozin, Phys. Lett. B267, 105(1991). 6. 7.
C. A. Dominguez and N. Paver, Phys. Lett. B276,179(1992). M. Neubert, Phys. Rev. D46, 1076(1992).
8. 9.
E. Bagan, P. Ball, V. M. Braun and H. G, Dosch, Phys. Lett. B278,457(1992). D. Broadhurst, Phys. Lett. BlOl, 423(1981), S. C. Genera& Ph.D. Thesis, Open University report No. OUT-4102-13(1984).
10. L. J. Reinders, S. Yazaki, and R. Rubinstien, 257( 1980). 11. L. Kisslinger and Zhenping Li to be published. 12. K-C. Yang, et al to appear in Phys. Rev. D.
Phys. Lett. B103,
63(1981),
B97,