- Email: [email protected]
D*Dπ and B*Bπ form factors from QCD sum rules
SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 96 (2001) 381-385
D*Dvr and B*BT form factors from QCD Sum Rules M. Nielsen a , F.S. Navarra a , M.E. Bracco b , M. Chiapparini b and C. L. Schat b
aInstituto de Fisica, Universidade de SBo Paulo, C.P. 66318, 05389-970 Sao Paulo, SP, Brazil bInstituto de Fisica, Universidade do Estado do Rio de Janeiro, Rua Sao Francisco Xavier 524 - 20559-900, Rio de Janeiro, RJ, Brazil The H’Hn form factor for H = B and D mesons is evaluated in a QCD sum rule calculation. We study the Bore1 sum rule for the three point function of two pseudoscalar and one vector meson currents up to order four in the operator product expansion. The double Bore1 transform is performed with respect to the heavy meson momenta. We discuss the momentum dependence of the form factors and two different approaches to extract the H’ HX coupling constant.
A dense partonic system, often called the quark-gluon plasma (QGP), is expected to be formed in heavy ion collisions at the relativistic heavy ion collider (RHIC), which will soon start to operate at the Brookhaven National Laboratory. As a matter of fact, very recently the NA50 Collaboration reported [l] that the anomalous suppression of J/$ observed in Pb + Pb collisions at CERN-SPS indicated already the formation of QGP, since these experimental data ruled out conventional hadronic models for the J/$Jsuppression, However, there are calculations that reproduce the new NA50 data reasonably well up to the highest transverse energies [2,3], based on hadronic J/I/J dissociation alone. Therefore, while there are suggestions that the anomalous suppression may be due to the formation of the QGP, other more conventional mechanisms based on J/I+LJ absorption by comovers and nucleons has to be still considered. Various approaches have been used in evaluating the charmonium absorption cross section by hadrons. One of them uses meson exchange models based on hadronic effective lagrangians. Some of the processes considered in these models involve the D’Dn vertex [4] where a monopole form factor, with a unknown parameter (the cut-off), is assumed. The results obtained for the corss section are very sensitive to the couplings and to 0920-5632/01/% - see front matter 0 2001 Elsevier Science B.V PI1 SO920-5632(01)01156-2
the form factor [4,5]. Therefore it is very important to estimate these couplings and form factors with a theoretically founded model. In this work we use the QCD sum rule (QCDSR) approach based on the three-point function to evaluate the D*Dr and B’Bn form factors and coupling constants. The D’Dr and B*Bx couplings have been studied by several authors using different approaches of the QCDSR: two point function combined with soft pion techniques [6], light cone sum rules [7], light cone sum rules including perturbative corrections [8], sum rules in a external field [9], double momentum sum rules [lo]. Unfortunately, the numerical results from these calculations may differ by almost a factor two. The advantage of using the three-point function approach with a double Bore1 transformation compared with the twopoint function with a single Bore1 transformation is the elimination of the terms associated with the pole-continuum transitions [7]. The three-point function associated with a H*Hr vertex, where H and H* are respectively the lowest pseudoscalar and vector heavy mesons, is given by WP,P’)
=
J &&, eiP’.l
,-i(P’-7d.Y
(OlT{j(z)js(Y)j;(O)]lO)
x
i
(1)
where j = i@n,u, js = iiiTsd and ji = dy,Q are All rights reserved.
M. Nielsen et al. /Nucleur
382
the interpolating fields for H, tively with u, d and Q being heavy quark fields. The phenomenological side tion, F,(p, p’), is obtained by H and H” state contribution ment in Eq. (1) [II]:
I?fphen)(p,p’) = CHH.
T- and H* respecthe up, down, and of the vertex functhe consideration of to the matrix ele-
‘,yf:’
p2
_lm2X H’
7r
1 P
r2_m2
Physics B (Proc. Suppl.)
m~*+m~-q2 -P:
+
2m2,*
H
higher resonances
p, >
,
+ (2)
where CHH. =
m$mH’mafHfH.
(mu +
fir
(3)
md)mQ
The QCD side, or theoretical side, function is evaluated by performing erator product expansion (OPE) of in Eq. (1). Writing Fp in terms of amplitudes: rp (P, P’) = rl (~2, pr2, 4%p+:2
of the vertex Wilson’s opthe operator the invariant
(A d2, 4.%;
,(4)
we can write a double dispersion relation for each one of the invariant amplitudes I’i (i = 1,2), over the virtualities p2 and p12 holding Q2 = -q2 fixed: ri(p2,p’2,Q2)
= -
O” dsdu pi(s, u, Q2) I m:, 47r2 (s - p2)(u - pt2)
Y(5)
where pi (3, u, Q2) equals the double discontinuity of the amplitude Ii (p2, p/2) Q2) on the cuts rni 5 s < 00, rn; 5 u < co, which can be evaluated u&g Cutkosky’s riles [12]. Finally we perform a double Bore1 transformation in both variables P2 = -p2 and PI2 = -p12 and equate the two representations described above. We get one sum rule for each invariant function. In this paper we focus on the structure p, which we found to be the more stable one. We get: m&. -CHH*
+ m&
+ Q2
gH*Hrh2)
2m2,,
2 Ml2 = -1 e -m~~lM2e-m~l p1 (s, u, Q2)e-sIMZe-UIM’2
x
Q2+mi
J0 / Q So ds ,2
4lr2 1
,
u” du [ m2
96 (2001) 381-385
where se and us are the continuum thresholds for the H* and H mesons respectively, which are, in general, taken from the mass sum rules. The two Bore1 masses M2 and M’2 are completely independent. However, for simplicity, we will study the sum rule as a function of M2 at a fixed ratio M2 rn&. M’2=y$.
(7)
We consider diagrams up to dimension four which include the perturbative diagram and the gluon condensate. The quark condensate term does not contribute since it depends only on one external momentum and, therefore, it is eliminated by the double Bore1 transformation. Higher dimension condensates are strongly suppressed in the case of heavy quarks [6,7,9,10]. The double discontinuity of the perturbative contribution reads:
and the integration (s - m$)(u
limit condition
- m$) 2 Q2m$ .
is (9)
For consistency we use in our analysis the QCDSR expressions for the decay constants up to dimension four in lowest order of os [7]. The parameter values used in all calculations are m,, + md = 14 MeV, m, = 1.5 GeV, mb = 4.7 GeV, m, = 140 MeV, mD = 1.87 GeV, mD* = 2.01 GeV, mB = 5.28 GeV, = 5.33 GeV, fn = 131.5 MeV, (ijq) = mBq -(0.23)3 GeV3, (g2G2) = 0.5 GeV4. The continuum thresholds for the sum rule Eq. (6) are of order SO = (mH* + A,)2 and uc = (mH + A,)2 with A, = A, = 0.5 GeV. In our study we will allow for a small variation in A, and A,, to test the sensitivity of our results to the continuum contribution. We first discuss the D’Dr form factor. In Fig. 1 we show the behavior of the perturbative and gluon condensate contributions to the form factor go.Dn(Q2) at Q2 = 1 GeV as a function of the Bore1 mass M2 using A, = A,, = 0.5 GeV. We can see that, in the case of the form factor, the gluon condensate is not negligible and it helps the
hf. Nielsen
et al. /Nuclear
Physics B (Proc. Suppl.)
6 I
383
96 (2001) 381-385
I
4
1-~~
-----
--“--‘-_
---~-..-.__.,___.__,____~_
1
. ._ __
:/,
L
0
5
4
3
2
(.
,
1
2
.
,JkJ
3
4
5
Q2 (GeV*)
M*(GeV’)
Figure 1. M2 dependence of the perturbative (longdashed line) and gluon condensate (dashed line) contributions to the D’Dn form factor at Q2 = 1 GeV’ (solid line) for A, = A, = 0.5 GeV.
of the curve, as a function of M2, providing a rather stable plateau for M2 > 3 GeV2. Fixing M2 = 3.5 GeV2 we show, in Fig. 2, the momentum dependence of the form factor (dots). Since the present approach cannot be used at Q2 = 0, to extract the gD*Dn coupling from the form factor we need to extrapolate the curve to Q2 = 0 (in the approximation rnz = 0). In order to do this extrapolation we fit the QCD sum rule results with an analytical expression. We tried to fit our results with a monopole form, since this is very often used for form factors [13], but the fit is very poor. We obtained good fits using both the gaussian form
Figure 2. Momentum dependence of the D’ DA form factor for A, = A, = 0.5 GeV (dots). The solid and dashed lines give the parametrization of the QCDSR results through Eqs. (11) and (10) respectively.
stability
AB = A,, ( GeV)
gD*Dx
A ( GeV)
a ( GeV)
0.5 0.6 A, = A, ( GeV)
5.3 6.0 &PDT
1.66 1.89 I’ ( GeV)
1.90 3.05
0.5 0.6
I TABLE
(
5.7 6.1
(
1.74 1.92
1
I: Values of the parameters in Eqs. (10)
and (11) which reproduce the QCDSR results for go*o=(Q’), for two different values of the continuum thresholds.
In view of the uncertainties involved, the results obtained with the two parametrizations are con-
and a curve of the form 9H*Hn(Q2)
=
gH*Hr
1 + (a/A)4 1 + (a/A)4e(Q2+m:)2/h
4 01)
The Q2 dependence of the form factor can be well reproduced by the parametrization in Eqs. (10) and (11) [ll]. The value of the parameters in Eqs. (10) and (11) are given in Table I for two different values of the continuum threshold.
sistent with each other, the systematic error being of the order of 10%. To test if our fit gives a good extrapolation to Q2 = 0 we can write a sum rule, based on the three-point function Eq. (l), but valid only at
Q2 =
0, as suggested in [14] for the pion-nucleon coupling constant. This method was also applied to the nucleon-hyperon-kaon coupling constant [15] and to the nucleon-A, - D coupling
6
384
M. Nielsett
7------
ei al. /Nuclear
Phwics
B (PRx.
Suppi.)
96 (2001) 3X1--385
tributes is the quark condensate given by
y---7
(13)
4:-
0
1
2
3
Equating Eqs. (12) and (13) and taking Q2 = 0 we obtain the sum rule for gH’ Hn +AM2, where A denotes the contribution from the unknown single poles terms. In Fig. 3 we show, for A, = A, = 0.5 GeV, the QCDSR results for gD*D, + AM2 as a function of M2 (dots) from where we see that, in the Bore1 region 2 5 M2 < 5 GeV2, they follow a straight line. The value of the coupling constant is obtained by the extrapolation of the line to M2 = 0. Fitting the QCDSR results to a straight line we get
4
M2 (GeV’)
Figure 3. D‘ DTT coupling constant as a function of the squared Bore1 mass M2 from the QCDSR valid at Q2 = 0 (dots). The straight line gives the extrapolation to M2 = 0.
constant [16]. It consists in neglecting the pion mass in the denominator of Eq. (2) and working at Q2 = 0, making a single Bore1 transformation to both P2 = PI2 -+ M2. The problem of doing a single Bore1 transformation is the fact that the single pole contribution, associated with the N -+ N* transition, is not suppressed [7,6]. However, the single pole contribution can be taken into account through the introduction of a parameter A, in the phenomenological side of the sum rule [7,15]. Therefore, neglecting rnz in the denominator of Eq. (2) and doing a single Bore1 transform in P2 = P’2, we get for the structure p,
pyen)
(i,,.f2,
Q2)
=
22*;2
e-mL/M2
_ pn2
H./M2
>
Q2
mkm;mtk-;
H
(gH*Hs
l
+ AM2)
H
, (12)
where CH*H in given in Eq. (3). On the OPE side only terms proportional to l/Q2 will contribute to the sum rule. Therefore, up to dimension four the only diagram that con-
QD-Dx
=
5.4 >
(14)
in excellent agreement with the values obtained with the extrapolation of the form factor to Q2 = 0, given in Table I. It is reassuring that both methods, with completely different OPE sides and Bore1 transformation approaches, give the same value for the coupling constant. In the case of B*Bn vertex, the Q2 = 0 sum rule results for gB.Ba + AM2 also follows a straight line in the Bore1 region 10 5 M2 < 25 GeV2, and the extrapolation to 2M2 = 0 gives gB*& N 10.6 [II]. In the case of the form factor gB-B,(Q2), the contribution of the gluon condensate is very small but it still goes in the right direction of providing a stable plateau for M2 2 15 GeV2. Fixing M2 = 17 GeV2 we evaluate the form factor which can still be well fitted by Eq. (11). However, the fit with Eq. (10) is not so good [ll]. In Table II we give the value of the parameters in Eqs. (10) and (11) that reproduce our results for two different choices of the continuum thresholds. In this case the agreement of the two different approaches to extract the coupling constant is not so good, but the numbers are still compatible. One possible reason for that is the fact that for heavier quarks the perturbative contribution (or hard physics) becomes more important. Since in the sum rule given by Eqs. (12) and (13) there is only soft physics information, we expect a, corrections to the sum rule to be more important in the case of g,+Bn(Q2) than for gD*DA(Q2).
M. Nielsen et al. /Nuclear Physics B (Proc. Suppl.) 96 (2001) 381-385
A, = A,, ( GeV)
gB*Bx
A ( GeV)
a ( GeV)
0.5 0.6 A, = A, ( GeV)
14.7 16.3 gB*Bx
1.62 1.81 r ( GeV)
1.37 1.67
0.5 0.6
17.2 1 18.4
1
1.79 1.97
TABLE II: Values of the parameters in Eqs. (10) and (11) which reproduce the QCDSR results for gB*Bx(Q*), for two different values of the continuum thresholds.
Comparing Table I with Table II we see that the cut-offs are of the same order in the two vertices and are very hard. Concerning the parameter a, it is smaller in the case of the B*Br vertex. This is because of the fact that the form factor gB.Bn(Q2) has a flatter peak around Q2 = 0 than gD.Drr(Q2). This can be interpreted as an indication that the spatial extension of the vertex is smaller for B*Br than for D*D?r. This is also the reason why the gaussian fit is not so good in the case of the B*Br vertex, and leads to bigger values for the coupling. It is interesting to notice that our results for the coupling constants are completely consistent with the QCDSR calculation of ref. [lo]. In conclusion, we extracted the H’Hn coupling constant using two different approaches of the QCDSR based on the three-point function. Considering the variation in the continuum thresholds and different approaches our results are: gD’DT = 5.7 f 0.4 , gB*Bx
=
14.5 f 3.9 .
(16)
(17) Using Eq. (15) we get
= 6.3 f 0.9 keV ,
1. 2.
4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
(15)
The D*Dr coupling is directly related with the D* + Dx decay width through
I’(D*- -+ ?%r-)
REFERENCES
3:
1
(18)
which is much smaller then the current upper limit [17] l?(D*- + @n-) < 89 keV.
385
NA50 Collaboration (M.C. Abreu et al.), Phys. Lett. B477, 28 (2000). W. C&sing, E.L. Bratkovskaya and S. Juchem, nucl-th/0001024. A. Capella, E.G. Ferreiro and A.B. Kaidalov, hep-phj0002300. Z. Lin and C.M. Ko, nucl-th/9912046. A. Sibirtsev, K. Tsushima and A.W. Thomas, nucl-th/0005041. P. Colangelo et al., Phys. Lett. B339, 151 (1994); V.L. Eletsky and Ya.1. Kogan, Z. Phys. C28, 155 (1985); A.A. Ovchinnikov, Sov. J. Nucl. Phys. 50, 519 (1989). V.M. Belyaev et al., Phys. Rev. D51, 6177 (1995). A. Khodjamirian et al., Phys. Lett. B457, 25 (1999). A.G. Grozin and 0.1. Yakovlev, Eur. Phys. J. C2, 721 (1998). H.G. Dosch and S. Narison, Phys. Lett. B368, 163 (1996). F.S. Navarra, M. Nielsen, M.E. Bracco, M. Chiapparini and C.L. Schat, hep-ph/0005026. R.E. Cutkosky, J. Math Phys. 1, 429 (1960). P. Ball, V.M. Braun and H.G. Dosch, Phys. Lett. B273, 316 (1991). L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127, 1 (1985). M.E. Bracco, F.S. Navarra and M. Nielsen, Phys. Lett. B454, 346 (1999). F.S. Navarra and M. Nielsen, Phys. Lett. B443, 285 (1998). Particle Data Group (C. Caso et al.), Eur. Phys. J. C3, 1 (1998).
Nuclear Physics B (Proc. Suppl.) 96 (2001) 381-385
D*Dvr and B*BT form factors from QCD Sum Rules M. Nielsen a , F.S. Navarra a , M.E. Bracco b , M. Chiapparini b and C. L. Schat b
aInstituto de Fisica, Universidade de SBo Paulo, C.P. 66318, 05389-970 Sao Paulo, SP, Brazil bInstituto de Fisica, Universidade do Estado do Rio de Janeiro, Rua Sao Francisco Xavier 524 - 20559-900, Rio de Janeiro, RJ, Brazil The H’Hn form factor for H = B and D mesons is evaluated in a QCD sum rule calculation. We study the Bore1 sum rule for the three point function of two pseudoscalar and one vector meson currents up to order four in the operator product expansion. The double Bore1 transform is performed with respect to the heavy meson momenta. We discuss the momentum dependence of the form factors and two different approaches to extract the H’ HX coupling constant.
A dense partonic system, often called the quark-gluon plasma (QGP), is expected to be formed in heavy ion collisions at the relativistic heavy ion collider (RHIC), which will soon start to operate at the Brookhaven National Laboratory. As a matter of fact, very recently the NA50 Collaboration reported [l] that the anomalous suppression of J/$ observed in Pb + Pb collisions at CERN-SPS indicated already the formation of QGP, since these experimental data ruled out conventional hadronic models for the J/$Jsuppression, However, there are calculations that reproduce the new NA50 data reasonably well up to the highest transverse energies [2,3], based on hadronic J/I/J dissociation alone. Therefore, while there are suggestions that the anomalous suppression may be due to the formation of the QGP, other more conventional mechanisms based on J/I+LJ absorption by comovers and nucleons has to be still considered. Various approaches have been used in evaluating the charmonium absorption cross section by hadrons. One of them uses meson exchange models based on hadronic effective lagrangians. Some of the processes considered in these models involve the D’Dn vertex [4] where a monopole form factor, with a unknown parameter (the cut-off), is assumed. The results obtained for the corss section are very sensitive to the couplings and to 0920-5632/01/% - see front matter 0 2001 Elsevier Science B.V PI1 SO920-5632(01)01156-2
the form factor [4,5]. Therefore it is very important to estimate these couplings and form factors with a theoretically founded model. In this work we use the QCD sum rule (QCDSR) approach based on the three-point function to evaluate the D*Dr and B’Bn form factors and coupling constants. The D’Dr and B*Bx couplings have been studied by several authors using different approaches of the QCDSR: two point function combined with soft pion techniques [6], light cone sum rules [7], light cone sum rules including perturbative corrections [8], sum rules in a external field [9], double momentum sum rules [lo]. Unfortunately, the numerical results from these calculations may differ by almost a factor two. The advantage of using the three-point function approach with a double Bore1 transformation compared with the twopoint function with a single Bore1 transformation is the elimination of the terms associated with the pole-continuum transitions [7]. The three-point function associated with a H*Hr vertex, where H and H* are respectively the lowest pseudoscalar and vector heavy mesons, is given by WP,P’)
=
J &&, eiP’.l
,-i(P’-7d.Y
(OlT{j(z)js(Y)j;(O)]lO)
x
i
(1)
where j = i@n,u, js = iiiTsd and ji = dy,Q are All rights reserved.
M. Nielsen et al. /Nucleur
382
the interpolating fields for H, tively with u, d and Q being heavy quark fields. The phenomenological side tion, F,(p, p’), is obtained by H and H” state contribution ment in Eq. (1) [II]:
I?fphen)(p,p’) = CHH.
T- and H* respecthe up, down, and of the vertex functhe consideration of to the matrix ele-
‘,yf:’
p2
_lm2X H’
7r
1 P
r2_m2
Physics B (Proc. Suppl.)
m~*+m~-q2 -P:
+
2m2,*
H
higher resonances
p, >
,
+ (2)
where CHH. =
m$mH’mafHfH.
(mu +
fir
(3)
md)mQ
The QCD side, or theoretical side, function is evaluated by performing erator product expansion (OPE) of in Eq. (1). Writing Fp in terms of amplitudes: rp (P, P’) = rl (~2, pr2, 4%p+:2
of the vertex Wilson’s opthe operator the invariant
(A d2, 4.%;
,(4)
we can write a double dispersion relation for each one of the invariant amplitudes I’i (i = 1,2), over the virtualities p2 and p12 holding Q2 = -q2 fixed: ri(p2,p’2,Q2)
= -
O” dsdu pi(s, u, Q2) I m:, 47r2 (s - p2)(u - pt2)
Y(5)
where pi (3, u, Q2) equals the double discontinuity of the amplitude Ii (p2, p/2) Q2) on the cuts rni 5 s < 00, rn; 5 u < co, which can be evaluated u&g Cutkosky’s riles [12]. Finally we perform a double Bore1 transformation in both variables P2 = -p2 and PI2 = -p12 and equate the two representations described above. We get one sum rule for each invariant function. In this paper we focus on the structure p, which we found to be the more stable one. We get: m&. -CHH*
+ m&
+ Q2
gH*Hrh2)
2m2,,
2 Ml2 = -1 e -m~~lM2e-m~l p1 (s, u, Q2)e-sIMZe-UIM’2
x
Q2+mi
J0 / Q So ds ,2
4lr2 1
,
u” du [ m2
96 (2001) 381-385
where se and us are the continuum thresholds for the H* and H mesons respectively, which are, in general, taken from the mass sum rules. The two Bore1 masses M2 and M’2 are completely independent. However, for simplicity, we will study the sum rule as a function of M2 at a fixed ratio M2 rn&. M’2=y$.
(7)
We consider diagrams up to dimension four which include the perturbative diagram and the gluon condensate. The quark condensate term does not contribute since it depends only on one external momentum and, therefore, it is eliminated by the double Bore1 transformation. Higher dimension condensates are strongly suppressed in the case of heavy quarks [6,7,9,10]. The double discontinuity of the perturbative contribution reads:
and the integration (s - m$)(u
limit condition
- m$) 2 Q2m$ .
is (9)
For consistency we use in our analysis the QCDSR expressions for the decay constants up to dimension four in lowest order of os [7]. The parameter values used in all calculations are m,, + md = 14 MeV, m, = 1.5 GeV, mb = 4.7 GeV, m, = 140 MeV, mD = 1.87 GeV, mD* = 2.01 GeV, mB = 5.28 GeV, = 5.33 GeV, fn = 131.5 MeV, (ijq) = mBq -(0.23)3 GeV3, (g2G2) = 0.5 GeV4. The continuum thresholds for the sum rule Eq. (6) are of order SO = (mH* + A,)2 and uc = (mH + A,)2 with A, = A, = 0.5 GeV. In our study we will allow for a small variation in A, and A,, to test the sensitivity of our results to the continuum contribution. We first discuss the D’Dr form factor. In Fig. 1 we show the behavior of the perturbative and gluon condensate contributions to the form factor go.Dn(Q2) at Q2 = 1 GeV as a function of the Bore1 mass M2 using A, = A,, = 0.5 GeV. We can see that, in the case of the form factor, the gluon condensate is not negligible and it helps the
hf. Nielsen
et al. /Nuclear
Physics B (Proc. Suppl.)
6 I
383
96 (2001) 381-385
I
4
1-~~
-----
--“--‘-_
---~-..-.__.,___.__,____~_
1
. ._ __
:/,
L
0
5
4
3
2
(.
,
1
2
.
,JkJ
3
4
5
Q2 (GeV*)
M*(GeV’)
Figure 1. M2 dependence of the perturbative (longdashed line) and gluon condensate (dashed line) contributions to the D’Dn form factor at Q2 = 1 GeV’ (solid line) for A, = A, = 0.5 GeV.
of the curve, as a function of M2, providing a rather stable plateau for M2 > 3 GeV2. Fixing M2 = 3.5 GeV2 we show, in Fig. 2, the momentum dependence of the form factor (dots). Since the present approach cannot be used at Q2 = 0, to extract the gD*Dn coupling from the form factor we need to extrapolate the curve to Q2 = 0 (in the approximation rnz = 0). In order to do this extrapolation we fit the QCD sum rule results with an analytical expression. We tried to fit our results with a monopole form, since this is very often used for form factors [13], but the fit is very poor. We obtained good fits using both the gaussian form
Figure 2. Momentum dependence of the D’ DA form factor for A, = A, = 0.5 GeV (dots). The solid and dashed lines give the parametrization of the QCDSR results through Eqs. (11) and (10) respectively.
stability
AB = A,, ( GeV)
gD*Dx
A ( GeV)
a ( GeV)
0.5 0.6 A, = A, ( GeV)
5.3 6.0 &PDT
1.66 1.89 I’ ( GeV)
1.90 3.05
0.5 0.6
I TABLE
(
5.7 6.1
(
1.74 1.92
1
I: Values of the parameters in Eqs. (10)
and (11) which reproduce the QCDSR results for go*o=(Q’), for two different values of the continuum thresholds.
In view of the uncertainties involved, the results obtained with the two parametrizations are con-
and a curve of the form 9H*Hn(Q2)
=
gH*Hr
1 + (a/A)4 1 + (a/A)4e(Q2+m:)2/h
4 01)
The Q2 dependence of the form factor can be well reproduced by the parametrization in Eqs. (10) and (11) [ll]. The value of the parameters in Eqs. (10) and (11) are given in Table I for two different values of the continuum threshold.
sistent with each other, the systematic error being of the order of 10%. To test if our fit gives a good extrapolation to Q2 = 0 we can write a sum rule, based on the three-point function Eq. (l), but valid only at
Q2 =
0, as suggested in [14] for the pion-nucleon coupling constant. This method was also applied to the nucleon-hyperon-kaon coupling constant [15] and to the nucleon-A, - D coupling
6
384
M. Nielsett
7------
ei al. /Nuclear
Phwics
B (PRx.
Suppi.)
96 (2001) 3X1--385
tributes is the quark condensate given by
y---7
(13)
4:-
0
1
2
3
Equating Eqs. (12) and (13) and taking Q2 = 0 we obtain the sum rule for gH’ Hn +AM2, where A denotes the contribution from the unknown single poles terms. In Fig. 3 we show, for A, = A, = 0.5 GeV, the QCDSR results for gD*D, + AM2 as a function of M2 (dots) from where we see that, in the Bore1 region 2 5 M2 < 5 GeV2, they follow a straight line. The value of the coupling constant is obtained by the extrapolation of the line to M2 = 0. Fitting the QCDSR results to a straight line we get
4
M2 (GeV’)
Figure 3. D‘ DTT coupling constant as a function of the squared Bore1 mass M2 from the QCDSR valid at Q2 = 0 (dots). The straight line gives the extrapolation to M2 = 0.
constant [16]. It consists in neglecting the pion mass in the denominator of Eq. (2) and working at Q2 = 0, making a single Bore1 transformation to both P2 = PI2 -+ M2. The problem of doing a single Bore1 transformation is the fact that the single pole contribution, associated with the N -+ N* transition, is not suppressed [7,6]. However, the single pole contribution can be taken into account through the introduction of a parameter A, in the phenomenological side of the sum rule [7,15]. Therefore, neglecting rnz in the denominator of Eq. (2) and doing a single Bore1 transform in P2 = P’2, we get for the structure p,
pyen)
(i,,.f2,
Q2)
=
22*;2
e-mL/M2
_ pn2
H./M2
>
Q2
mkm;mtk-;
H
(gH*Hs
l
+ AM2)
H
, (12)
where CH*H in given in Eq. (3). On the OPE side only terms proportional to l/Q2 will contribute to the sum rule. Therefore, up to dimension four the only diagram that con-
QD-Dx
=
5.4 >
(14)
in excellent agreement with the values obtained with the extrapolation of the form factor to Q2 = 0, given in Table I. It is reassuring that both methods, with completely different OPE sides and Bore1 transformation approaches, give the same value for the coupling constant. In the case of B*Bn vertex, the Q2 = 0 sum rule results for gB.Ba + AM2 also follows a straight line in the Bore1 region 10 5 M2 < 25 GeV2, and the extrapolation to 2M2 = 0 gives gB*& N 10.6 [II]. In the case of the form factor gB-B,(Q2), the contribution of the gluon condensate is very small but it still goes in the right direction of providing a stable plateau for M2 2 15 GeV2. Fixing M2 = 17 GeV2 we evaluate the form factor which can still be well fitted by Eq. (11). However, the fit with Eq. (10) is not so good [ll]. In Table II we give the value of the parameters in Eqs. (10) and (11) that reproduce our results for two different choices of the continuum thresholds. In this case the agreement of the two different approaches to extract the coupling constant is not so good, but the numbers are still compatible. One possible reason for that is the fact that for heavier quarks the perturbative contribution (or hard physics) becomes more important. Since in the sum rule given by Eqs. (12) and (13) there is only soft physics information, we expect a, corrections to the sum rule to be more important in the case of g,+Bn(Q2) than for gD*DA(Q2).
M. Nielsen et al. /Nuclear Physics B (Proc. Suppl.) 96 (2001) 381-385
A, = A,, ( GeV)
gB*Bx
A ( GeV)
a ( GeV)
0.5 0.6 A, = A, ( GeV)
14.7 16.3 gB*Bx
1.62 1.81 r ( GeV)
1.37 1.67
0.5 0.6
17.2 1 18.4
1
1.79 1.97
TABLE II: Values of the parameters in Eqs. (10) and (11) which reproduce the QCDSR results for gB*Bx(Q*), for two different values of the continuum thresholds.
Comparing Table I with Table II we see that the cut-offs are of the same order in the two vertices and are very hard. Concerning the parameter a, it is smaller in the case of the B*Br vertex. This is because of the fact that the form factor gB.Bn(Q2) has a flatter peak around Q2 = 0 than gD.Drr(Q2). This can be interpreted as an indication that the spatial extension of the vertex is smaller for B*Br than for D*D?r. This is also the reason why the gaussian fit is not so good in the case of the B*Br vertex, and leads to bigger values for the coupling. It is interesting to notice that our results for the coupling constants are completely consistent with the QCDSR calculation of ref. [lo]. In conclusion, we extracted the H’Hn coupling constant using two different approaches of the QCDSR based on the three-point function. Considering the variation in the continuum thresholds and different approaches our results are: gD’DT = 5.7 f 0.4 , gB*Bx
=
14.5 f 3.9 .
(16)
(17) Using Eq. (15) we get
= 6.3 f 0.9 keV ,
1. 2.
4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
(15)
The D*Dr coupling is directly related with the D* + Dx decay width through
I’(D*- -+ ?%r-)
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3:
1
(18)
which is much smaller then the current upper limit [17] l?(D*- + @n-) < 89 keV.
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