- Email: [email protected]
How is the charmonium splitting in QCD? predictions from exponential moments as limit of power moments
Volume 106B, number 4
PHYSICS LETTERS
12 November 1981
HOW IS THE CHARMONIUM SPLITTING IN QCD? PREDICTIONS FROM EXPONENTIAL MOMENTS AS LIMIT OF POWER MOMENTS R.A. BERTLMANN Institut fffr Theoretische Physik, Universiti/t Wien, Vienna, Austria Received 2 July 1981
Using the SVZ moment procedure to predict resonance masses within QCD we have calculated exponential moments as a limit of the QCD formulae given by Reinders, Rubinstein and Yazaki. Applied to charmordum we reproduce their results (besides 3Po) very well.
Shifman, Vainshtein and Zakharov [1,2] (SVZ hereafter) have introduced an original method, the moment procedure, to discuss resonance properties within QCD. In introducing power moments, derivatives of the vacuum polarization function rr(Q 2) (1) c.gfn(Q2) = ~.. _
zr(Q2) =
Im ~r(s) ds (s + a2)n+ l '
they could relate the physical Im n(s), the resonance structure to the vacuum properties of QCD. Their novel idea in proceeding from short to long distances was to include nonperturbative terms, gluonic vacuum fluctuations. Applied to charmonium SVZ have estimated (at Q2 = 0) the charmed quark mass mc, the strong coupling constant a s and the gluon condensate ((as/r0GG). In a series of papers Reinders, Rubinstein and Yazaki [3-5 ] (RRY hereafter) generalized the calculations of SVZ to several types of heavy quark currents at arbitrary Q2. In fitting a few parameters only the lowest lying charmonium states have been reproduced extremely well. Whereas SVZ and RRY use the power moments to analyze heavy quark states we work with exponential moments. They appear to be an improvement within a nonrelativistic approach [6]. In this paper we want to present results given by a limit of the QCD formulae of RRY [5] which provides the exponential moments. According to SVZ the power moments are calculated 336
in the following way C~n(Q2 ) =An(Q2)[1 + asan(Q 2) + Cbbn(Q2)] ,
(2)
where A n is the free quark contribution, asan represents the perturbative gluon corrections, Ob n originates from the nonperturbative gluonic vacuum fluctuations. The gluon condensate parameter • is conventionally defined by • = O1/(4m2)2 ,
(3)
with 01 = ] 7r2 ((as/Tr)GG).
(4)
We regard • 1 to be the flavour independent (mass normalization independent) quantity (see ref. [1]). In an elaborate work RRY have calculated the functions An, a n, b n for a great variety of currents, for the pseudoscalar 1S0 and scalar 3P0, for the vector 3S1, axial vector 3P 1 and 1P1, and for the tensor current 3P 2. On their formulae listed on table 1 of ref. [5] we are going to rely. We now take the limit n~oo,
Q2 ~ o o ,
with n/Q 2 = o fixed,
(5) (6)
lim (a2)n+ l Trqffn(O 2) = c~ ( o ) ,
and obtain the exponential moments
0 031-9163/81/0000-0000[$ 02.75 © 1981 North-Holland
Volume 106B, number 4
PHYSICS LETTERS
(o) = f d ~ e-O, Im zr(s).
12 November 1981
(7)
Splitting the moments into c ' ~ ( o ) = e-4m2orrA(o)[1 + asa(O) + qbb(o)] ,
(8)
we can calculate for the above currents the free quark contribution rrA(o), the perturbative asa(o) and nonperturbative correction ~b (o) from the RRY formulae by taking the above limit. For the 3S 1 current this has been demonstrated explicitly by Bell and the author [7]. For all the other currents the calculations proceed very similar, but are somewhat lengthy and will be presented in detail somewhere else [8]• To obtain the mass of the ground state we apply the logarithmic derivative to c/~ (o) yielding a ratio of moments [9] c~ (o) = -(d/do) logC~ (a).
~0~
MMMMM 0
O O O O O O +1 +1 +1 +1 ÷1
/I
+1
~, . ~
(9)
Whereas the exact ratio in the limit
q~(o) -~ ~ 2 ,
(1o) .
just approaches the mass of the ground state, we re. gard the minimum of the approximated cR (a) min c~ (o) = M 2 ,
(11)
2 ~,.2.
(12)
~.~
~
o
~
(7
to be an approximation to the ground state. In fact we now perturb this ratio c~(o) = F(o)[I + %P(o) + q~Q(tr)] , where
~A'(~)_
F ( o ) = 4m 2 -- teA(o) '
Q(o)-
_
a'(o)
P(o) - - F(o) '
b'(o) F(o)
We argue that the ratio ~'~ (o) is the more stable quantity. Whereas the corrections asa(O), ~b (o) blow up for increasing o, their derivatives -Ctsa'(o), - ~ b ' ( o ) stay reasonably small• In order to lower the gluonic corrections SVZ and RRY have introduced an off-shell mass m = m(p 2 = -m2),
o q ~ # .
(13)
.
~s ~.~
"~'ddd~ ~ d ~ M M
(14)
providing in Otsan an extra term proportional to 337
Volume 106B, number 4
~s47r-1 In 2 ,
PHYSICS LETTERS
(15)
which reduces the net correction in ~ nAs we already discussed in refs. [7,9] this is not the case in the ratio c~ (o). There the mass correction (15)just adds to the quark mass (14). Therefore we work with an on-shell mass
m = rn(p 2 = m 2 ) ,
(16)
which is related to the euclidean mass by N2 = m2(1 + as4rr-1 In 2).
(17)
Now, in order to compare our results for charmonium very directly with the ones o f RRY or SVZ we have calculated both versions: (i) using an off-shell mass m and retaining terms proportional to (15) in the corrections asa(O ) and
-asa'(o ), (ii) using an on-shell mass ~ and dropping terms proportional to (15). The results of both calculations differ only by a few MeV! Before we present our results we want to mention an important feature of RRY's approach in comparison to ours. RRY have chosen Q2 quite differently for sand p-states thus yielding a significant change in m c and a s when going from s- to p-states. In our approach, however, by using exponential moments as limit of power moments the Q2 dependence disappeared and quark mass and coupling constant remain fixed in moving from s- to p-states. For the parameters m c (or r~c), as, q~l we did not try to find a best fit. For ease of comparison we used values as they were determined by SVZ [1], RRY [4,5] and Miller and Olsson [10] (first set in each colunm of table 1). With these values the 351 states always emerge below the experimental J / $ (3.10). Therefore, in assuming that m c and a s have been determined correctly from low moments we just increase q~l to obtain the correct 351(3.10 ) (second set in each column of table 1). Note that Bell and myself conjectured before [6,7,9] that the condensate parameter ~b1 has been underestimated by previous authors. This is now confirmed by QCD 2 model calculations of Bradley et al. [ 11 ]. Now, let us discuss our results on table 1. We have the best parameter set collected in column 1. These are parameter values advocated by RRY [5] for Q2
338
12 November 1981
= 0. We reproduce their results for charmonium splitting very well, except for the scalar 3P 0 which lies about 60 MeV too high. Lowering a s and increasing q~l a bit does not yield much change (second column). But for parameter values of the third column which RRY [4] claim for Q2 = 4m 2 the p-states emerge too high [when d#1 is adjusted to 351 (3.10)]. This remains so for parameter values of SVZ [1 ] (column 4) and of Miller and Olsson [10] (column 5). The reason may be that we cannot vary m c and a s in moving from sto p-states. We regard the first parameter set of table 1 as significant to produce a charmonium splitting in agreement with experiment. We want to thank J.S. Bell for his valuable advice. We profited much from discussions with E. de Rafael and V.I. Zakharov, and we obtained helpful information from correspondance with L.J. Reinders and M.A. Shifman.
References [1 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448,519; Phys. Rev. Lett. 42 (1979) 297. [2] M.A. Shifman, Z. Phys. 4 (1980) 345. [3] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 94B (1980) 203; 97B (1980) 257. [4] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 95B (1980) 103. [5] L.J. Reinders, H.R. Rubinstein and S. Yazaki, QCD sum rules for heavy quark systems, Rutherford preprint, RL-80-088 (1980). [6] J.S. Bell and R.A. Berflmann, Nucl. Phys. B177 (1981) 218. [7] J.S. Bell and R.A. Berflmann, Shifman-VainshteinZakhazov moments and quark-antiquark potentials, CERN preprint TH 2986 (1980), to be published in Nucl. Phys. B. [8] R.A. Bertlmann, to be published. [91 R.A. Bertlmannt Duality between resonances and asymptotia, Univ. of Vienna preprint, UWThPh-80-38 (1980) to be published in Acta Phys. Austr. [10] K.J. Miller and M.G. Olsson, Testing QCD with moment sum rules, Univ. of Wisconsin-Madison preprint, DOE-
ER/00881-189 (1981). [11 ] A. Bradley, C.S. Langensiepen and G. Shaw, Magic moments in QCD2, National Institute for Nuclear Physics preprint, NIKHEF-H/81-15 (Amsterdam, 1981). [12] For a review of the experimental eharmonium situation see: K. Berkelman, Cornell Univ. preprint, CLNS 80/ 470 (1980).
PHYSICS LETTERS
12 November 1981
HOW IS THE CHARMONIUM SPLITTING IN QCD? PREDICTIONS FROM EXPONENTIAL MOMENTS AS LIMIT OF POWER MOMENTS R.A. BERTLMANN Institut fffr Theoretische Physik, Universiti/t Wien, Vienna, Austria Received 2 July 1981
Using the SVZ moment procedure to predict resonance masses within QCD we have calculated exponential moments as a limit of the QCD formulae given by Reinders, Rubinstein and Yazaki. Applied to charmordum we reproduce their results (besides 3Po) very well.
Shifman, Vainshtein and Zakharov [1,2] (SVZ hereafter) have introduced an original method, the moment procedure, to discuss resonance properties within QCD. In introducing power moments, derivatives of the vacuum polarization function rr(Q 2) (1) c.gfn(Q2) = ~.. _
zr(Q2) =
Im ~r(s) ds (s + a2)n+ l '
they could relate the physical Im n(s), the resonance structure to the vacuum properties of QCD. Their novel idea in proceeding from short to long distances was to include nonperturbative terms, gluonic vacuum fluctuations. Applied to charmonium SVZ have estimated (at Q2 = 0) the charmed quark mass mc, the strong coupling constant a s and the gluon condensate ((as/r0GG). In a series of papers Reinders, Rubinstein and Yazaki [3-5 ] (RRY hereafter) generalized the calculations of SVZ to several types of heavy quark currents at arbitrary Q2. In fitting a few parameters only the lowest lying charmonium states have been reproduced extremely well. Whereas SVZ and RRY use the power moments to analyze heavy quark states we work with exponential moments. They appear to be an improvement within a nonrelativistic approach [6]. In this paper we want to present results given by a limit of the QCD formulae of RRY [5] which provides the exponential moments. According to SVZ the power moments are calculated 336
in the following way C~n(Q2 ) =An(Q2)[1 + asan(Q 2) + Cbbn(Q2)] ,
(2)
where A n is the free quark contribution, asan represents the perturbative gluon corrections, Ob n originates from the nonperturbative gluonic vacuum fluctuations. The gluon condensate parameter • is conventionally defined by • = O1/(4m2)2 ,
(3)
with 01 = ] 7r2 ((as/Tr)GG).
(4)
We regard • 1 to be the flavour independent (mass normalization independent) quantity (see ref. [1]). In an elaborate work RRY have calculated the functions An, a n, b n for a great variety of currents, for the pseudoscalar 1S0 and scalar 3P0, for the vector 3S1, axial vector 3P 1 and 1P1, and for the tensor current 3P 2. On their formulae listed on table 1 of ref. [5] we are going to rely. We now take the limit n~oo,
Q2 ~ o o ,
with n/Q 2 = o fixed,
(5) (6)
lim (a2)n+ l Trqffn(O 2) = c~ ( o ) ,
and obtain the exponential moments
0 031-9163/81/0000-0000[$ 02.75 © 1981 North-Holland
Volume 106B, number 4
PHYSICS LETTERS
(o) = f d ~ e-O, Im zr(s).
12 November 1981
(7)
Splitting the moments into c ' ~ ( o ) = e-4m2orrA(o)[1 + asa(O) + qbb(o)] ,
(8)
we can calculate for the above currents the free quark contribution rrA(o), the perturbative asa(o) and nonperturbative correction ~b (o) from the RRY formulae by taking the above limit. For the 3S 1 current this has been demonstrated explicitly by Bell and the author [7]. For all the other currents the calculations proceed very similar, but are somewhat lengthy and will be presented in detail somewhere else [8]• To obtain the mass of the ground state we apply the logarithmic derivative to c/~ (o) yielding a ratio of moments [9] c~ (o) = -(d/do) logC~ (a).
~0~
MMMMM 0
O O O O O O +1 +1 +1 +1 ÷1
/I
+1
~, . ~
(9)
Whereas the exact ratio in the limit
q~(o) -~ ~ 2 ,
(1o) .
just approaches the mass of the ground state, we re. gard the minimum of the approximated cR (a) min c~ (o) = M 2 ,
(11)
2 ~,.2.
(12)
~.~
~
o
~
(7
to be an approximation to the ground state. In fact we now perturb this ratio c~(o) = F(o)[I + %P(o) + q~Q(tr)] , where
~A'(~)_
F ( o ) = 4m 2 -- teA(o) '
Q(o)-
_
a'(o)
P(o) - - F(o) '
b'(o) F(o)
We argue that the ratio ~'~ (o) is the more stable quantity. Whereas the corrections asa(O), ~b (o) blow up for increasing o, their derivatives -Ctsa'(o), - ~ b ' ( o ) stay reasonably small• In order to lower the gluonic corrections SVZ and RRY have introduced an off-shell mass m = m(p 2 = -m2),
o q ~ # .
(13)
.
~s ~.~
"~'ddd~ ~ d ~ M M
(14)
providing in Otsan an extra term proportional to 337
Volume 106B, number 4
~s47r-1 In 2 ,
PHYSICS LETTERS
(15)
which reduces the net correction in ~ nAs we already discussed in refs. [7,9] this is not the case in the ratio c~ (o). There the mass correction (15)just adds to the quark mass (14). Therefore we work with an on-shell mass
m = rn(p 2 = m 2 ) ,
(16)
which is related to the euclidean mass by N2 = m2(1 + as4rr-1 In 2).
(17)
Now, in order to compare our results for charmonium very directly with the ones o f RRY or SVZ we have calculated both versions: (i) using an off-shell mass m and retaining terms proportional to (15) in the corrections asa(O ) and
-asa'(o ), (ii) using an on-shell mass ~ and dropping terms proportional to (15). The results of both calculations differ only by a few MeV! Before we present our results we want to mention an important feature of RRY's approach in comparison to ours. RRY have chosen Q2 quite differently for sand p-states thus yielding a significant change in m c and a s when going from s- to p-states. In our approach, however, by using exponential moments as limit of power moments the Q2 dependence disappeared and quark mass and coupling constant remain fixed in moving from s- to p-states. For the parameters m c (or r~c), as, q~l we did not try to find a best fit. For ease of comparison we used values as they were determined by SVZ [1], RRY [4,5] and Miller and Olsson [10] (first set in each colunm of table 1). With these values the 351 states always emerge below the experimental J / $ (3.10). Therefore, in assuming that m c and a s have been determined correctly from low moments we just increase q~l to obtain the correct 351(3.10 ) (second set in each column of table 1). Note that Bell and myself conjectured before [6,7,9] that the condensate parameter ~b1 has been underestimated by previous authors. This is now confirmed by QCD 2 model calculations of Bradley et al. [ 11 ]. Now, let us discuss our results on table 1. We have the best parameter set collected in column 1. These are parameter values advocated by RRY [5] for Q2
338
12 November 1981
= 0. We reproduce their results for charmonium splitting very well, except for the scalar 3P 0 which lies about 60 MeV too high. Lowering a s and increasing q~l a bit does not yield much change (second column). But for parameter values of the third column which RRY [4] claim for Q2 = 4m 2 the p-states emerge too high [when d#1 is adjusted to 351 (3.10)]. This remains so for parameter values of SVZ [1 ] (column 4) and of Miller and Olsson [10] (column 5). The reason may be that we cannot vary m c and a s in moving from sto p-states. We regard the first parameter set of table 1 as significant to produce a charmonium splitting in agreement with experiment. We want to thank J.S. Bell for his valuable advice. We profited much from discussions with E. de Rafael and V.I. Zakharov, and we obtained helpful information from correspondance with L.J. Reinders and M.A. Shifman.
References [1 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448,519; Phys. Rev. Lett. 42 (1979) 297. [2] M.A. Shifman, Z. Phys. 4 (1980) 345. [3] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 94B (1980) 203; 97B (1980) 257. [4] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 95B (1980) 103. [5] L.J. Reinders, H.R. Rubinstein and S. Yazaki, QCD sum rules for heavy quark systems, Rutherford preprint, RL-80-088 (1980). [6] J.S. Bell and R.A. Berflmann, Nucl. Phys. B177 (1981) 218. [7] J.S. Bell and R.A. Berflmann, Shifman-VainshteinZakhazov moments and quark-antiquark potentials, CERN preprint TH 2986 (1980), to be published in Nucl. Phys. B. [8] R.A. Bertlmann, to be published. [91 R.A. Bertlmannt Duality between resonances and asymptotia, Univ. of Vienna preprint, UWThPh-80-38 (1980) to be published in Acta Phys. Austr. [10] K.J. Miller and M.G. Olsson, Testing QCD with moment sum rules, Univ. of Wisconsin-Madison preprint, DOE-
ER/00881-189 (1981). [11 ] A. Bradley, C.S. Langensiepen and G. Shaw, Magic moments in QCD2, National Institute for Nuclear Physics preprint, NIKHEF-H/81-15 (Amsterdam, 1981). [12] For a review of the experimental eharmonium situation see: K. Berkelman, Cornell Univ. preprint, CLNS 80/ 470 (1980).