How to use heavy quarks to probe the QCD vacuum

Volume 98B, number 6 PHYSICS LETTERS 29 January 1981

HOW TO USE HEAVY QUARKS TO PROBE THE QCD VACUUM ~

H. LEUTWYLER Institute for Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland

Received 7 November 1980

We calculate the leading nonperturbative contributions to the energy levels and wavefunctions of a heavy qcl pair.

For sufficiently heavy quarks the qq bound state picture based on a nonrelativistic positronium-like description is consistent: one gluon exchange leads to small bound state velocities of order/3 = 4 as = g2/37r. The root mean square radius of a state with principal quantum number n is of order n2(m/3) -1 ; if n is not too large heavy quarks move at short dis- tances where a perturbative analysis of QCD makes sense. If the quarks are not very heavy the description fails not so much because higher order effects in a s (relativistic corrections) become sizeable, but because it neglects nonperturbative effects which are important if the size of the system is not small in comparison to the scale of QCD. We have argued [1] that the leading nonperturbative effects can be seen in the short distance behaviour of the wavefunctions. The purpose of the present paper is to show that for a heavy q?q pair the leading nonperturbative effect is governed by the gluon pair density

(B 2) = ¼ (O[GavGUVa[o), ( l )

which produces a shift of the Schr6dinger levels by

(B 2) 1 m{32 +m - - n6enl . (2)

3,11 = 2m - ~- n~ (m/3) 4

In this formula n is the principal quantum number, l the orbital angular momentum of the state, and/3 = ] a s. [We include the strong coupling constant in the definition of the gluon field: V u = 8 u + iG~ xa/2, G a - a G a ~ G a f a b c G b ~ e 1 The dimension-

l a y -- ~ p -- I: ~ -- a Id ~ P " l

Supported in part by the Schweizerischer Nationalfonds.

less quantity enl is of order one (see below). The formula (2) is accurate up to terms of order m/3 4, m × (V G 2 )(m/3) -6, m(G 3 )(m/3) -6 and up to effects

# Oo po due to the pair density of light quarks.

The gluon condensate also modifies the wavefunctions. In radiation gauge the leading nonperturbative effect on the ground state wavefunction is given by

~(x -- y ) = (81r)- 1/2(m/3)3/2 e -p/2

× [1 1 , ) , (1704+15603+936p2 26712)]

7 = m-413-6(B2), (3)

where/9 = m/31x - Y l . In particular, the wavefunction at the origin, measured in e+e - -decay is modified as follows:

t~(O) = (8n)-l/2(m/3)3/2(1 + 6.983'). (4)

Finally, we have calculated the leading nonperturbative correction to the hyperfine splitting of the ground state with the result

1 4 Mtt - M t ~ = ~m/3 (1 + 18.33,). (5)

After this work was essentially completed I learned that Voloshin [2] has carried out a similar analysis in the framework of QCD sum rules. I am indebted to Professor S. Iwao for bringing this work to my at- tention.

To derive the above results we observe that the quarks move in a medium containing soft gluons. If the radius r of the orbit is small compared to the average wave length of the fluctuating chromoelectric and chromomagnetic vacuum fields we have rl V u Goa I ~ IGvl. In this situation the space-t ime variation of

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company 447


the fields may be neglected: terms proportional to (0IV G 2 10) are suppressed by two inverse powers of m3 in comparison to those retained in (2). In leading order we are therefore faced with the problem of calculating the level shift due to constant random fields. Lorentz invariance of the vacuum requires that the electric and magnetic pair densities are of equal magnitude and opposite sign (E i = Goi , B i = --Gkl):

a b 1 ab 2 (0[Ei Ek[0) = -~4~ik 6 ( B ) ,

(0]Bt.aBb[0) __ 1 ab 2 ~6ik6 ( B ) . (6)

Since we are dealing with a nonrelativistic system this implies that the level shift due to the magnetic field is smaller by two powers of 3 than the shift due to the electric field: the leading nonperturbative contribution is the quadratic Stark effect due to the fluctuating chromoelectric vacuum field (there is no linear Stark effect, because the expectation value of the field vanishes). For a colour singlet state the interaction hamiltonian is

H 1 ~O(x ,y ) = - E a" (x - y ) ½ X a ~O(x ,y ) .

To lowest order the level shift is therefore given by

-(4°l , tr. (/4 0 -En) - i ¢°l). The state E . z d/Onl is a colour octet. When acting on such a state one gluon exchange is repulsive:

H 0 -+[10 : - rn - 1 A + ~ 3/[zh. (7)

Using (6) we get

AMnl = ~ (B 2 ) ( gg Onl , z i([10 - E n ) - 1 z i V Onl ) .

To evaluate the matrix element we make use of a method described in standard textbooks [3] " the equation

. i _z ia .O ([1o - En) X n l - ~nl'

can be solved explicitly with an ansatz of the form

X = polynomial × exp (--~ [z [rn3/n).

The result for the level shift is given in (2) with

2 1 {(l + 1)[F(n, l) - F ( - n , t)]

enl - 9 n3(2l + 1)

+ l[F(n, - l - 1) - F ( - n , - l - 1)] },

Table 1 Coefficients enl relevant for the leading nonperturbative mass shift.

n l enl

1 0 1.468 2 0 1.585 2 1 0.998 3 0 1.661 3 1 1.361 3 2 0.804

F ( n , l ) = 2 n [ n 2 - ( l + 1 ) 2 ] + ( n + l + 2 ) ( n + l + l )

x [ ( n l ) ( n + l + 3) + 4 ( 2 n - 1 ) 2] 9n + 16 9n + " (8)

For the lowest levels the numerical values are given in table 1. The main feature of the result is that the shift explodes with increasing principal quantum number: AMnl ~ n 6. This implies that even for very heavy quarks the nonperturbative effects are large on excited levels. The calculation described above is of course only valid if the nonperturbative effect is smaller than the SchriSdinger binding energy, i.e. 4 × 3 - 6 r n - 4 n 8 ( B 2) < 1. What counts is the ratio m/n 2. A quark four times as heavy as the b-quark suffers the same (relative) perturbation by the gluon condensate in the first excited state the b-quark experiences in the ground state. (The reason for the rapid growth with n is easy to understand: the perturbation is proportional to z ~ n2(m3) -1 while the energy denomi- nator is of order m32n -2 . ) Inserting the estimate (B 2) = 0.34 m 4 ~ (600 MeV) 4 given by Shifman et al. [4] one fin"ds that for the ground state of the bb- system the nonperturbative effect reduces the Schr6- dinger binding energy by 13% i f a s = 0.35, by 34% i f a s = 0.3 and by 100% i f a s = 0.25. For the cg-sys- tern for which the quark mass is smaller by more than a factor 3 the nonperturbative effect completely over- whelms the Coulomb force even for the ground state. This observation explains why the Balmer formula fails even for the bb-system: only the ground state sits in the region of the potential for which a short distance expansion is adequate. The leading nonperturbative effect lifts the degeneracy of the states be- longing to the same principal quantum number. For n = 2 e.g. we find

M20 - M21 = 37.6 m(m3) -a (B2) . (9)

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The effect does indeed have the sign observed in the ca-system (M~, > Mx) provided (B 2) is positive as it is indicated by the estimate of Shifman et al. [4] and as it is required by the stability of the vacuum [5], For the reason given above the prediction (9) cannot be used quantitatively for the first excited levels of the bb-system. Since the state n = 2, l = 1 is the lowest lying P-wave one may hope to nevertheless obtain a reliable prediction for its mass by using variational methods instead of the above perturbative scheme. Indeed Voloshin [2] predictsM21 - M 1 0 = 370 + 30 MeV on the basis of his analysis of the QCD sum rules.

The gluon pair density of course also modifies the wavefunctions. Again the first order effect vanishes upon averaging over the stochastic electric field. The second order perturbation is given by

+P(II 0 - En)-IpE'z(H 0 - En)-I E'z ~Ol, (10)

where P projects on the subspace orthogonal to @0 l. The physical wavefunction is the average of this expression over the electric field. It is important to realize that the first order contribution to the wavefunction ~ (H 0 En)-IE. z ~0 is of no physical sig- nificance. In the language of the Bethe-Salpeter equation this contribution corresponds to diagrams containing an arbitrary number of ladder exchanges plus one solft gluon insertion - the physical ampli- tude contains only diagrams with none or with two such insertions. In the analysis described above the fictitious diagrams containing only one soft gluon insertion play the role of a mere calculational device. It would be wrong to include the contribution from these diagrams in the normalization of the state.

The expression (10) may be evaluated explicitly by applying the technique described above once more. The resulting ground state wavefunction is given in (3). This expression shows that the nonperturbative effects increase the value of the wavefunction at the origin and reduce the radius of the orbit. Since the wavefunction at the origin is measured in the decay into an e+e --pair

['T -,e+e - = (16rr/9) (ee2/M 2 )l t~(0)12( 1 - 16C~s/37r ).

(11)

We may use the known decay rate , l i, qc ~e+e - = 1.16

,1 The value quoted represents an average value of the data reviewed at the Madison Conference (1980).

-+ 0.15 keV to extract the value of the b-quark mass and an estimate of (B2). Unfortunately, the result for (B 2) is sensitive to the value o f % . For % = 0.35, 0.3, 0.25 we get m b = (4.805 -+ 0.010, 4.765 -+ 0.010, 4.730 + 0.005) GeV and (B 2) = (0.85 + 0.20, 0.52 -+ 0.10, 0.27 -+ 0.05)m 4, respectively. Unless % is below 0.25 we invariably obtain a positive binding ener-

gy, m b >½MT. Note that the leading nonperturbative contribu-

tions cannot be described by a potential V(Izl). They amount to an attractive nonlocal interaction of the form

o o ^ .

V= a~(B 2) f d~tzi e-?~H°z I e Mf°, (12)

0

where H 0 is the attractive hamiltonian relevant in the color singlet channel whereas//0 is the repulsive hamiltonian (7) governing the color octet channel.

We now turn to the calculation of the leading nonperturbative contribution to the hyperfine splitting of the ground state. Since this splitting is of order m/34 we have to analyze the relativistic corrections to the bound state hamiltonian. These corrections may be derived from the Bethe Salpeter equation

( iTuVu +m)~x,y)(iTV*Vv +m)

b ab =-ig27u½xa•(x,y)TV½X O v(x,y), (13)

where V u is the covariant derivative with respect to the slowly varying "external" vacuum field and

ab Duu (x,y) denotes the gluon propagator in the pres- ence of this external field. The propagator may be worked out from the differential equation

Vu(V ua v - gvau) - i[Guv,aU ] =iv,

aa u(x)= f dy ab .vb - D v ( x , y ) / ( y ) . (14)

In radiation gauge Via i = 0 the instantaneous term D00 only involves the magnetic field, whereas the electric field appears in Doi , Dio (the components Dik are only needed to lowest order, i.e. for E = R = 0). The calculation simplifies considerably for S- wave states for which the spin-orbi t interaction is absent. It turns out that in this case the magnetic effects come exclusively from the magnetic term whereas the electric effects arise from the interference of the familiar local spin-spin interaction term ~ s 1 .s 2

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X 6 3 ( x - y ) with the electric potential E . z . The electric and magnetic terms have the same sign, the electric contribution is numerically more important. The result is given in (5), a detailed account of the calculation will be given elsewhere. Using the value of (B 2) given by Shifman et al. [4] we predict M~ r - M n

= 90 + 15 MeV. The result is remarkably stable witbh

respect to uncertainties in %. Note that the nonperturbative contribution is larger than the perturbative term.

If there is a t quark with a mass of order 20 GeV or more the tLsystem will provide an ideal probe to measure perturbative as well as nonperturbative QCD effects. The lowest lying states should provide enough information to extract a reliable value of % as well as an accurate value for the leading nonperturbative parameter (B2).

It is a pleasure to thank J. Glasser, S. Iwao, M. Liischer, P. Minkowski and P. Schwab for stimulat- ing discussions as well as for help with numerical prob-

lems.

R e f e r e n c e s

[1 ] H. Leutwyler, Proc. Intern. Summer Institute on Theo- retical physics (Kaiserslautern, 1979), ed. W. Riihl (Plenum, New York, 1980).

[2] M.B. Voloshin, Nonrelativistic bottomium in the physical vacuum of QCD. Mass of the 1P level, preprint ITEP 21 (Moscow, 1980); Nucl. Phys. B154 (1979) 365.

[3] M. Kotani, Quantum mechanics, Vol. 1 (Yuwanami, Tokyo, 1951) p. 127; L.I. Schiff, Quantum mechanics (McGraw-Hill, New York, 1968) p. 263.

[4] M.A. Shifman et al., Nucl. Phys. B147 (1979) 385,448; A.I. Vainshtein et al., Soy. J. Nucl. Phys. 27(2) (1978); V.A. Novikov et al., Phys. Rep. 41 (1978) 1.

[5] H. Leutwyler, Phys. Lett. 96B (1980) 154.

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How to use heavy quarks to probe the QCD vacuum