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Heavy quarkonium decays and the renormalization group
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
HEAVY QUARKONIUM DECAYS AND THE RENORMALIZATION GROUP ~ A. DUNCAN and A. MUELLER Department of Physics, Columbia University, New York, N Y 10027, USA
Received 11 January 1980
A renormalization group approach to heavy quarkonium decays is developed. Exclusive two-meson decays and inclusive decays are considered.
1. Introduction. Perhaps the most striking property of the charmonium family of resonances [I ] is the narrow hadronic decay width (typically, a fraction of an MeV) for hadronic states in a mass regime where such decay widths are generally hundreds of MeV. It has been customary to regard the dynamical suppression at work here as evidence for a phenomenological Zweig rule which forbids annihilation of the initial state quark-antiquark pair. It has also been suggested [2] that the Zweig rule is in some way a reflection of the asymptotic freedom of the underlying nonabelian gauge theory. However, a systematic demonstration of renormalization group control in heavy quarkonium decays has so far been lacking. The object of this note is to point out that for sufficiently heavy quarkonia, the renormalization group provides us with firm predictions for various exclusive, semi-inclusive and total inclusive decay modes. In particular, we shall assume that the heavy quark mass M is much greater than all other scales (light quark masses, renormalization scale of the theory, etc.). In this limit the bound-state dynamics will be coulombic. Consequently, we should not expect the results to be quantitatively.relevant to the charmonium system. We also calculate explicitly in this note only those decays which can proceed via a two-ghion intermediate state; however, the techniques described may equally be applied to three-gluon decay modes. In the next section we discuss the exclusive decay of a normal-parity heavy quarkonium state into two pions. The exclusive amplitude is shown to factorize and Satisfy a renormalization group equation which leads to a firm prediction for the decay rate in an asymptotically free theory. The structure of the factorization is very similar to that manifested'in the vector form factor of the pion [3] at large Q2. Next, the total inclusive hadronic decay rate is computed and seen to be proportional to exactly the same heavy meson matrix element as the exclusive decay: thus, the branching ratio into two pions depends only on renormalization group quantities and pion matrix elements. Finally, we discuss briefly the single-particle inclusive decay: the relevant amplitude will be seen to factorize into a simple coefficient function times a matrix element of a two-gluon cut vertex (time-like gluon fragmentation function). 2. Decay into two mesons. Consider the Green's function ~ ( P , K ; P l , k 1 ,P2,k2), illustrated in fig. 1, with all external fermion legs truncated. 5r also depends implicitly on a gauge coupling g, renormalized at a scale/a; all wave function renormalizations are also performed at this scale. The parameters r n , M locate, respectively, the poles of the light and heavy quark propagators. Thus, 9" satisfies a renormalization group equation
(it 2 a/a/a 2 +/3 alag - 2 ~ / ~ - 4-[~ ) s r = 0 ,
(2.1)
¢~This research was supported in part by the US Department of Energy. 119
2 June 1980
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~
kl -Pl Pl+kl
P2+k2 2 -P2
7 Fig. 2. Quarkonium decay amplitude to four light quaxks.
Fig. 1. Two heavy-, four light-quark Green's function.
with ~, 7a,, 7¢ functions of g,/22/M2, and m2//22. Eventually, we shall take m//2 ~ 0; the remaining dependence of ~ and 3' on/2/M is a reflection of the change in the renormalization group properties as the momenta in the process pass through a new quark threshold, and will be neglected for simplicity. Going to the heavy meson pole, we obtain the decay amplitude 9 illustrated in fig. 2; formally (ai, [3i are Dirac indices) 9(P; P l , k l , P2, k 2) = ig1 (3'53'+)31oq (3'53'_)~2a2 X
fd4Xld4yl d4x 2 d4y2 e i(p1+kD .xl ei(p2+k2)'x2 e-i(kl-pl)'yl
e-i(k2-p2)"Y2
X (0[ T ~ a 1(x 1) ~a2(x 2) 531 (Y 1) ~th(Y2) 12P)tr •
(2.2)
We have taken the Breit frame for the decay: the heavy meson decays at rest, producing two mesons with momentum in the z-direction, and with P l - ~ k l - ~P2+ ~ k2+~ M. 9 satisfies the renormalization group equation (the location of the bound-state pole is an invariant of the renormalization group): (/22 D/0U 2 +/3 a/~g - 43'¢ ) 9 = 0 ,
(2.3)
For M >>/2, m, it is straightforward to obtain a factorization of 9 similar to that which holds" for meson vector form factors at large Q2. First, we observe that all the gluons connecting the heavy quark line to the final twomeson state must be off-shell of order M 2 to obtain a dominant contribution (up to powers of m[M). The regime in which one of these gluons is close to shell and collinear with one of the final state mesons, for example, is seen to be suppressed when a collinear Ward identity is used to sum all possible insertions of the collinear polarized gluon on the heavy quark line. There are also the usual wee-parton cancellations among exchanges of soft gluons between the f'mal meson states. All other regimes in which some of the emitted gluons are close to shell are suppressed by simple phase space. It should now be obvious tl~at the topology of large m o m e n t u m flows in this process is such that the subtraction scheme described in ref. [3] also serves here to extract the leading behavior for large M. Namely, -4'' r d4"' g 1 cl g 2 , ~ -greg , , (P1,kl,kl)(3'53'-)c~i#i
l
~(P;Pl,kl,P2, k2)Ma, m,ta r~(3"s3"+)e,,~,(3"s3"-)~=,,=,l
0el/~llG10e1 o
o
o
o
X 9(P, Pl,kl,P2, k2)(3"S3'+)o{=#iKreg
t
(P2,k2,k2),
(2.4)
/}20t20t2/32 o
o
o
j~
where (/131, kl) u = 0 unless # = - , (P2, 2)u = 0 unless/a = +, and Kreg is an oversubtracted four-point function defreed as in ref. [3], except that it contains the extra finite renormalization needed to fix the following matrix elements at a renormalization scale/2:
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~,~, t~'i,
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2 June 1980
rd4k ', * 1, - d (2~) 4 Kreg , , (Pl, 1'1, I'1) (~53'-)-i~i ( ~ _ ) , 1 O~lfllflla 1
k~=l
=fd4Xld4yl ei(pl+kl)'Xl e-i(k1-P1)'Y'(O]Tt~.(Yl)~(xl)i~(~T53"_(½
i/~_) nl 4)[0)
× s ~ l ( k i - Pl)S~{(PI + k l ) . o
Expanding
o
o
(2.5)
o
( P ; P l , k l , P 2 , k2):
o o o 9 (P; Po l , kl, P2' k2) = ~ ( k l - / P l - ) n a 9nln2(M2 ) (k2+/P2+)n:, npn2 we obtain the usual double light-cone expansion on the final meson legs:
(2.6)
9(P;Pl,kl,P2,k2)
~ ~a --1 tr[3"53"+vnl(Pl,kl)19__ (M2) l-J--tr[3"53"_vn2(P2,k2)]. M~u,m n~,n2 4pnla__ ,,1,,2 4p~2+ The matrix elements Vn(p, k) defined in eq. (2.5) satisfy a renormalization group equation:
(2.7)
(/12 31a//2 +/33lag) Vn(p, k; m, la, g) = - ~
(2.8)
3",'n (g, m/la) vn'(p, k; m, la, g) .
FIr
For m/la -~ O, the anomalous dimensions 3"n'n are easily seen to reduce to those obtained by the soft mass-insertion generated by the Callan-Symanzik operator, as in ref. [3]. Using eqs. (2.3) and (2.8), we find
022 O/~p 2 + 133[3g - 43'~p) 9nln: (M 2) = ~ 7nlnl On'in2(M2) + n~ 3"n2n~Onln'2(M2) •
(2.9)
At this point we observe that, in consequence of our off-shell renormalization and the smoothness of the m -+ 0 limit for the hard part 9 n l n2 (M2; m,/~), we may set m to zero in eq. (2..9) and obtain a renormalization group equation which, in an asymptotically free theory, determines the large M 2 behavior of 9nln2. The smoothness of the zero-mass limit is a consequence of the absence of double flow momentum configurations around a soft region of the graph connecting the two outgoing mesons [3]. It would not hold in a scalar theory such as ~3. The immediate consequence of eq. (2.9) is that we are now allowed, in an asymptotically free theory, to compute the large M 2 behavior of 9 by considering the asymptotic expansion of the hard part 9 nln2 in the small quantity geff(M), the running coupling subtracted at the scale of the heavy quark mass. In a massless gauge theory, this expansion may be systematically extracted by isolating [4] the graphs with the strongest threshold singularities, in perturbation theory, as p2_+ 4M2. The leading term is obtained by taking the nonrelativistic reduction of the heavy quark propagators, and exchange of Coulomb gluons only. (Corrections beyond the dominant contribution are straightforward. In calculating these corrections the la/m dependence of t3 and 3' should, in principle, be kept. In practice such la/m dependences are negligible in QCD.) Thus, the full Bethe-Salpeter wavefunction of the heavyquark system is replaced by its Coulomb limit. Furthermore, one takes only the minimum allowed number of hard glUons exchanged between the heavy- and light-quark systems. Finally, it may be necessary, depending on the quantum numbers of the initial heavy meson state, to take a relativistic correction on some of the heavy-quark propagators to obtain the leading contribution to the hard part. Let us illustrate these considerations by computing the exclusive decay into two pions (or kaons) of a heavy. quark P-state (J= 0, L = 1, S = I) in the dominant order of QCD. The leading contributions to 9 (P; Pl, kl, P2' k2) for small coupling are indicated in fig. 3. Since the Coulomb wavefunction of a P-state will vanish at the origin in coordinate space, it is necessary to keep (at least) a term linear in the spatial relative momentum K of the bound state. Such terms arise from (a) the K" ~ terms in the numerators of the K+P, (K+ k 2 - kl) heavy quark propagators of fig. 3, or (b) the cross term K" (k 2 - k l ) in the denominator of the propagator carrying momentum K+ •
o
c
o
o
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kl-Pl iJ~l ,.'.
o-
K-Pc~ ~ 1
2 June 1980
o
P+K kj~/k./~./Ij'*..~...~oP2+=k:::'.., "-
Fig. 3. Hard part for two-mesonexclusivedecay Ofa 0÷ quarkonium state.
k2 -- kl" After a short calculation, we find (with x 1 --- k l _ / P l _ , x 2 ~ k2+/P2+) , after including color factors, o o o o 1 I[- 4 +~'2\7~4'T(F)C2(G)(oIT¥iV'I"P1/2P/)c"ouI,l 9(P;Pl'tq'P2'k2)-3(l_x~)(l_x2 ) Ll~XlX 2 U+xlx2] J
M6
i
(2.10)
where T(F), C2(G ) are Casimir invariants for the fundamental and adjoint representations of the color group SU(Nc). The matrix element in eq. (2.10) is to be evaluated in the Coulomb limit: it can be expressed directly in terms of the radial wavefunction. Namely, if ~(K) = G(IK[)Ylm ([~) is the normalized Schr6dinger wavefunction in momentum space, then ._, V 2 M N c / - d3K (0l~½i¥'V xI'12P)coul = r ~ " ( 2 ~ IKIG(IKI).
(2.11)
However, the same matrix element appears, as we shall see below, in the calculation of the inclusive decay and hence cancels in the branching ratio. To dominant order, the renormalization group equation (2.9) is diagonalized as for the pion form factor [3,4], by projecting onto the Gegenbauer polynomials. With the notation of ref. [3], we find the following asymptoiic behavior for 9: 9 (P; Pl, kl, Pl, k2)
×
2 2) TfF)C2(G~(Ol~t , , .~ i ¥ . ~ (2rr)464( 2 P - 2Pl - 2P2) 3~2 - ~ aeff(M MZ>u2
e(NO¢ e(N2) ~ l_~_tr[?57+vnl(P1, k l ) ] nl SN1N2 n2 1 tr[757_vn2(P2,k2)] n2 n1,.2 pnll tCNI+CN2 P2+ N1,N2
(2.12)
where t - In (M2//~2), the e(nN) are the eigenvectors of the anomalous dimension matrices (with CN the corresponding eigenvalues), and (2N1+3)!!(2N2+3)!! 1 1 I 4 (Xl+X 2 )2] fN'N:~=-4(NI+2)!(N2+2)' -fl dxl-lfdx2C~/2'(x')c2/::(x2) l+XlX2+ l+XlX2 "
(2.13)
We may now go to the pion pole on each of the final meson legs and obtain the large M 2 behavior of the decay amplitude. Thus {2p 1 , 2P212P) = (2rr)4~4(2p - 2Pl - 2P2) c/g,
(2.14a)
with ¢(nOe(N1),; ~(N2)c(n2) 32rr2 NT(F)C2(F)a2t'*2~(01'v'ti¥'~I'12P)eoulefgj,l) ,2 . ~ ~.a J~r n 1 JN1N2~n2 J , M 2~ u2 3 N2M 4 n 1,n = tCNt +CN2 ' N1 ,N2 122
(2.14b)
Volume 93B, number 1, 2
2 June 1980
PHYSICS LETTERS
f(n) = (4p~+l)-I (2Pl 1Nt,(~753,_ ( } ij~_)n ~)10), and f(0) is the usual pion decay constant. 3. Inclusive and semi-inclusive decay. The calculation of inclusive and semi-inclusive decay of a heavy quarkonium state proceeds as follows. The relevant amplitude for total inclusive decay is displayed in fig. 4. Formally:
W(P,K, K') =fdax d4y d4z ei(K-P) "xe-i(P+K)"Yei(P+K')"Y(0IT [~(z) ~(0)] T[~(x)gI'(y)] 10) -+
1(2p)2- M2 + ieI-2x(P+K,K-P)~((P+K',K'-P)~ (2rr)464(Ps- 2P)lClg S 12 .
(2p)2-+M~
(3.1)
S
In e q. (3.1), c/g S represents the decay amplitude of the heavy meson (of mass MB) into a physical state S, while X, X are Bethe-Salpeter wave functions. The hard part of this process is shown in fig. 5. Once again, for decay of a 0 + P-state, we must keep terms linear in spatial relative momentum on the heavy quark line. The calculation is straightforward and we find for M >>/a, m,
~S (2rr)464(PS- 2P)lcff~S[2~
2rroe2ff(M2)T(F)2f d4K ~NTc J (~)4 tr[¥'KXc°ul(P+K,K-P)]
xf d4K' tr [¥-K' ~cotd(P+ K', K ' (2rr)4
P)] =
2rra2ff(M2)T(F) 2 (0[+½ i ¥" V ~12P)coul i2 • M4Nc
(3.2)
The total decay rate is thus (1) 7- tot
~_a2ff(M2) T(F) 2 2 MSN2 (N2- 1)[(0[C~i~" ~ xlt[2e)c°ul[2'
(3.3)
while from eq. (2.14) we obtain the partial decay rate '1)
~7 ..-,2~r
32rr3 a4ff(M2)T(F)2C2(G) 2 -1 "-" ( 9 M9N4 I(OIq~2i¥'V q+12P)c°ul]2
e(N~)c(n2) 2 f(nl)e(N1)¢ nl JN1N2 n2 s~ tCNI+CN2
/71 ,/72 N1 ,N2
!
(3.4)
Observe, as stated previously, that the nonrelativistic matrix element cancels in the branching ratio. The single-particle inclusive decay of the heavy quarkonium state may also be computed in terms of the usual gluon fragmentation functions for inclusive e+-e - annihilation as follows. The hard part 9uv(P; k) for this process is shown in fig. 6. Once again, we shall illustrate the procedure by considering decay of a P-state. After extracting the terms linear in the relative momentum from the heavy quark propagators adjacent to the gluon insertions, we find, in the frame where P+>>P_(say):
K-P
P+K ~
K'-P
P+ K' '1
Fig. 4. Amplitude for inclusivequarkonium decay.
P÷K
I
X
P÷K'
Fig. 5. Hard part for inclusivequarkonium decay. 123
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+ EXCHANGE GRAPHS
k ~ 2 P
-k Fig. 6. Hard part for semi-inclusive quarkonium decay.
f
~7
L
2NeM3_]
_
1
~)~v(P; k) - -27r| g2T(F j_126(w-1)l(O[~i¥" ~l ~ol2P)12 ~-~ [g~vk2- - k_(k~gv + kvgu_ ) + k2g~_gv_ ] (3.5) where w =P'k[P 2. The tensor in square brackets in eq. (3.5) is instantly recognized as proportional to the bare cut vertex for the two-gluon operator [6]. The matrix dements of this cut vertex (say, in a pion state) are just those which occur in inclusive e + - e - annihilation. Also, by considering the branching ratio into a single-pion + anything, we may eliminate the explicit dependence on the heavy meson wavefunction, as in the case of the exclusive decay.
References [1] For an extensive discussion of the charmonium model, see E. Eichten et al., Phys. Rev. D17 (1978) 3090. [2] A. De Rujula and S.L. Glashow, Phys. Rev. Lett. 34 (1975) 46; T. Appelquist and H.D. Politzer, Phys. Rev. D12 (1975) 1404; R. Barbieri, E. d'Emilio, G. Curci and R. Remiddi, CERN report TH 2622 (1979); I.J. Munizich and F.E. Paige, BNL Report 26660 (1979). [3] A. Duncan and A. MueUer, Phys. Rev. D21, to be published. [4] A. Duncan, Phys. Rev. D13 (1976) 2866. [5] A. Efremov and A. Radyushkin, Dubna report E2-11983 (1979); S. Brodsky and G.P. Lepage, Phys. Lett. 87B (1979) 359. [6] See, for example, S. Gupta and A. MueUer, Phys. Rev. D20 (1979) 118.
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HEAVY QUARKONIUM DECAYS AND THE RENORMALIZATION GROUP ~ A. DUNCAN and A. MUELLER Department of Physics, Columbia University, New York, N Y 10027, USA
Received 11 January 1980
A renormalization group approach to heavy quarkonium decays is developed. Exclusive two-meson decays and inclusive decays are considered.
1. Introduction. Perhaps the most striking property of the charmonium family of resonances [I ] is the narrow hadronic decay width (typically, a fraction of an MeV) for hadronic states in a mass regime where such decay widths are generally hundreds of MeV. It has been customary to regard the dynamical suppression at work here as evidence for a phenomenological Zweig rule which forbids annihilation of the initial state quark-antiquark pair. It has also been suggested [2] that the Zweig rule is in some way a reflection of the asymptotic freedom of the underlying nonabelian gauge theory. However, a systematic demonstration of renormalization group control in heavy quarkonium decays has so far been lacking. The object of this note is to point out that for sufficiently heavy quarkonia, the renormalization group provides us with firm predictions for various exclusive, semi-inclusive and total inclusive decay modes. In particular, we shall assume that the heavy quark mass M is much greater than all other scales (light quark masses, renormalization scale of the theory, etc.). In this limit the bound-state dynamics will be coulombic. Consequently, we should not expect the results to be quantitatively.relevant to the charmonium system. We also calculate explicitly in this note only those decays which can proceed via a two-ghion intermediate state; however, the techniques described may equally be applied to three-gluon decay modes. In the next section we discuss the exclusive decay of a normal-parity heavy quarkonium state into two pions. The exclusive amplitude is shown to factorize and Satisfy a renormalization group equation which leads to a firm prediction for the decay rate in an asymptotically free theory. The structure of the factorization is very similar to that manifested'in the vector form factor of the pion [3] at large Q2. Next, the total inclusive hadronic decay rate is computed and seen to be proportional to exactly the same heavy meson matrix element as the exclusive decay: thus, the branching ratio into two pions depends only on renormalization group quantities and pion matrix elements. Finally, we discuss briefly the single-particle inclusive decay: the relevant amplitude will be seen to factorize into a simple coefficient function times a matrix element of a two-gluon cut vertex (time-like gluon fragmentation function). 2. Decay into two mesons. Consider the Green's function ~ ( P , K ; P l , k 1 ,P2,k2), illustrated in fig. 1, with all external fermion legs truncated. 5r also depends implicitly on a gauge coupling g, renormalized at a scale/a; all wave function renormalizations are also performed at this scale. The parameters r n , M locate, respectively, the poles of the light and heavy quark propagators. Thus, 9" satisfies a renormalization group equation
(it 2 a/a/a 2 +/3 alag - 2 ~ / ~ - 4-[~ ) s r = 0 ,
(2.1)
¢~This research was supported in part by the US Department of Energy. 119
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~
kl -Pl Pl+kl
P2+k2 2 -P2
7 Fig. 2. Quarkonium decay amplitude to four light quaxks.
Fig. 1. Two heavy-, four light-quark Green's function.
with ~, 7a,, 7¢ functions of g,/22/M2, and m2//22. Eventually, we shall take m//2 ~ 0; the remaining dependence of ~ and 3' on/2/M is a reflection of the change in the renormalization group properties as the momenta in the process pass through a new quark threshold, and will be neglected for simplicity. Going to the heavy meson pole, we obtain the decay amplitude 9 illustrated in fig. 2; formally (ai, [3i are Dirac indices) 9(P; P l , k l , P2, k 2) = ig1 (3'53'+)31oq (3'53'_)~2a2 X
fd4Xld4yl d4x 2 d4y2 e i(p1+kD .xl ei(p2+k2)'x2 e-i(kl-pl)'yl
e-i(k2-p2)"Y2
X (0[ T ~ a 1(x 1) ~a2(x 2) 531 (Y 1) ~th(Y2) 12P)tr •
(2.2)
We have taken the Breit frame for the decay: the heavy meson decays at rest, producing two mesons with momentum in the z-direction, and with P l - ~ k l - ~P2+ ~ k2+~ M. 9 satisfies the renormalization group equation (the location of the bound-state pole is an invariant of the renormalization group): (/22 D/0U 2 +/3 a/~g - 43'¢ ) 9 = 0 ,
(2.3)
For M >>/2, m, it is straightforward to obtain a factorization of 9 similar to that which holds" for meson vector form factors at large Q2. First, we observe that all the gluons connecting the heavy quark line to the final twomeson state must be off-shell of order M 2 to obtain a dominant contribution (up to powers of m[M). The regime in which one of these gluons is close to shell and collinear with one of the final state mesons, for example, is seen to be suppressed when a collinear Ward identity is used to sum all possible insertions of the collinear polarized gluon on the heavy quark line. There are also the usual wee-parton cancellations among exchanges of soft gluons between the f'mal meson states. All other regimes in which some of the emitted gluons are close to shell are suppressed by simple phase space. It should now be obvious tl~at the topology of large m o m e n t u m flows in this process is such that the subtraction scheme described in ref. [3] also serves here to extract the leading behavior for large M. Namely, -4'' r d4"' g 1 cl g 2 , ~ -greg , , (P1,kl,kl)(3'53'-)c~i#i
l
~(P;Pl,kl,P2, k2)Ma, m,ta r~(3"s3"+)e,,~,(3"s3"-)~=,,=,l
0el/~llG10e1 o
o
o
o
X 9(P, Pl,kl,P2, k2)(3"S3'+)o{=#iKreg
t
(P2,k2,k2),
(2.4)
/}20t20t2/32 o
o
o
j~
where (/131, kl) u = 0 unless # = - , (P2, 2)u = 0 unless/a = +, and Kreg is an oversubtracted four-point function defreed as in ref. [3], except that it contains the extra finite renormalization needed to fix the following matrix elements at a renormalization scale/2:
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~,~, t~'i,
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rd4k ', * 1, - d (2~) 4 Kreg , , (Pl, 1'1, I'1) (~53'-)-i~i ( ~ _ ) , 1 O~lfllflla 1
k~=l
=fd4Xld4yl ei(pl+kl)'Xl e-i(k1-P1)'Y'(O]Tt~.(Yl)~(xl)i~(~T53"_(½
i/~_) nl 4)[0)
× s ~ l ( k i - Pl)S~{(PI + k l ) . o
Expanding
o
o
(2.5)
o
( P ; P l , k l , P 2 , k2):
o o o 9 (P; Po l , kl, P2' k2) = ~ ( k l - / P l - ) n a 9nln2(M2 ) (k2+/P2+)n:, npn2 we obtain the usual double light-cone expansion on the final meson legs:
(2.6)
9(P;Pl,kl,P2,k2)
~ ~a --1 tr[3"53"+vnl(Pl,kl)19__ (M2) l-J--tr[3"53"_vn2(P2,k2)]. M~u,m n~,n2 4pnla__ ,,1,,2 4p~2+ The matrix elements Vn(p, k) defined in eq. (2.5) satisfy a renormalization group equation:
(2.7)
(/12 31a//2 +/33lag) Vn(p, k; m, la, g) = - ~
(2.8)
3",'n (g, m/la) vn'(p, k; m, la, g) .
FIr
For m/la -~ O, the anomalous dimensions 3"n'n are easily seen to reduce to those obtained by the soft mass-insertion generated by the Callan-Symanzik operator, as in ref. [3]. Using eqs. (2.3) and (2.8), we find
022 O/~p 2 + 133[3g - 43'~p) 9nln: (M 2) = ~ 7nlnl On'in2(M2) + n~ 3"n2n~Onln'2(M2) •
(2.9)
At this point we observe that, in consequence of our off-shell renormalization and the smoothness of the m -+ 0 limit for the hard part 9 n l n2 (M2; m,/~), we may set m to zero in eq. (2..9) and obtain a renormalization group equation which, in an asymptotically free theory, determines the large M 2 behavior of 9nln2. The smoothness of the zero-mass limit is a consequence of the absence of double flow momentum configurations around a soft region of the graph connecting the two outgoing mesons [3]. It would not hold in a scalar theory such as ~3. The immediate consequence of eq. (2.9) is that we are now allowed, in an asymptotically free theory, to compute the large M 2 behavior of 9 by considering the asymptotic expansion of the hard part 9 nln2 in the small quantity geff(M), the running coupling subtracted at the scale of the heavy quark mass. In a massless gauge theory, this expansion may be systematically extracted by isolating [4] the graphs with the strongest threshold singularities, in perturbation theory, as p2_+ 4M2. The leading term is obtained by taking the nonrelativistic reduction of the heavy quark propagators, and exchange of Coulomb gluons only. (Corrections beyond the dominant contribution are straightforward. In calculating these corrections the la/m dependence of t3 and 3' should, in principle, be kept. In practice such la/m dependences are negligible in QCD.) Thus, the full Bethe-Salpeter wavefunction of the heavyquark system is replaced by its Coulomb limit. Furthermore, one takes only the minimum allowed number of hard glUons exchanged between the heavy- and light-quark systems. Finally, it may be necessary, depending on the quantum numbers of the initial heavy meson state, to take a relativistic correction on some of the heavy-quark propagators to obtain the leading contribution to the hard part. Let us illustrate these considerations by computing the exclusive decay into two pions (or kaons) of a heavy. quark P-state (J= 0, L = 1, S = I) in the dominant order of QCD. The leading contributions to 9 (P; Pl, kl, P2' k2) for small coupling are indicated in fig. 3. Since the Coulomb wavefunction of a P-state will vanish at the origin in coordinate space, it is necessary to keep (at least) a term linear in the spatial relative momentum K of the bound state. Such terms arise from (a) the K" ~ terms in the numerators of the K+P, (K+ k 2 - kl) heavy quark propagators of fig. 3, or (b) the cross term K" (k 2 - k l ) in the denominator of the propagator carrying momentum K+ •
o
c
o
o
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kl-Pl iJ~l ,.'.
o-
K-Pc~ ~ 1
2 June 1980
o
P+K kj~/k./~./Ij'*..~...~oP2+=k:::'.., "-
Fig. 3. Hard part for two-mesonexclusivedecay Ofa 0÷ quarkonium state.
k2 -- kl" After a short calculation, we find (with x 1 --- k l _ / P l _ , x 2 ~ k2+/P2+) , after including color factors, o o o o 1 I[- 4 +~'2\7~4'T(F)C2(G)(oIT¥iV'I"P1/2P/)c"ouI,l 9(P;Pl'tq'P2'k2)-3(l_x~)(l_x2 ) Ll~XlX 2 U+xlx2] J
M6
i
(2.10)
where T(F), C2(G ) are Casimir invariants for the fundamental and adjoint representations of the color group SU(Nc). The matrix element in eq. (2.10) is to be evaluated in the Coulomb limit: it can be expressed directly in terms of the radial wavefunction. Namely, if ~(K) = G(IK[)Ylm ([~) is the normalized Schr6dinger wavefunction in momentum space, then ._, V 2 M N c / - d3K (0l~½i¥'V xI'12P)coul = r ~ " ( 2 ~ IKIG(IKI).
(2.11)
However, the same matrix element appears, as we shall see below, in the calculation of the inclusive decay and hence cancels in the branching ratio. To dominant order, the renormalization group equation (2.9) is diagonalized as for the pion form factor [3,4], by projecting onto the Gegenbauer polynomials. With the notation of ref. [3], we find the following asymptoiic behavior for 9: 9 (P; Pl, kl, Pl, k2)
×
2 2) TfF)C2(G~(Ol~t , , .~ i ¥ . ~ (2rr)464( 2 P - 2Pl - 2P2) 3~2 - ~ aeff(M MZ>u2
e(NO¢ e(N2) ~ l_~_tr[?57+vnl(P1, k l ) ] nl SN1N2 n2 1 tr[757_vn2(P2,k2)] n2 n1,.2 pnll tCNI+CN2 P2+ N1,N2
(2.12)
where t - In (M2//~2), the e(nN) are the eigenvectors of the anomalous dimension matrices (with CN the corresponding eigenvalues), and (2N1+3)!!(2N2+3)!! 1 1 I 4 (Xl+X 2 )2] fN'N:~=-4(NI+2)!(N2+2)' -fl dxl-lfdx2C~/2'(x')c2/::(x2) l+XlX2+ l+XlX2 "
(2.13)
We may now go to the pion pole on each of the final meson legs and obtain the large M 2 behavior of the decay amplitude. Thus {2p 1 , 2P212P) = (2rr)4~4(2p - 2Pl - 2P2) c/g,
(2.14a)
with ¢(nOe(N1),; ~(N2)c(n2) 32rr2 NT(F)C2(F)a2t'*2~(01'v'ti¥'~I'12P)eoulefgj,l) ,2 . ~ ~.a J~r n 1 JN1N2~n2 J , M 2~ u2 3 N2M 4 n 1,n = tCNt +CN2 ' N1 ,N2 122
(2.14b)
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f(n) = (4p~+l)-I (2Pl 1Nt,(~753,_ ( } ij~_)n ~)10), and f(0) is the usual pion decay constant. 3. Inclusive and semi-inclusive decay. The calculation of inclusive and semi-inclusive decay of a heavy quarkonium state proceeds as follows. The relevant amplitude for total inclusive decay is displayed in fig. 4. Formally:
W(P,K, K') =fdax d4y d4z ei(K-P) "xe-i(P+K)"Yei(P+K')"Y(0IT [~(z) ~(0)] T[~(x)gI'(y)] 10) -+
1(2p)2- M2 + ieI-2x(P+K,K-P)~((P+K',K'-P)~ (2rr)464(Ps- 2P)lClg S 12 .
(2p)2-+M~
(3.1)
S
In e q. (3.1), c/g S represents the decay amplitude of the heavy meson (of mass MB) into a physical state S, while X, X are Bethe-Salpeter wave functions. The hard part of this process is shown in fig. 5. Once again, for decay of a 0 + P-state, we must keep terms linear in spatial relative momentum on the heavy quark line. The calculation is straightforward and we find for M >>/a, m,
~S (2rr)464(PS- 2P)lcff~S[2~
2rroe2ff(M2)T(F)2f d4K ~NTc J (~)4 tr[¥'KXc°ul(P+K,K-P)]
xf d4K' tr [¥-K' ~cotd(P+ K', K ' (2rr)4
P)] =
2rra2ff(M2)T(F) 2 (0[+½ i ¥" V ~12P)coul i2 • M4Nc
(3.2)
The total decay rate is thus (1) 7- tot
~_a2ff(M2) T(F) 2 2 MSN2 (N2- 1)[(0[C~i~" ~ xlt[2e)c°ul[2'
(3.3)
while from eq. (2.14) we obtain the partial decay rate '1)
~7 ..-,2~r
32rr3 a4ff(M2)T(F)2C2(G) 2 -1 "-" ( 9 M9N4 I(OIq~2i¥'V q+12P)c°ul]2
e(N~)c(n2) 2 f(nl)e(N1)¢ nl JN1N2 n2 s~ tCNI+CN2
/71 ,/72 N1 ,N2
!
(3.4)
Observe, as stated previously, that the nonrelativistic matrix element cancels in the branching ratio. The single-particle inclusive decay of the heavy quarkonium state may also be computed in terms of the usual gluon fragmentation functions for inclusive e+-e - annihilation as follows. The hard part 9uv(P; k) for this process is shown in fig. 6. Once again, we shall illustrate the procedure by considering decay of a P-state. After extracting the terms linear in the relative momentum from the heavy quark propagators adjacent to the gluon insertions, we find, in the frame where P+>>P_(say):
K-P
P+K ~
K'-P
P+ K' '1
Fig. 4. Amplitude for inclusivequarkonium decay.
P÷K
I
X
P÷K'
Fig. 5. Hard part for inclusivequarkonium decay. 123
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+ EXCHANGE GRAPHS
k ~ 2 P
-k Fig. 6. Hard part for semi-inclusive quarkonium decay.
f
~7
L
2NeM3_]
_
1
~)~v(P; k) - -27r| g2T(F j_126(w-1)l(O[~i¥" ~l ~ol2P)12 ~-~ [g~vk2- - k_(k~gv + kvgu_ ) + k2g~_gv_ ] (3.5) where w =P'k[P 2. The tensor in square brackets in eq. (3.5) is instantly recognized as proportional to the bare cut vertex for the two-gluon operator [6]. The matrix dements of this cut vertex (say, in a pion state) are just those which occur in inclusive e + - e - annihilation. Also, by considering the branching ratio into a single-pion + anything, we may eliminate the explicit dependence on the heavy meson wavefunction, as in the case of the exclusive decay.
References [1] For an extensive discussion of the charmonium model, see E. Eichten et al., Phys. Rev. D17 (1978) 3090. [2] A. De Rujula and S.L. Glashow, Phys. Rev. Lett. 34 (1975) 46; T. Appelquist and H.D. Politzer, Phys. Rev. D12 (1975) 1404; R. Barbieri, E. d'Emilio, G. Curci and R. Remiddi, CERN report TH 2622 (1979); I.J. Munizich and F.E. Paige, BNL Report 26660 (1979). [3] A. Duncan and A. MueUer, Phys. Rev. D21, to be published. [4] A. Duncan, Phys. Rev. D13 (1976) 2866. [5] A. Efremov and A. Radyushkin, Dubna report E2-11983 (1979); S. Brodsky and G.P. Lepage, Phys. Lett. 87B (1979) 359. [6] See, for example, S. Gupta and A. MueUer, Phys. Rev. D20 (1979) 118.
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