

 29 Sep 95

HYBRIDS : DYNAMICS AND DISGUISES

P.R. PAGE Theoretical Physics, University of Oxford,

1 Keble Road, Oxford OX1 3NP, UK

We introduce selection rules arizing from flux-tube dynamics with non-relativistic and adiabatic quark motion. Specifically, we indicate how states can disguise themselves by not decaying to S-wave states.|

q(x)..q(x) |

B r

B y y^ rC C|

Q(r). rA .Q(r)|

A

Mesons can be regarded as Q _Q systems connected by a flux-tube. As a first orientation we conceptually seperate the dynamics of the "quark" and "flux" components of the system. This is called the adiabaticity assumption, and is valid for large quark masses. As the quarks move, the flux-tube is considered to re-assemble itself spontaneously. Hybrids are Q _Q systems with an excited flux-tube. The flux-tube can possess angular momentum \Lambda H around the Q _Q- axis rH, which is conserved as the flux-tube j rH Hi must a priori be invariant under rotations around the Q _Q-axis. The quarks move adiabatically in an effective potential generated by the flux-tube dynamics, and hence obey a Schr"odinger equation which determines the dependence on Q _Q seperation of the wave function H (rH) of the system. The above ideas for mesons and hybrids have been implemented in lattice gauge theory1 and in simulations thereof2.

We proceed to discuss the decay dynamics of states with adiabatically and non-relativistically1 moving quarks. The term "states" refers to both mesons and hybrids. We assume that a q _q pair is created with quark mass m at position y with spin Sq_q = 1, orbital angular momentum Lq_q = 1 and total angular momentum Jq_q = 0 (see Figure for the decay topology). The decay process is called "3P0 pair creation". Its pre-eminence as a decay model for mesons is based on its surprising phenomenological success3 especially for light mesons2, in contrast to 3S1 pair creation for example. We adopt it.

Given the success of 3P0 pair creation for meson decays to mesons, we extrapolate this mechanism to all states. We shall develop selection rules valid for arbitrary wave functions. By the conservation of spin (since the decay operator creates a Sq_q = 1 pair) we firstly obtain the following selection rule :

1

Decays of net spin S = 0 states to two S = 0 states are forbidden.

The 3P0 pair creation amplitude can be shown2;4 to be proportional toZ

d3rA d3y A(rA) exp(i Mm + M pB \Delta rA) fl(rA; y)

\Theta (irrB + irrC + 2mm + M pB) B\Lambda (rB) C\Lambda (rC) (1) for a stationary state A with quarks of mass M decaying to the outgoing states B and C. The quark and flux degrees of freedom are seperated adiabatically. The initial flux-tube would have to re-assemble into the two flux-tubes of the final states. This has a certain re-arrangement amplitude, which we call the flux-tube overlap fl(rA; y).

The pair creation position y can be decomposed into the transverse "component" y? j \Gamma (y \Theta ^rA) \Theta ^rA and the parallel "component" yk j (y \Delta ^rA)^rA (with magnitude yk j y \Delta ^rA). Defining OE as the angle around the Q _Q-axis, we introduce a result related to the conservation of angular momentum around the Q _Q-axis.

Theorem 1 The most general form of the flux-tube overlap in the limit where the pair creation is near to the initial Q _Q-axis is

fl(rA; y) = ei\Lambda OE f(rA; y2?; yk) where \Lambda j \Lambda A \Gamma \Lambda B \Gamma \Lambda C (2) Proof The full decay configuration can be described2 by the six variables rA; y. By rotational invariance the overlap fl(rA; y) cannot depend on the direction ^rA of the Q _Q-axis. This leaves the dependence to be on the four variables rA; y2?; OE and yk. We can reveal the OE-dependence by considering a rotation R of an initial pair creation position y corresponding to OE = 0 by an angle OE around the Q _Q-axis. Denoting the effect of pair creation by ^O

fl(rA; Ry) j hrBB rCC j ^O(Ry) j rA Ai

= hrBB rCC j R+ ^O(y)R j rA Ai = hRrBB RrCC j ^O(y) j RrA Ai (3)

giving the desired result since R jrH Hi = exp(iOE\Lambda H ) jrH Hi, where H 2 fA; B; Cg. 2

We note that when the y-integral in Eq. 1 is performed the OE-dependence exp i\Lambda OE in the flux-tube overlap must be matched by a factor exp \Gamma i*OE arising from the y-dependent part of the decay amplitude which only contributes when * = \Lambda using R 2ss0 dOE e\Gamma i*OEei\Lambda OE = 2ssffi*\Lambda

2

Theorem 2 For pair creation near the initial Q _Q-axis, decay is forbidden for (1) j\Lambda Aj * 2 state ! two S-wave states; (2) j\Lambda Aj = 1 state ! two identical S-wave states, if fl(rA; y) is even under yk ! \Gamma yk; (3) j\Lambda Aj = 0 state ! two identical S-wave states, if fl(rA; y) is odd under yk ! \Gamma yk. Proof Because H(rH), H 2 fB; Cg, is an S-wave wave function it only depends on rH2, i.e. H (rH) j ~H(r2H ), and hence in the last line in Eq. 1 on the OE-independent variable rH2 = r2A=4 \Sigma rA:y + y2 when we substitute2 rB = rA=2 + y, rC = rA=2 \Gamma y. The non-derivative term in Eq. 1 has * = 0 due to its OE-independence. Moreover, the derivative terms in the last line of Eq. 1 equals

2i n( rA2 + y) rrB2 ~\Lambda B(r2B) ~\Lambda C(r2C) + ( rA2 \Gamma y) ~\Lambda B(r2B) rrC2 ~\Lambda C(r2C)o (4) by the chain rule. In the above * = \Gamma 1; 0; 1 because rrB2 ~\Lambda (r2B) ~\Lambda C(r2C) and ~\Lambda B(r2B) rrC2 ~\Lambda (r2C) are OE-independent. So clearly if j\Lambda j * 2 there is no contribution, establishing the first result. For the remaining results define the common wave function j B = C. The last line of Eq. 1 equals

(2irrA + 2mm + M pB) \Lambda ( rA2 + y)\Lambda ( rA2 \Gamma y) (5) If j\Lambda j = 1 Eq. 2 implies that fl(rA; y) is odd under y? ! \Gamma y?, and by assumption it is even under yk ! \Gamma yk, so that it is odd under exchange of y ! \Gamma y. Analogously, if j\Lambda j = 0, fl(rA; y) is even under y? ! \Gamma y?, and by assumption odd under yk ! \Gamma yk, so that it is odd under y ! \Gamma y. It is sufficient to show that Eq. 5 is even under y ! \Gamma y. But this is manifest since the derivative term in Eq. 5 is independent of y and \Lambda ( rA2 + y) \Lambda ( rA2 \Gamma y) is symmetric under y ! \Gamma y. 2

We expect (partial) breaking of the above rule when the outgoing states do not have identical wave functions, making the decay amplitude proportional to the "difference of the final state wave functions". Theorem 2 also has the consequence that modes not suppressed by this rule might in the absence of arguments to the contrary be considered potentially significant.

References

1. C. Michael and S. Perantonis, Nucl. Phys. B 347, 854 (1990). 2. N. Isgur et al, Phys. Rev. Lett. 54, 869 (1985); F.E. Close and P.R.

Page, Nucl. Phys. B 443, 233 (1995); Phys. Rev. D 52, 1706 (1995). 3. P. Geiger and E.S. Swanson, Phys. Rev. D 50, 6855 (1994). 4. P.R. Page, Nucl. Phys. B 446, 189 (1995); NATO ASI "Hadron Spectroscopy and the Confinement Problem" (Plenum Press, 1996).

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