Difference between revisions of "GlueX Physics Strong Decay Models"

Revision as of 22:36, 19 April 2010

Strong Decay Models

SU(3) Decay Calculations

SU(3) Rules

A simple starting point for looking at decays of mesons is to consider SU(3) flavor symmetry in the light-quark (u,d,s) sector. As we saw in the earlier section on SU(3), mesons are arranged in nonets (nine) of $q{\bar q}$ pairs, with these in turn being broken into an SU(3) singlet state and eight SU(3) octet states. If we consider the decay of a meson to pairs of mesons, where both daughters are members of the same nonet (e.g. the pseudoscalar mesons), then we can use SU(3) algebra to determine which decays are possible.

In SU(3), we note that the product of an 8 times an 8 is given as the sum:

$8\otimes 8=27\oplus 10\oplus 10\oplus 8\oplus 8\oplus 1$

Thus, it is possible for a pair of octet mesons to couple to an octet meson or a singlet meson (the 27 and the 10 do not represent simple $q{\bar q}$ systems). Similarly, under SU(3)

$1\otimes 1=1$

and

$1\otimes 8=8$ .

Putting all of this together, there are four types of decays that we see could be allowed under SU(3). Each of these could, in principle, have its own decay constant

$8\rightarrow 8\otimes 8$	g_T	(1)
$8\rightarrow 8\otimes 1$	g₁₈	(2)
$1\rightarrow 8\otimes 8$	g₁	(3)
$1\rightarrow 1\otimes 1$	g₁₁	(4)

Decay Rates

In the following, we will show that under the assumption that some decays are forbidden, we can reduce the four constants g_T, g₁₈, g₁, g₁₁ down to one unknown constant per nonet.

The decay rate $\Gamma$ can be expressed as

$\Gamma =\gamma ^{2}\cdot q\cdot f_{L}(q^{2})$ (5)

where q is the break-up momentum from the decay of $A\rightarrow BC$ . In the rest-frame of A, both B and C have momentum q, with

$q={\frac {[(m_{A}^{2}-(m_{B}+m_{C})^{2})\cdot (m_{A}^{2}-(m_{B}-m_{C})^{2})]^{{(1/2)}}}{2m_{A}}}$

The momentum q is related to the available phase-space for the decay to occur,

\rho =2q/m

. In addition to the phase-space factor, q, we can also have a form factor for the decay, f_L(q). This can depend on the break-up momentum as well as the orbital angular momentum between B and C (L).

SU(3) Clebsch-Gordan Coefficients

In order to compute the decay rates, we will now take advantage of the SU(3) Clebsch-Gordan (c.g.) coefficients. For the decay $1\rightarrow 8\otimes 8$ , we can express the individual decays as

(

\eta _{1}

)

\rightarrow

[(K^{+},K^{0}){\bar K},(\pi ^{+},\pi ^{0},\pi ^{-})\pi ,\eta _{8}\eta _{8},(K^{-},{\bar K}^{0})K]

where the corresponding four c.g coefficients are given by

{\frac {1}{{\sqrt {8}}}}\cdot (2,3,-1,-2)^{{1/2}}

,

where it is implicitly taken that the minus sign inside the square root is factored out front. Thus

$[c.g.](\eta _{1}\rightarrow \eta _{8}\eta _{8})=-1/{\sqrt {8}}$

</P>

For the decays $1\rightarrow 1\otimes 1$ and $8\rightarrow 1\otimes 8$ , the c.g. coefficients are all 1. Finally, for the decays $8\rightarrow 8\otimes 8$ , there are 17 c.g. coefficients given as follows

$K$	$\rightarrow$	$K\pi$		$K\eta _{8}$	$\pi K$	$\eta _{8}K$
$\pi$	$\rightarrow$	$K{\bar K}$	$\pi \pi$	$\eta _{8}\pi$	$\pi \eta _{8}$	${\bar K}K$
$\eta _{8}$	$\rightarrow$	$K{\bar K}$	$\pi \pi$	$\eta _{8}\eta _{8}$		${\bar K}K$
${\bar K}$	$\rightarrow$	$\pi {\bar K}$		$\eta _{8}{\bar K}$	${\bar K}\pi$	${\bar K}\eta _{8}$

${\frac {1}{{\sqrt {20}}}}$	$\left(\right.$	${\sqrt {9}}$		$-{\sqrt {1}}$	$-{\sqrt {9}}$	$-{\sqrt {9}}$	$\left.\right)$
${\frac {1}{{\sqrt {20}}}}$	$\left(\right.$	$-{\sqrt {6}}$	${\sqrt {0}}$	${\sqrt {4}}$	${\sqrt {4}}$	$-{\sqrt {6}}$	$\left.\right)$
${\frac {1}{{\sqrt {20}}}}$	$\left(\right.$	$-{\sqrt {2}}$	$-{\sqrt {12}}$	$-{\sqrt {4}}$		$-{\sqrt {2}}$	$\left.\right)$
${\frac {1}{{\sqrt {20}}}}$	$\left(\right.$	${\sqrt {9}}$		$-{\sqrt {1}}$	$-{\sqrt {9}}$	$-{\sqrt {1}}$	$\left.\right)$

Decays of Mesons

We can now use these c.g. coefficients to compute relative decay rates. As an example, let us consider the decay of an $\eta _{1}$ .

$\gamma (\eta _{1}\rightarrow \pi \pi )$	$=$	${\sqrt {3/8}}\cdot g_{1}$
$\gamma (\eta _{1}\rightarrow K{\bar K})$	$=$	${\sqrt {2/8}}\cdot g_{1}$
$\gamma (\eta _{1}\rightarrow \eta _{8}\eta _{8})$	$=$	$-{\sqrt {1/8}}\cdot g_{1}$

This yields the following ratios for the $\gamma ^{2}$ for the decays of the $\eta _{1}$ to pairs of pseudoscalar mesons.

$\gamma ^{2}\eta _{1}\rightarrow (\pi \pi :K{\bar K}:\eta _{8}\eta _{8})=(4:3:1)\cdot g_{{1}}^{{2}}$

We can also compute the decay to a pair of SU(3) singlets as

$\gamma ^{2}\eta _{1}\rightarrow (\eta _{1}\eta _{1})=(1)\cdot g_{{11}}^{{2}}$

While pure singlet and octet $\eta$ s don't exist, we can use these to expand our calculation to more physical mixing of states. In particular, we will eventually be able to say something about the nonet mixing angles. We consider an f and f^' states that are physical mixtures of the octet and singlet states.

$f=cos(\theta )\cdot f_{1}+sin(\theta )\cdot f_{8}$
$f^{{'}}=cos(\theta )\cdot f_{8}-sin(\theta )\cdot f_{1}$

We can now write out several general decays to pairs of pseudoscalar mesons. We first consider the decays of f to pairs of octet mesons.

$\gamma (f\rightarrow \pi \pi )=\cos(\theta )\gamma (f_{1}\rightarrow \pi \pi )+\sin(\theta )\gamma (f_{8}\rightarrow \pi \pi )$
$\gamma (f\rightarrow \pi \pi )={\sqrt {3/8}}\cdot g_{1}\cdot \cos(\theta )-{\sqrt {3/5}}\cdot g_{T}\cdot \sin(\theta )$ (6)

Similarly, we can find that

\gamma (f\rightarrow K{\bar K})=1/2\cdot g_{1}\cdot \cos(\theta )+{\sqrt {1/10}}\cdot g_{T}\cdot \sin(\theta )

(7)

The ³P₀ Model

Attempts at modeling strong decays date from 1969, when Micu suggested that hadron decay proceeds through $q\bar q$ pair production with vacuum quantum numbers, $J^{{PC}}=0^{{++}}$ . Since this corresponds to a ³P₀ $q{\bar q}$ state, it is now generally referred to as the 3p0 decay model. This suggestion was developed and applied extensively by Le Yaouanc et al. in the 1970s. Studies of hadron decays using the 3p0 model have been concerned almost exclusively with numerical predictions, and have not led to any fundamental modifications to the original model. Recent studies have considered changes in the spatial dependence of the pair production amplitude as a function of quark coordinates but the fundamental decay mechanism is usually not addressed; this is widely believed to be a nonperturbative process, involving ``flux tube breaking". There have been some studies of the decay mechanism which consider an alternative phenomenological model in which the $q{\bar q}$ pair is produced with ³S₁ quantum numbers; this possibility however appears to disagree with experiment.

References

Meson Decay References

Higher quarkonia, T.Barnes, F.E. Close, P.R. Page and E.S. Swanson, Phys. Rev. D55 (1997), 4157-4188 PRD.
On the mechanism of open flavor strong decays, E.S. Ackleh, T. Barnes and E.S. Swanson, Phys. Rev. D54 (1996), 6811-6829, PRD.
Hybrid and conventional mesons in the flux tube model: Numerical studies and their phenomenological implications, T. Barnes, F.E. Close and E.S. Swanson, Phys. Rev. D52 (1995), 5242-5256, PRD.
The Quenched Approximation In The Quark Model, P. Geiger and N. Isgur, Phys. Rev. D41 (1990), 1595, PRD.
Meson Decays by Flux Tube Breaking, R. Kokoski and N. Isgur, Phys. Rev. D35 (1987), 907, PRD.
A Flux Tube Model for Hadrons in QCD, N. Isgur, J. E. Paton, Phys. Rev. D31 (1985), 2910, PRD.
A Flux Tube Model for Hadrons N. Isgur and J. E. Paton, Phys. Lett. B124, (1983), 247, Science Direct.

Hybrid Meson Decay References

The "forbidden" decays of hybrid mesons to π ρ can be large, F.E. Close and J.J. Dudek, Phys. Rev. D70 (2004), 094015, arXiv.
- Hybrid Meson Decay Phenomenology, P.R. Page, E.S. Swanson and A.P. Szczepaniak, Phys. Rev. D59 (1999), 034016, arXiv.
Why Hybrid Meson Coupling to Two S-wave Mesons is Suppressed, P.R. Page, Phys. Lett. B402 (1996), 183-188, arXiv.
Q anti-Q G Hermaphrodite Mesons in the MIT Bag Model, T. Barnes, F.E. Close, F. de Viron and J. Weyers, Nucl. Phys. B224 (1983), 241, Science Direct.
Gluonic Excitations of Mesons: Why They Are Missing and Where to Find Them, N. Isgur, R. Kokoski and J. E. Paton, Phys. Rev. Lett. 54 (1985), 869, PRL.

@@ Line 195: / Line 195: @@
 <math>
 \gamma (f\rightarrow \pi\pi) = \cos (\theta) \gamma (f_1\rightarrow \pi\pi) + \sin (\theta) \gamma (f_8\rightarrow \pi\pi)</math><br><math>
-\gamma (f\rightarrow \pi\pi) = \sqrt{3/8} \cdot g_1 \cdot \cos (\theta) - \sqrt{3/5} \cdot g_T \cdot \sin (\theta )
+\gamma (f\rightarrow \pi\pi) = \sqrt{3/8} \cdot g_1 \cdot \cos (\theta) - \sqrt{3/5} \cdot g_T \cdot \sin (\theta )</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (6)
-</math>
 </center>
 Similarly, we can find that
 <center><math>
-\gamma (f\rightarrow K\bar K) = 1/2 \cdot g_1 \cdot \cos (\theta) + \sqrt{1/10} \cdot g_T \cdot \sin (\theta )
+\gamma (f\rightarrow K\bar K) = 1/2 \cdot g_1 \cdot \cos (\theta) + \sqrt{1/10} \cdot g_T \cdot \sin (\theta )</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (7) </center>
-</math></center>
 ===The <sup>3</sup>P<sub>0</sub> Model===

Difference between revisions of "GlueX Physics Strong Decay Models"

Revision as of 22:36, 19 April 2010

Contents

GlueX Physics

Review Papers

Mesons in the Quark Model

Exotic Quantum Number Mesons

Lattice QCD Calculations

Photoproduction

Strong Decay Models

SU(3) Decay Calculations

SU(3) Rules

Decay Rates

SU(3) Clebsch-Gordan Coefficients

Decays of Mesons

The ³P₀ Model

References

Meson Decay References

Hybrid Meson Decay References

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Difference between revisions of "GlueX Physics Strong Decay Models"

Revision as of 22:36, 19 April 2010

Contents

GlueX Physics

Review Papers

Mesons in the Quark Model

Exotic Quantum Number Mesons

Lattice QCD Calculations

Photoproduction

Strong Decay Models

SU(3) Decay Calculations

SU(3) Rules

Decay Rates

SU(3) Clebsch-Gordan Coefficients

Decays of Mesons

The 3P0 Model

References

Meson Decay References

Hybrid Meson Decay References

Navigation menu

Search

The ³P₀ Model