Pentaquarks
James D. Olsen and Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(April 8, 2005)
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Problem
a) Diquarks are combinations of two quarks (or two antiquarks). What are the possible
spin, flavor, and color configurations of ground-state (orbital angular momentum L =
0) diquarks composed of the three light quarks (u, d and s)? Present your results for
the flavor wavefunctions in the form of states in I3 -Y space, labeled by their quark
content, where I3 is the third component of isospin and Y = B + S is the hypercharge
= baryon number + strangeness.
Can diquarks exist as free particles?
b) Experimental data (of erratic quality) suggests that there is a (baryon) resonance in
K + n scattering at 1540 MeV center-of-mass energy. This cannot correspond to a particle in any plausible three-quark model of baryons (as no appropriate companion states
have been observed), which has led to speculations as to the existence of pentaquark
baryon states consisting of two diquarks plus an antiquark: (qq)(qq)q̄.
One pentaquark model, due to Jaffe and Wilczek, Phys. Rev. Lett. 91, 232003 (2003),
supposes that the constituent diquark states are of the type considered in part a), with
the additional restriction that the diquark wavefunctions are antisymmetric separately
with respect to quark interchange in each of the three degrees of freedom: color, flavor
and spin. Furthermore, the overall space and spin wavefunctions of the pentaquarks
are symmetric with respect to quark interchange.
What flavor multiplets of pentaquarks are permitted in this model? Present your
results in the form of states in I3 -Y space, labeled by their quark content. Which
pentaquark state could correspond to the K + n resonance, which is sometimes called
the θ+ (1540)?
c) A possible merit of the Jaffe-Wilczek model is that it can also accommodate the nucleon
resonance N (1440), the so-called Roper resonance, another state for which the evidence
is not strong. Supposing this identification to be correct, predict the masses of the other
pentaquark states in a simple constituent quark model that assumes isospin invariance
and neglects binding energies.
d) Despite the many restrictions stated above for pentaquark model, there still remains
considerable freedom in the symmetries of the wavefunctions. Jaffe and Wilczek argued
further that the dynamics of the interactions of diquark pairs favors orbital angular
momentum 1 between them, while the orbital angular momentum of the antiquark
relative to the tetraquark system is favored to be 0. Assuming further that the total
angular momentum of the pentaquarks is the minimum consistent with the postulated
additional constraints, what are the total angular momentum and parity, J P , of the
pentaquark states?
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Solution
a) Quarks have spin 1/2, so diquarks can have spin 0 or spin 1. The spin part of the wave
function of the spin-0 (spin-1) states is antisymmetric (symmetric)
with respect√to
√
interchange of the two quarks. That is, (↑↓)0 = (↑↓ − ↓↑)/ 2, (↑↓)1 = (↑↓ + ↓↑)/ 2.
If we restrict our attention to the light quarks, u, d and s, we consider each quark
to be a member of a flavor triplet 3f , commonly represented as a triangle in I3 -Y
(isospin-hypercharge) space,
noting that hypercharge Y = B + S, where B = baryon number = 1/3 for quarks, and
S = strangeness = 0 for the u and d quarks but S = −1 for the s quark.
The flavor structure of a diquark is therefore described as the product
3f ⊗ 3f = 6f ⊕ 3̄f ,
(1)
which can be represented graphically as
The exchange symmetry of the 3̄f and 6f multiplets follows a pattern like that for the
spin multiplets, namely the wavefunctions of the smaller multiplet, the 3̄f , is
√ antisymmetric with respect to interchange of the quarks (i.e., ud3̄f = (ud − du)/ 2), while
wavefunctions in the larger
√ multiplet are symmetric with respect to quark interchange
(i.e., ud6f = (ud + du)/ 2).
Each quark is also a member of a color triplet, 3̄c . Diquarks are therefore members of
either a color triplet or a color sextet, according to the color version of eq.(1),
3c ⊗ 3c = 6c ⊕ 3̄c ,
(2)
And similarly, the wave functions of the multiplet 3̄c are antisymmetric with respect to
interchange of the quarks, while those of the multiplet 6c are symmetric with respect
to quark interchange.
An apparent fact of QCD (quantum chromodynamics) is that free states (at least at
presently accessible energies) must be color singlets (1c ). Since there are no colorsinglet diquark states, we do not expect to observe free diquarks. They can, however,
exist as substates of larger particles; for example, the proton has quark content uud
and can be thought of as containing a ud diquark.
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b) The premise that the diquark color, flavor and spin wavefunctions are antisymmetric
with respect to quark interchange implies that these diquark states are members of the
spin-zero singlet 1S , the flavor triplet 3̄f , and the color triplet 3̄c .
We digress briefly to consider the overall color wavefunction of the pentaquarks, which
should be a color singlet if the pentaquarks are to exist as free particles. First, the
tetraquark color multiplets follow from the relation
3̄c ⊗ 3̄c = 6̄c ⊕ 3c ,
(3)
which is the color conjugate of eq. (2). Combining the tetraquark with an antiquark
(from a 3̄c triplet), the overall color multiplets are given by
(6̄c ⊕ 3c ) ⊗ 3̄c = (10c ⊕ 8c ) ⊕ (80c ⊕ 1c ).
(4)
As desired, a color singlet appears among the 27 possible states, and this is the one
state that would be realized in Nature. The color singlet wavefunction is, of course,
antisymmetric with respect to quark interchange.
Turning to the pentaquark flavor wavefunctions, the flavor multiplet structure is given
by the flavor version of eq. (4),
(6̄f ⊕ 3f ) ⊗ 3̄f = (10f ⊕ 8f ) ⊕ (80f ⊕ 1f ).
(5)
Since quarks are fermions, the overall wavefunction of a pentaquark must be antisymmetric with respect to quark interchange. QCD requires that the color wavefunction be
antisymmetric, and the Jaffe-Wilczek model postulates that the space-spin wavefunction is symmetric. Hence, the flavor wave function must be symmetric with respect to
quark interchange as well.
The largest multiplet in eq. (5), the 10f decuplet is symmetric with respect to quark
interchange, while the two the octets, 8f and 80f are of mixed symmetry. However,
these two octets can be combined to form one symmetric and one antisymmetric octet.
The overall antisymmetry requirement leads us to suppose that physical pentaquarks
are members of the symmetric flavor decuplet 10f or the symmetric octet 8f . The
structure of these multiplets is readily given graphically, whether or not we have been
guided by the formal relation (5).
It is useful to give first a graphic rendition of the flavor analog of eq. (3),
3̄f ⊗ 3̄f = 6̄f ⊕ 3f ,
3
(6)
To obtain pentaquarks whose flavor wavefunctions are symmetric with respect to quark
interchange, we combine the symmetric sextet 6̄f shown above (in the middle), with
the antiquark triplet 3̄f ,
which leads to the 18 states represented as
These states are regrouped as the decuplet 10f and the octet 8f ,
where the state at the center of the octet is actually doubled according to the graphical
construction.
The separation of the 18 states in the 10f + 8f multiplets is accomplished in an orderly
fashion by first noting that the outermost 3 states must belong to the decuplet. Then,
as we move from one state to its nearest neighbor within the decuplet, we change
the flavor of only one of the quarks. Having identified the states of the decuplet, the
remaining 8 states belong to the octet. Nearest-neighbor states in the octet sometimes
differ in the flavor of two or even three quarks.
The θ+ (1540) resonance of the K + n system is obtained from a combination of us̄
and udd quarks, and so would correspond to the (ud)(ud)s̄ pentaquark, which is the
uppermost state of the decuplet 10f .
c) The N (1440) state could correspond to the upper two states of the octet, (ud)(ud)ū
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¯ which form an isospin doublet.
and (ud)(ud)d,
In a constituent quark model of pentaquark masses, assuming isospin invariance and
neglecting binding energies, we suppose that the u and d quarks have equal masses,
mu = md . Then from the θ+ (1540) we infer that
2md + 2md + ms = 4mu + ms = 1540 MeV/c2 ,
(7)
while from the N (1440) we infer that
5mu = 1440 MeV/c2 .
(8)
In this approximation, we find
mu = md = 288 MeV/c2 ,
ms = 388 MeV/c2 .
(9)
Isospin invariance implies that all states on the same horizontal line within a multiplet
have the same mass. We now predict the masses of the four isospin multiplets within
the decuplet 10f to be
m10f (Y = 2) = 1540 MeV/c2 ,
m10f (Y = 1) = 1640 MeV/c2 ,
m10f (Y = 0) = 1540 MeV/c2 ,
m10f (Y = −1) = 1740 MeV/c2 ,
(10)
and the masses of the three (actually four, as there are two Y = 0 multiplets) of the
octet 8f to be
m8f (Y = 1) = 1440 MeV/c2 ,
m8f (Y = 0) = 1540 MeV/c2 ,
m8f (Y = −1) = 1640 MeV/c2 .
(11)
d) If the orbital angular momentum between the two diquarks is L = 1, and the diquarks
are each spin singlets, the total angular momentum and parity of the tetraquark system
is J P = 1− .
The tetraquark system combines with the spin-1/2 antiquark, whose intrinsic parity is
− relative to the quarks. If the orbital angular momentum of the antiquark relative to
the tetraquark is 0, then the total spin of the pentaquark can be either 1/2 or 3/2, and
the overall parity is +. The lowest-spin pentaquark states therefore have spin-parity
+
J P = 12 in the Jaffe-Wilczek model, as does the nucleon. This is consistent with the
possibility that the Roper resonance is a pentaquark state.
Many, but not all, of the pentaquark states in the decuplet 10f and octet 8f involve
diquarks of identical flavor composition. If the color, flavor and spin wavefunctions of
these states are all antisymmetric under quark interchange, then these diquarks have
spin 1/2, and so they are fermions. In this case, the wavefunction of tetraquark states
consisting of identical diquarks must be antisymmetric with respect to interchange of
the diquarks. Therefore, the spatial wavefunction of the two diquarks must incorporate
odd orbital angular momentum, which is consistent with the dynamical argument of
Jaffe and Wilczek that the orbital angular momentum of the two diquarks is L = 1.
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