Large $N_c$ Universality of The Baryon Isgur--Wise Form Factor: The Group Theoretical Approach

10 Pages·0.11 MB·English

by  Chi-Keung Chow

CLNS 96/1392
Large N Universality of The Baryon Isgur–Wise Form Factor:
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The Group Theoretical Approach
Chi-Keung Chow
6 Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853.
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(February 1, 2008)
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Abstract
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Inapreviousarticle, ithas beenproved undertheframeworkof chiral soliton
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4
2 model that the same Isgur–Wise form factor describes the semileptonic Λb
→
1
0 Λ and Σ(∗) Σ(∗) decays in the large N limit. It is shown here that
6 c b → c c
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this result is in fact independent of the chiral soliton model and is solely the
/
h
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consequence of the spin-flavor SU(4) symmetry which arises in the baryon
-
p
e sector in the large N limit.
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Typeset using REVTEX
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Inapreviousarticle[1],ithasbeenprovedthatthesemileptonicΛ Λ andΣ(∗) Σ(∗)
b → c b → c
decays are controlled by the same Isgur–Wise form factor in the large N limit. Recall that
c
the semileptonic Λ Λ decay depends on a universal form factor η(w) [2–6], which is
b c
→
defined by
Λ (v′,s′) c¯Γb Λ (v,s) = η(w)u¯ (v′,s′)Γu (v,s), (1)
h c | | b i Λc Λb
where w = v v′. For the semileptonic Σ(∗) Σ(∗) decay we have two Isgur–Wise form
· b → c
factors, ζ (w) and ζ (w) [3–6].
1 2
Σ(∗)(v′,s′) c¯Γb Σ(∗)(v,s)
h c | | b i
= (ζ (w)g +ζ (w)v v′)u¯ν (v′,s′)Γuµ (v,s), (2)
1 µν 2 ν µ Σ(c∗) Σ(b∗)
where uν (v′,s′) is the Rarita–Schwinger spinor vector for a spin-3 particle and uµ (v,s) is
Σ∗b 2 Σb
defined by
(γµ +vµ)γ
uµ (v,s) = 5u (v,s) (3)
Σb √3 Σb
and similarly for uν (v′,s′). In Ref. [1], it has been shown that
Σ(c∗)
ζ (w) = (1+w)ζ = η(w). (4)
1 2
−
i.e., the same form factor describes both semileptonic transitions.
The largeN limit wasstudied inRef.[1]under theframework ofthechiralsolitonmodel,
c
whichisgenerallybelievedtobetherealizationoflargeN QCDinthebaryonsector, though
c
the equivalence has not yet been rigorously proved. Naturally it raises the question whether
the same universality of baryon Isgur–Wise form factor in the large N limit can be obtained
c
without reference to the chiral soliton model. In a recent paper [7] it was attempted to get
constraints on the Isgur–Wise form factors from unitarity in 1-pion loop renormalization of
the Λ Λ decay and weak decays with 1 or 2 pion emission. Since pion-baryon Yukawa
b c
→
couplings are of order N1/2, many individual weak decay graphs involving pion lines diverge
c
in the large N limit. To preserve unitarity, non-trivial cancelation must take place between
c
2
(∗)
graphs with intermediate Λ and Σ states, and hence giving non-trivial relations between
Q Q
(∗)
the Λ and Σ Isgur–Wise form factors. Ref. [7] found that the relations obtained by this
Q Q
method are ”consistent with, but not as powerful as” those obtained in Ref. [1] through the
chiral soliton model. In particular, no constraint can be placed on ζ (w) in Ref. [7]. It is not
2
surprising by noting that ζ (w) contributes only away from the point of zero recoil, where
2
the Isgur–Wise form factors vanish like exp( N3/2) in the large N limit [8], faster than
− c c
N−n for any finite positive n.
c
In this article, it will be attempted to reproduce relation (4) without using the chiral
soliton model. We will use the formalism of large N baryon developed in Ref. [9], which
c
depends on group theoretical considerations and is completely model independent. It is
found that relation (4) can indeed be reproduced and hence the result of Ref. [1] follows
solely from the large N limit without any additional assumptions.
c
We will first review the SU(4) spin-flavor symmetry for large N baryons developed in
c
Ref. [9]. The symmetry is defined by the commutation relations,
[Ji,Jj] = iǫijkJk, [Ia,Ib] = iǫabcIc, [Ia,Ji] = 0, (5a)
[Ji,Xjb] = iǫijkXkb, [Ia,Xjb] = iǫabcXjc, (5b)
0 0 0 0
[Xia,Xjb] = 0, (5c)
0 0
where Ia and Ji are the isospin and spin operators respectively, and Xia is the baryon axial
0
current matrix element in leading order of the 1/N expansion, defined by
c
hB′|q¯γiγ5τaq|Bi = Ncg(X0ia)B′B +higher order in 1/Nc. (6)
Eq. (5a) is just the usual commutation relations for SU(2) SU(2) , while Eq. (5b) states
I J
⊗
that fact that the axial current couplings are of spin 1 and isospin 1. Lastly, Eq. (5c) follows
from unitary constraint of pion-baryon scattering and is the starting point of the formalism
developed in Ref. [9].
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The induced representation of this spin-flavor SU(4) has been discussed in detail in
Ref. [9] and will not be repeated here. A vector under the induced representation is an
eigenvector of Xia (we will denote the eigenvalue by Xia as well) and also labeled by its
0 0
transformation properties under the little group. For the case of physical interest, the little
group is SU(2) Z , and the state would be denoted as Xia,K,k, where (K,k) and
× 2 | 0 ±i ±
are the representations under the little groups SU(2) and Z respectively. It can be shown
2
that K~ = I~ + J~ and Z = + and for bosonic and fermionic states respectively. The
2
−
prime example in Ref. [9] are the four nucleon states with (I,J) = (1, 1) and the sixteen
2 2
Delta states with (I,J) = (3, 3) which fall under the 20 representation under the spin-flavor
2 2
SU(4). These states can be constructed out of the basis Xia,0,0, , as K~ = I~+J~ = 0 and
| 0 −i
both the nucleon and the Delta are fermions.
To apply this formalism to heavy quark states, a straightforward application will be to
follow the treatment of hyperons in Ref. [9] and consider the induced representation with
K~ = I~+J~ = 1 and Z = . But it will be much more convenient and illuminating to study
2 2 −
the induced representation describing just the “brown mucks” of the heavy baryons without
the heavy quark. More exactly, instead of considering the spin-flavor SU(4) generated by
(Ia,Ji,Xia), one can consider instead that generated by (Ia,si,Xia), with s the spin of the
0 ℓ 0 ℓ
“brown muck”1. The SU(4) commutation relations stay unchanged, and the “brown mucks”
of Λ and Σ(∗), with (I,s ) = (0,0) and (1,1) respectively, form an SU(4) 10 representation.
Q Q ℓ
Now K~ = I~+~s = 0 and Z = + as the “brown mucks” are bosonic for odd N . Hence the
ℓ 2 c
heavy quark “brown muck” states can be constructed out of Xia,0,0,+ . Since we are not
| 0 i
going to concern about the transformation properties under the little group for the rest of
1The spin of light degrees of freedom s is well defined and conserved, a consequence of heavy
ℓ
quark symmetry. Recall that J~=~s +~s . The conservation of s follows from the conservation of
Q ℓ ℓ
the heavy quark spin s in the heavy quark limit. A similar treatment in the hyperon sector will
Q
be problematic as the spin of the strange quark is not conserved.
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our discussion, these state will be denoted simply as Xia below.
| 0 i
It is in place to discuss the properties of the states Xia . Since a change of the nor-
| 0 i
malization of Xia is equivalent to a redefinition of the axial current coupling constant g in
0
Eq. (6), we can, without loss of generality, impose the renormalization that
XiaXia = TrX2 = 3. (7)
0 0 0
The states with different Xia eigenvalues can be rotated or iso-rotated into each other.
0
U (g) Xia = Dij(g)Xja , (8a)
sℓ | 0 i | 0 i
U (h) Xia = Dab(h)Xib , (8b)
I | 0 i | 0 i
where U (g) is the unitary transformation corresponding to a finite spin rotation by the
sℓ
g SU(2) , Dij(g) is the usual rotation matrix in 3-dimensions, while U (h) and Dab(h)
∈ sℓ I
are the counterparts in isospace for h SU(2) . Moreover, since K~ = I~+~s = 0, for any
∈ I ℓ
state Xia and any g SU(2) , there exist a certain h SU(2) such that
| 0 i ∈ sℓ ∈ I
U (g) Xia = U (h) Xia , (9)
sℓ | 0 i I | 0 i
i.e., an isorotation is equivalent to a rotation. In particular, if we choose Xia = X
0 0 ≡
diag(1,1,1), Eq. (9) is satisfied by g = h.
This opens up the possibility of labeling the set of states Xia : TrX2 = 3 by SU(2)
{| 0 i 0 }
elements. With the definition
X = U (h) X , (10)
h I 0
| i | i
the set X : h SU(2) are orthogonal.
h I
{| i ∈ }
Xh′ Xh = δ(h′h−1), (11)
h | i
where δ(g) is a δ-function on the SU(2) group normalized so that dgδ(g) = 1. This
R
association of the states to SU(2) elements is crucial to our proof, as will be shown below.
I
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So far the heavy quark has not yet appeared in our discussion. In the heavy quark
limit, the heavy quark is just the source of a static color field in which the “brown mucks”
appear as eigenstates. During a b c transition, all the “brown muck” feels is the change
→
of the velocity of the color source. In this language, the Isgur–Wise form factors are just the
overlap of the initial and final “brown mucks” [10]. Now, let the state X discussed above
0
| i
be one moving with velocity v. Of all the normalized baryon “brown mucks” moving with
velocity v′, we will denote the one which overlaps maximally with X as X′ . Analogous
| 0i | 0i
to Eq. (10), we define
X′ = U (h) X′ . (12)
| hi I | 0i
Then it is trivial to prove that
X′ X δ(h′). (13)
h h′| 0i ∼
(To prove this, assume the contrary and there exist a certain non-trivial h′ for which
X′ X > 0. Then some normalized linear combination of X′ and X′ will have a
h h′| 0i | 0i | hi
′
larger overlap with X than X , violating the assumption.) It follows that
| 0i | 0i
X′ X = X′ U (h) X
h h′| hi h h′| I | 0i
= X′ X δ(h′h−1). (14)
h h′h−1| 0i ∼
We can repeat the procedure and define a X′ for all v′. The overlap as a function of
| 0i
w = v v′ is denoted by η(w) and will turns out to be the universal Isgur–Wise form factor.
·
′
X X = η(w). (15)
h 0| 0i
A rotation in isospace gives
X′ X = η(w). (16)
h h| hi
Combining Eqs. (14) and (16), we end up with
X′ X = η(w)δ(h′h−1). (17)
h h′| hi
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This is the central result of this article, the overlap of any state in X : h SU(2) with
h
{| i ∈ }
any state in X′ : h′ SU(2) can be expressed in terms of a single form factor η(w). All
{| h′i ∈ }
remains to be done is to express the result in terms of the eigenstates of Ia and si.
ℓ
Following the notation of Ref. [1], I,a;s ,m will denote a state with isospin I and
ℓ
| i
“brown muck” spin s , while a and m are the third components of the (iso)spin. Then
ℓ
(∗)
0,0;0,0 and 1,a;1,m are the “brown mucks” of Λ and Σ respectively. We can express
| i | i Q Q
0,0;0,0 in terms of the X basis.
h
| i
0,0;0,0 = dh X , (18)
| i Z | hi
and
0,0;0,0(v′) 0,0;0,0(v) = dhdh′ X′ X
h | i Z h h′| hi
= η(w) dhdh′δ(h′h−1)
Z
= η(w), (19)
justifying the notation of η(w). On the other hand,
1,a;1,m = dhDam(h) X , (20)
| i Z | hi
and
1,a′;1,m′(v′) 1,a;1,m(v) = dhdh′ X′ Da′m′†(h′)Dam(h) X
h | i Z h h′| | hi
= η(w) dhdh′δ(h′h−1)Da′m′†(h′)Dam(h)
Z
= η(w)δaa′δmm′, (21)
by the orthonormality of the rotational matrices Dam’s. This is exactly (the second equality
of) Eq. (16) of Ref. [1]. In terms of the full baryon states Σ∗ , the result is
| Qi
η(w)
Σ∗(v′,ǫ′,s′) c¯Γb Σ∗(v,ǫ,s) = [(1+w)g v v′]ǫ′∗νǫµu¯ Γu , (22)
h c | | b i 1+w µν − ν µ c b
where ǫ and ǫ′ are the polarization vectors and s and s′ are the heavy quark spins2. And
when compared to Eq. (2), the main result of Ref. [1] is recovered.
2As noted in Ref. [7], the “east coast” metric ( ,+,+,+) is used.
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ζ (w) = (1+w)ζ (w) = η(w). (23)
1 2
−
Since the proof above is quite complicated, let’s consider an simple but problematic
alternative proof which may help to bring out the essence of the correct proof above. Note
that
1,a;1,m = Xma 0,0;0,0 , (24)
| i 0 | i
and hence
1,a′;1,m′(v′) 1,a;1,m(v) = 0,0;0,0(v) X′m′a′†Xma 0,0;0,0(v)
h | i h | 0 0 | i
= 0,0;0,0(v′) 0,0;0,0(v) , (25)
h | i
if one sloppily identifies the operators X and X′. But such sloppiness is problematic. The
0 0
operators Xma carries a spatial index m, and the operator may get non-trivially transformed
0
when boosted from the v frame to the v′ frame. The isospin operator Ia, on the other
hand, is manifestly independent of Lorentz frames. That is why the X basis is used: for
h
| i
these states, a rotation in the real space is equivalent to a rotation in the isospace. So,
after identifying one state with velocity v with one with velocity v′ (through the criterion of
maximal overlap), one can establish a one-one correspondence between the two sets of states
just by isorotations, which are frame-independent operations. These one-one corresponded
states all have the same overlap, and that is the Isgur–Wise form factor.
It must be emphasized that this study does not question the computational correctness
of Ref. [7]. While the authors of Ref. [7] try to obtain constraints on the form factor through
unitarity, the present work depends mainly on the SU(4) symmetry structure of baryons in
the large N limit. Since this SU(4) spin-flavor symmetry is completely model independent,
c
and all existing approaches to large N baryons (chiral soliton, Hartree–Fock, etc.) exhibit
c
this symmetry, our result is truly model independent. In fact, this universality of baryon
Isgur–Wise form factor should hold in any formalism exhibiting this spin-flavor symmetry,
no matter it is large N motivated or not. One such example is the constituent quark model
c
developed in Ref. [11], which is not directly related to 1/N expansion but embodies the
c
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same symmetry. In fact, the universality of baryon Isgur–Wise form factor made its first
appearance in their work, which chronologically precedes Ref. [1].
In conclusion, it is found that the large N universality of baryon Isgur–Wise form factor
c
discussed in Ref. [1] is solely the consequence of the SU(4) spin-flavor symmetry and is
independent of any dynamical assumptions. This universality can be put to experimental
test in the future, when more data are available on η(w) (from Λ Λ decays) and
b c
→
the ζ(w)’s (from Ω Ω decays). The deviation from universality is a measure of the
b c
→
(in)applicability of the large N expansion for heavy baryons.
c
ACKNOWLEDGMENTS
I am grateful to Tung–Mow Yan for discussions. This work is supported in part by the
National Science Foundation.
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REFERENCES
[1] C.K. Chow, Phys. Rev. D51 1224 (1995).
[2] N. Isgur and M.B. Wise, in “B Decay,” ed. S. Stone.
[3] N. Isgur and M.B. Wise, Nucl. Phys. B348 276 (1991).
[4] H. Georgi, Nucl. Phys. B348 293 (1991).
[5] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B355 38 (1991).
[6] F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phy C51 321 (1991).
[7] D.E. Brahm and J. Walden, hep-ph 9511309 (1995).
[8] E. Jenkins, A.V. Manohar and M.B. Wise, Nucl. Phys. B396 38 (1993).
[9] R. Dashen, E. Jenkins and A.V. Manohar, Phys. Rev. D49 4713 (1994).
[10] C.K. Chow, Phys. Rev. D51 3587 (1995).
[11] J.G. Korner, M. Kramer and D. Pirjol, Prog. Part. Nucl. Phys. 33 787 (1994).
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