Table Of ContentHigher Resonance Contributions
to the Adler–Weisberger Sum Rule
in the Large N Limit
c
4
A. Wirzba1, M. Kirchbach1 and D.O. Riska 2
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n 1 Institut fu¨r Kernphysik, Technische Hochschule Darmstadt,
a
J
D–64289 Darmstadt, Germany
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1
2 Department of Physics, SF–00014 University of Helsinki, Finland
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9
2
1
1
3
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h
p
Abstract
-
p
e We determine the N –dependence of the resonance contributions to the Adler–
c
h
: Weisberger sumrulefortheinverse square1/g2 oftheaxialchargecouplingconstant
v A
i
X and show that in the large N limit the contributions of the Roper-like excitations
c
r
a scale as (1/N ). Consistency with the 1/N2 scaling of the 1/g2 term in the sum
O c c A
rule requires these contributions to cancel against each other.
hep-ph/9311299
1
1 Introduction
If the the axial charge commutator in the Adler-Weisberger relation for the axial
vector coupling constant is saturated by a discrete set of pion-nucleon resonances
(with spin j, parity P and isospin I) the sum rule takes the following form [1], [2]
1 g(±)2 (m2 m2 )2j−3 (m +ηm )2 [(j + 1)!]2
= 1+ f jI m2 jI − N jI N 2 . (1)
gA2 (jXP,I) I gπ2NN N (mNmjI)2j−1 2j−27 (2j +1)!
In this expression g stands for the pseudoscalar pion nucleon coupling constant,
πNN
(±)
g is the (dimensionless) coupling constant for the spin–j (isospin–I) resonance
jI
coupling to the pion-nucleon system in the relativistic Rarita–Schwinger representa-
tion [3], and m and m denote the the nucleon and resonance masses, respectively.
N jI
Subsequently, f denotes the pion weak decay coupling constant.
π
The quantities g , m and m scale with the number of color degrees of free-
A N jI
3
dom [4, 5] as (N ), whereas f (√N ) and g (N2). The mass differ-
c π c πNN c
O ∼ O ∼ O
ences m m are either of order (N0) or of order (1/N ). The former scaling
jI − N O c O c
behavior is the usual situation for resonances in meson-baryon scattering [5] and
applies e.g. for the Roper-like resonances as well as for the odd-parity states. The
latterscaling behaviorisspecialfore.g.the∆(1232)anditspartnersinthe(j+,I(j))
tower that belongs to the contracted SU(4)-symmetry, which applies for the baryon
states in strong coupling models [6, 7] and the 1/N expansion [8, 9]. The isospin
c
factor f takes the value 1 for the I = 1 resonances and the value 1 for the I = 3
I 2 −3 2
ones. (Note that resonances with I > 3 cannot couple to the pion-nucleon system).
2
The parameter η in (1) takes the values and corresponds to the structure in the
± ±
exponent of the parity P = ( 1)j±1/2 of the intermediate resonance. For example,
−
the sign “+” corresponds to the odd-parity resonances (with ( 1)j+12 = 1) as well
− −
as for the ∆–resonance and its Roper like excitation, whereas the “ ” sign has to
−
be used for the Nucleon-Roper, the Delta-odd-parity resonant excitations etc.
(±)
With the exception of the coupling constants g the scaling with N of all
jI c
the quantities that appear in eq.(1) is known. The left hand side of eq.(1) is of
order (1/N2). The right hand side, however, because of its first summand, has
O c
an apparent order (N0) term. The resonance contributions on the right hand side
O c
of eq.(1) therefore have to combine so as to (a) exactly cancel the first summand
2
(1) and to (b) give the required (1/N2) behavior. This requirement was used in
O c
(±)
ref.[10] to get the N scaling of the coupling constants g . We here show that the
c jI
quark model implies that the coupling constants for the Roper-like resonances scale
as N and not as N3/2 as inferred from the sum rule in ref.[10]. This N dependence
c c c
implies that these resonances give contributions of order 1/N to the sum rule, and
c
thus consistency with the r.h.s. requires these contributions to cancel against each
other. That cancellation is made possible by the opposite signs of the contributions
from the Roper and ∆-Roper resonances in the sum rule.
2 N scaling of the coupling constants
c
In the relativistic Rarita-Schwinger representation πN-resonance vertices are [3]:
hN(p)|χβ|j(p+q)i = mgj(j+I−)12u¯(p)uµ1···µj−12(p+q) q′µ1 ···q′µj−21 ξ+21 τβξξβ21 , (2)
N 3
2
hN(p)|χβ|j(p+q)i = migjj(−−I)12u¯(p)γ5uµ1···µj−12(p+q) q′µ1 ···q′µj−21 ξ+21 τβξξβ21 ,(3)
N 3
2
where u(p) and uµ1···µj−21(p+q) are in turn the spinor and and totally symmetric
tensor-spinor wave functions that describe the proton and the spin–j resonance, ξ ,
1
2
ξβ arethewave functions of theisospin–1 or 3 resonances, respectively, andξ+ is the
3 2 2 1
2 2
isospin–1 wave function of the proton. The two equations correspond to resonances
2
of parity ( 1)j+21 and ( )j−12, respectively. The four-momentum q′ is defined as
− − µ
q(p+q)
′
q = q (p+q) . (4)
µ µ − m2 µ
jI
Compare now the N behavior of the non-relativistic reduction (in the baryon
c
degrees of freedom) of eqs.(2) and (3) with the results for the non-relativistic quark
model in the large N limit. In this limit the non-relativistic reduction is well
c
justified, since the baryon masses scale as (N ) and can be made arbitrarily heavy.
c
O
The naive quark model can also be relied upon in this limit as we only need the
leading N behavior of the resonance couplings, which is model-independent [8]. In
c
the non-relativistic limit the spin-isospin-structure of the one-body vertex operators
3
sandwiched between the proton state N and a (by the transition allowed) baryon
| i
state B can take only one of the following four forms: a scalar-isoscalar sum
j
| i
G Nc (√Nc)
N 1 B O , (5)
h |fπ νX=1 ν| ji ∼ (N0)
O c
a pseudoscalar-isovector sum
G Nc (√Nc)
N ~σ ~qτβ B O , (6)
h |fπ νX=1 ν · ν| ji ∼ (N0)
O c
a scalar-isovector sum
G Nc (1/√Nc)
N τβ B O , (7)
h |fπ νX=1 ν| ji ∼ (1/N )
c
O
or a pseudoscalar-isoscalar sum
G Nc (1/√Nc)
N ~σ ~q B O . (8)
h |fπ νX=1 ν · | ji ∼ (1/N )
c
O
Here the index ν counts the N different quarks of the proton and the spin–j reso-
c
nance. In (8) G is a pion-quark coupling strength which scales as (N0). The pion
O c
decay constant f appears through the normalization of the pion field in the vertex
π
operator. The spatial vector ~q is the pion three-momentum in the resonance rest
frame which also scales as (N0) for fixed kinematics. In the equations above the
O c
upper scaling behavior on the r.h.s. applies to the case when B is the nucleon, or
j
a state in which all quarks are in their spatial ground state (e.g. the lowest ∆ res-
onance). The lower scaling behavior applies for such baryon states with one or two
quarks in an excited spatial state (e.g. the Nucleon- or ∆-Roper ). In that case the
symmetrization of the color-singlet baryon state introduces an additional normaliza-
tion factor 1/√N [11], which leads to a similar extra factor in the N dependence
c c
of the matrix element. This is consistent with the fact that – because of unitarity –
the πN πN scattering amplitude can at most scale as (N0): The direct Born
→ O c
diagram with a nucleon as intermediate state scales by itself as (N ) since both
c
O
vertices scale as (√N ). In ref.[8] it was shown that this leading N scaling cancels
c c
O
4
when the crossed Born diagram and the corresponding direct and crossed diagrams
with the (in the large N limit degenerate SU(4) partner) ∆ are included. However,
c
in case the intermediate state in the πN πN scattering is a (with the nucleon
→
non-degenerate) baryon-resonance, the direct Born diagram cannot be cancelled any
longer in this way. Thus, in order to be consistent with unitarity, the vertex itself
can at most scale as (N0) in agreement with the scaling behavior derived from the
O c
normalization of the excited baryon state [11].
Note that in the vertex sum (6) the single quark ~σ ~qτβ contributions add
ν · ν
up coherently and lead to the the same N scaling as in eq.(5), whereas the other
c
two vertex sums (7) and (8) are suppressed by 1/N since the quark contributions
c
only add up destructively. In fact only the vertex sums (6) and (7) contribute
here because of the isovector nature of the pion coupling. The N behavior of the
c
pion-proton vertices in the naive quark model is thus determined.
The non-relativistic reduction of the Rarita-Schwinger-type vertex matrix ele-
ments (2) and (3) is standard. Using the totally symmetric nature of the j-1 tensor
2
spinors one can identify – in the non-relativistic limit and in spin-isospin space – the
matrix elements (2) with the pseudoscalar-isovector structure, eq.(6), of the naive
quark model, if j-1 is odd, and with the scalar-isovector one, eq.(7), if j-1 is even.
2 2
For the vertex matrix element (3) the fact has to be taken into account that the γ
5
Dirac matrix links one upper and one lower component of the proton and the spin–j
resonance and therefore induces an extra factor ~σ (p~ + ~q)/2m ~σ p~/2m =
jI N
· − ·
~σ ~q/2m + (1/N2) #1. Therefore the vertex matrix element (3) leads to the
· N O c
pseudoscalar-isovector structure, eq.(6), for j-1 even and to the scalar-isovector one,
2
eq.(7), for j-1 odd. Thus, the N behavior of the g(±)’s can now be predicted.
2 c jI
The scaling of the Roper resonance coupling g(−) #2 follows via eqs.(3) and (6)
11
22
from
(−)
g
11 G
22 N (1) (9)
c
2m ∼ f √N × ∼ O
N π c
(+)
and is thus (N ). The scaling of the odd-parity-nucleon-like coupling g results
c 11
O
22
#1Note that in the rest frame of the resonance ~q =~q′.
#2The “ ” sign in the notation of ref.[3] is reflecting the “ ” sign in j 1 and should not be
− − − 2
mixed up with the parity of the Roper state which of course is positive.
5
from
G
(+)
g (1/N ) (10)
1212 ∼ fπ√Nc ∼ O c
(see eqs.(2) and (7)) and is (1/N ). The scaling behavior of the ∆ (ground state)
c
O
(+)
coupling g can be deduced from
33
22
(+)
g
33 G
22 N (√N ) (11)
c c
m ∼ f ∼ O
N π
3
as N2, whereas the coupling g(+) of the Roper-like ∆ scales – because of the 1/√N
c 3′3 c
2 2
normalization – only as N :
c
(+)
g
3′3 G
2 2 N (N0) (12)
m ∼ f √N × c ∼ O c
N π c
(−)
(see eqs.(2) and (6)). The coupling constant g of the odd-parity ∆–resonance also
33
22
behaves as N (see eqs.(3) and (7)), since
c
(−)
g
33 G
22 (1/N ) . (13)
m2 ∼ f √N ∼ O c
N π c
In general, comparing the non-relativistic reduction of eqs.(2) or (3) with the N -
c
counting of eqs.(6) or (7) one can derive the following relations for the coupling
constants of the (j±,I) states with I=1 or 3: #3
2 2
gj(+I) = O(Ncj−12) for P = (−1)j+21 = +
gj(+I) = O(Ncj−32) for P = (−1)j+21 = − (14)
gj(−I) = O(Ncj+12) for P = (−1)j−21 = +
gj(−I) = O(Ncj−12) for P = (−1)j−21 = − ,
(see the 5th row in Table 1).
3 N scaling of the resonance contributions to the
c
sum rule
The N scaling of all the quantities that enter on the right hand side of the Adler-
c
Weisberger sum rule (1) is now determined. In the 7th row of Table 1 the contri-
3
#3With the exception of the lowest ∆ state for which the coupling constant behave as (Nc2)
O
and not as (Nc).
O
6
butions of the single resonances with j 7/2 are listed. Using the results of Table
≤
1 one finds that (a) the ∆(1232) contributes to leading order (N0) to the r.h.s. of
O c
the sum rule, (b) the N-Roper, the ∆-Roper, the (3+, 1), the (5+, 1), the (1+, 3)
2 2 2 2 2 2
etc. contribute to next-to-leading order, (1/N ), and (c) the contribution of e.g.
c
O
−
the odd-parity (1 , 1) nucleon excitation is suppressed by (1/N3).
2 2 O c
Since the contributions of the the radial excitations of the nucleon- and the ∆-
resonancescontributetonext-to-leadingorder, (1/N ),theyhavetocancelinorder
c
O
to keep consistency with the scaling of the l.h.s. of the sum rule. The contributions
from these states therefore have to be considered with some care when the Adler-
Weisberger sum-rule is applied to large N models as the Skyrme-model (or its
c
extensions) or large N quark models. The Roper-like states, on the one side, and
c
the ∆(1232) resonance, on the other side, contribute for rather different reasons
significantly to the r.h.s. of the sum rule in eq.(1). In the large N limit the latter is
c
degenerate with the nucleon (the mass splitting goes as (1/N )) and has a width
c
O
that vanishes with 1/N2. The Ropers, on the other hand, do not become degenerate
c
with the nucleon in the large N limit as the splitting is (N0), but their widths are
c O c
also of order (N0) and therefore large enough to leave strength left at the nucleon
O c
pole. The masses of the other resonances are order (N0) above the nucleon pole
O c
and have widths that are only of order (1/N2). Therefore they have no strength
O c
at the nucleon pole in the large N limit and may be neglected.
c
The widths of the spin–j isospin–I resonances are given by the expression [1, 3]:
Γ(±) = 3 f gj(±I)2 2j−21[(j − 12)!]2EN +ηmN(p~2)j . (15)
jI | I|4πm2j−1 (2j)! m
N jI
Here ~p and E are the three-momentum and the energy of the nucleon in the j-
N
resonance rest frame, respectively. Forfixed kinematics E scales as (N ), whereas
N c
O
p~ normally scales as (N0). Note, however, that the nucleon kinetic energy E
| | O c N −
m scales as (1/N ), which is obvious in the non-relativistic limit. Furthermore
N c
O
for the ∆(1232) (and its partners in the (j+,j) tower) the three-momentum ~p does
not scale as (N0), but rather as 1/N . From the expression (15), it then follows
O c c
that the widths of all the Roper like excitations scale as (N0), whereas those of
O c
the odd-parity resonances scale as (1/N2) (see Table 1). Finally, as pointed out
O c
above, the width of the ∆(1232) falls with N at least as (1/N2).
c O c
7
4 Conclusions
The contribution C of the ∆ resonance to the right hand side of the Adler–
∆
Weisberger sum rule (1) is:
(+)2
2 g33 (m +m )2
C = 22 ∆ N . (16)
∆ −9g2 m2
πNN ∆
In the large N limit when m m one finds that C 1 [2, 12] in accordance
c ∆ N ∆
→ → −
with the prediction of the contracted SU(4) spin–flavor symmetry. The contribution
C alone is therefore sufficient to cancel the terms of order N0 in the sum rule eq.(1)
∆ c
exactly. However, as shown in section 3, the contributions from the Roper–like, as
well as the D negative parity resonances in the large N limit scale as 1/N and
13 c c
consistency therefore requires these to cancel against each other #4. This behavior
of the Roper-like states is in line with the unitarity limit, (N0), of the total cross
O c
sections σπ±p in π±-proton scattering which appear on the r.h.s. of the integral form
tot
of the Adler-Weisberger relation
1 = 1+ 2m2N ∞ dν ν2 m2 σπ−p(ν) σπ+p(ν) , (17)
gA2 πgπ2NN Zmπ ν q − π(cid:16) tot − tot (cid:17)
(ν = p q/m ). Note that the I = 1 resonances can only show up in the π−p
· N 2
scattering. We have shown – using unitarity – that the corresponding resonance
cross sections can at most scale as (N0) and that in fact for the Roper-like res-
O c
onances the upper bound is saturated. Taking into account the prefactor of the
integral one easily finds that the total contribution of the Roper-like states scales
as (1/N ) in agreement with our derivation presented above. In the case of the
c
O
I = 3 resonances both σπ±p cross sections contribute. The net result of the I = 3
2 tot 2
Roper-like resonances is still of order (1/N ) and negative in order to remove the
c
O
positive (1/N ) contribution of their corresponding I = 1 partners. For the non-
O c 2
±
resonant π -proton scattering, however, it can be shown [15] that the cross sections
−
cancel to order (1/N ) since the π meson is counting the (N + 1)/2 u quarks
c c
O
of the proton, whereas the π+ is counting the respective N /2 d quarks. The total
c
contribution of the non-resonant π-proton scattering to the Adler-Weisberger inte-
gral (17) is therefore of order (1/N2) [15]. The contributions of the S and D
O c 11 15
#4The importance of the Nucleon-Roper resonance for the saturation of the chiral axial charge
commutator has already been emphasized in [14].
8
odd-parity resonances and the other resonances whose decay width are of the order
(1/N2) appear suppressed by the order of (1/N3) as compared to the leading
O c O c
and next-to-leading effects discussed above. The predicted behavior is qualitatively
in line with the one extracted from the data on the πN decay widths of the j < 7
2
resonances below 2.2 GeV.
Acknowledgements
We thank Dr. W. Broniowski for pointing out an error in the first version of this
note. A.W. would like to thank Ismail Zahed for discussions.
9
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