Baryon masses in large Nc chiral perturbation theory

36 Pages·0.3 MB·English

by  Yongseok Oh

TUM/T39-98-10
SNUTP-98-037
hep-ph/9901354
Baryon masses in large N chiral perturbation theory
c ∗
Yongseok Oh†
Physik-Department, Technische Universita¨t Mu¨nchen, D-85747 Garching, Germany
and Research Institute for Basic Sciences and Department of Physics, Seoul National University,
9
Seoul 151-742, Korea
9
9
1

n Wolfram Weise
a
J Physik-Department, Technische Universita¨t Mu¨nchen, D-85747 Garching, Germany

0
2

1 Abstract
v
4
5
3
1
We analyse the baryon mass spectrum in a framework which combines
0
9 the1/Nc expansionwithchiralperturbationtheory. Mesonloopcontributions
9
involvingthefullSU(3)octetofpseudoscalarGoldstonebosonsareevaluated,
/
h and the influence of explicit chiral and flavor symmetry breaking by non-zero
p
and unequal quark masses is investigated. We also discuss sigma terms and
-
p
the strangeness contribution to the nucleon mass.
e
h
:
v
i
X
r
a
Typeset using REVTEX
∗Work supported in part by BMBF
†Alexander von Humboldt fellow, E-mail address: [email protected]
1
I. INTRODUCTION
The large N limit, where N is the number of colors, is a useful device to understand
c c
many systematic features of baryon properties [1,2], such as the 1/N scaling of various
c
physical quantities. In a series of papers, Dashen and Manohar [3,4], and Jenkins [5] have
discussed the 1/N structure of baryon properties, and the framework for combining chiral
c
symmetry with the large N aspects of QCD has been developed by many authors [6–12]. In
c
Refs. [7,10], the baryon octet and decuplet mass spectra were discussed in this framework
and the baryon mass relations were derived. However, although those works successfully
reproduce mass relations at tree level, they do not compute all possible terms allowed by
the chiral and large N expansions.
c
The baryon mass spectrum was re-examined in conventional baryon chiral perturbation
theory by Borasoy and Meissner [13,14]. To compute the baryon masses to order m2, where
q
m is the quark mass, the decuplet degrees of freedom are integrated out to give counter
q
terms, and some low-energy constants are determined from resonance saturation. However,
when we work with the 1/N expansion, the octet and decuplet states are degenerate at the
c
leading order, and the decuplet fields must be treated explicitly.
In this paper, we re-examine the baryon masses in chiral perturbation theory taking into
account the 1/N counting based on the techniques developed in the literature, e.g., in Refs.
c
[9–11]. This enables us to investigate the 1/N structure of the baryon properties and the
c
meson-baryon interactions in a systematic way. The baryon axial current matrix elements
and the strangeness contribution to the nucleon mass are computed as well. Some of these
topics were discussed in the literature [7,10,11] focusing on the leading order terms in 1/N
c
expansion (up to one loop corrections). In this paper, we perform the calculations to the
next orders and discuss a difficulty which arises in computing the one loop corrections in a
way which is consistent with the 1/N expansion. This paper is organized as follows. In the
c
next Section, we briefly discuss the formalism of this approach. In Section III, we compute
the baryon axial current up to one-loop corrections. The baryon mass formulas are given
in Section IV. The one-loop corrections to the baryon masses are calculated in Section V.
We discuss the strangeness contribution to the nucleon mass and the sigma term in Section
VI and then finish with a summary and conclusion in Section VII. Explicit expressions of
baryon wave functions and some detailed formulas are given in Appendices.
II. FORMALISM
We start with a brief review of the construction of baryon states in the large N limit,
c
referring to Refs. [9,10,15] for further details. The baryon states with N quarks can be
c
written as follows:
B a1α1...aNcαNcεA1...ANcq† ...q† 0), (2.1)
| i ≡ B a1α1A1 aNcαNcANc|
in particle number space, where a are the flavor indices, α the spin indices and A the
i i i
color indices. The quark creation and annihilation operators q† and q satisfy the usual anti-
commutator relations for fermions. The symmetric tensor is characteristic of each given
B
2
baryon wave function. Since the baryons are in color-singlet states, however, it is more
convenient to work with
B) a1α1...aNcαNcα† ...α† 0), (2.2)
| ≡ B a1α1 aNcαNc|
by dropping the explicit color indices, where the operators α† and α are bosonic operators
satisfying the usual commutator relations. For short-hand notation, we label the quark
operators as
α α , α α , α α ,
1 u↑ 2 u↓ 3 d↑
≡ ≡ ≡
α α , α α , α α , (2.3)
4 d↓ 5 s↑ 6 s↓
≡ ≡ ≡
so that α† creates u-quark with spin-up, and so forth.
1
There is an ambiguity when we extrapolate the physical baryon states to large N . As
c
in the literature, we keep the spin, isospin and strangeness of baryons as O(1) in the large
N limit. For example, the nucleon state in large N limit has spin 1/2, isospin 1/2 and no
c c
strangeness. This can be done by acting with spin-flavor singlet operators on the physical
baryon states. For example, the proton spin-up state can be written as
p,+1) = C α†(A†)n 0), (2.4)
| 2 N 1 s |
where n = (N 1)/2 and C is the normalization constant. The spin-isospin singlet
c N
−
operator A† is defined as
s
A† = α†α† α†α†. (2.5)
s 1 4 − 2 3
One can easily verify that this state reduces to the usual quark model state in the real world
with N = 3. The complete list of the baryon octet and decuplet states can be found in
c
Appendix A.
Next we define a one-body operator X in spin-flavor space as
{ }
X = α† Xαβα , (2.6)
{ } aα ab bβ
so that its expectation value on baryon states is at most of O(N ). In a similar way, one
c
can define 2-body operators X Y and 3-body operators X Y Z , and so on. Then,
{ }{ } { }{ }{ }
it is found that the coefficient of an r-body operator is at most O(1/Nr−ℓ−1), where ℓ is
c
the number of inner quark loops [9,16]. This enables us to treat the coupling constants as
O(1) quantities in the large N expansion by writing the N -dependence of the operators
c c
explicitly.
By direct evaluation, one can see the well-known commutator relation,
[ X , Y ] = [X,Y] . (2.7)
{ } { } { }
Note that the left-hand side is naively a two body operator whose expectation values can
be of O(N2), whereas the right-hand side is a one-body operator whose expectation values
c
are of O(N ) at most. This means that the order of an operator in 1/N counting reduces
c c
when we have a commutator structure as in Eq. (2.7). This plays an important role in the
large N analyses of the baryon properties.
c
We will discuss the explicit forms of some operators which appear in the calculation of
baryon axial currents and masses in the next Sections.
3
III. BARYON AXIAL CURRENTS
A. Tree level
Our starting point is the chiral meson-baryon effective Lagrangian. Baryon matrix ele-
ments of this Lagrangian involve the meson-baryon interaction in the following form:
h
= g(B Aµσ B)+ (B Aµ σ B)+..., (3.1)
eff µ µ
hL i |{ }| N |{ }{ }|
c
where σµ is the baryon spin matrix,1 and the dots denote higher order terms. The axial field
A is defined as
µ
i
A = (ξ∂ ξ† ξ†∂ ξ), (3.2)
µ µ µ
2 −
where ξ = exp(iΠ/f) with the meson decay constant f. The SU(3) matrix field Π represents
the octet of pseudoscalar Goldstone bosons. It is defined as
1
Π = λ π , (3.3)
a a
2
withtheusualGell-Mannmatricesλ (a = 1,...,8). InEq. (3.1),theN factorsofoperators
a c
are given explicitly, and the coupling constants g and h are of O(1) in the 1/N counting.
c
Then the baryon axial current Ja reads
5,µ
g h
Ja = T˜aσ + T˜a σ , (3.4)
5,µ 2 { µ} 2N { }{ µ}
c
from the Lagrangian (3.1) with
1
T˜a = (ξλaξ† +ξ†λaξ). (3.5)
2
This gives its matrix elements as
(B′|J5a,µ|B) = αBa′Bu¯B′(σµ)uB. (3.6)
where u is the Dirac spinor of the baryon and
B
h
αa = g(B′ 1λaσ3 B)+ (B′ 1λa σ3 B), (3.7)
B′B |{2 }| N |{2 }{ }|
c
at the tree level.
By using the wave functions given in Appendix A, we can compute the baryon axial
current straightforwardly. A naive investigation of each term gives that, despite the 1/N
c
1In the baryon rest frame, σµ = (0,σ) with the usual Pauli matrices σi.
4
factor, the h term contribution is expected to be of the same order as that coming from the
g term. This is because the h term contains a 2-body operator whose expectation value can
be O(N2), thus the leading order of the h term contribution can be of N . However, close
c c
inspection shows that the g term contribution dominates, because the h term contains the
operator σ and our baryon wave functions satisfy σ O(1).
µ µ
{ } { } ∼
The explicit forms of the relevant operators are
1λ1+i2σ3 = α†α α†α , 1λ4+i5σ3 = α†α α†α ,
{2 } 1 3 − 2 4 {2 } 1 5 − 2 6
1λ1+i2 σ3 = α†α +α†α , 1λ4+i5 σ3 = α†α +α†α , (3.8)
{2 }{ } 1 3 2 4 {2 }{ } 1 5 2 6
so that we obtain
g h
α1+i2 = (N +2)+ ,
pn 3 c N
c
g
α1+i2 = α1+i2 = (N 1)(N +3),
ΛΣ− Σ+Λ 3√2 c − c
q
N g h
α1+i2 = c ,
Ξ0Ξ− 9 − N
c
g √2h
α1+i2 = α1+i2 = (N +1)+ , (3.9)
Σ0Σ− − Σ+Σ0 3√2 c N
c
and
g h
α4+i5 = N +3 N +3,
pΛ −2 c − 2N c
q cq
√N g √3h
α4+i5 = c + N 1,
ΛΞ− 2√3 2N c −
c q
1 g h
α4+i5 = α4+i5 = N 1 N 1,
pΣ0 √2 nΣ− 6 c − − 2N c −
q cq
1 5g h
α4+i5 = α4+i5 = N +3+ N +3. (3.10)
Σ0Ξ− √2 Σ+Ξ0 6√3 c 2√3N c
q cq
These results show that the h term contributions are suppressed as compared to those of
the g terms as we discussed above. We can also find that the leading order of α1+i2 is
BB′
O(N ), whereas α4+i5, which changes the baryon strangeness, is O(√N ). This shows that
c BB′ c
the strangeness-changing (i.e., ∆S = 0) baryon axial currents are suppressed as compared
6
to the strangeness-conserving (i.e., ∆S = 0) baryon axial currents by O(√N ). This can
c
be understood from Eq. (3.8) by noting that the number of u,d quarks in the baryon wave
functions is O(N ) whereas that of s quark, i.e., strangeness, is O(N0). For example, in the
c c
case of α1+i2, acting with α (or α ) on the baryon state gives the factor N , and the inner
pn 3 4 c
product of initial and final baryon wave functions with the proper normalization constants
givesO(1),sothatα1+i2 isO(N ). However, forα4+i5, theactionwithα (orα )givesO(N0)
pn c pΛ 5 6 c
because our baryon wave functions have the strangeness of O(N0). Since the normalization
c
constants of nucleon and Λ are O(1/N ) and O(1/√N ), respectively, we have an additional
c c
factor √N in the calculation of α4+i5, which implies that the order of α4+i5 is O(√N ).
c pΛ pΛ c
5
Since the contributions of the h term are suppressed as compared to those of the g term
by O(1/N2) for the ∆S = 0 axial currents and by O(1/N ) for the ∆S = 1 axial currents,
c c
we can neglect the h term up to next to leading order. At this order, when we fix N = 3,
c
we can recover the quark model relation [10],
2
D = g, F = g, (3.11)
3
by comparing with the results of the baryon chiral perturbation theory [17,18] in addition
to the quark model predictions
= 2g, = 3g, (3.12)
C − H −
for the octet-decuplet-meson and decuplet-decuplet-meson coupling constants, and ,
C H
defined as in Ref. [19]. When we go further in the 1/N expansion, we must include the h
c
term, and we get the modified relations,
2g +h
D = g, F = , (3.13)
3
as found in Ref. [10].
We also compute α8 by using
B′B
1
1λ8σ3 = (N N +N N 2N +2N ),
{2 } 2√3 1 − 2 3 − 4 − 5 6
1
1λ8 σ3 = (N +N +N +N 2N 2N ), (3.14)
{2 }{ } 2√3 1 2 3 4 − 5 − 6
where we have introduced N = α†α . This leads to
i i i
g h
α8 = + ,
pp 2√3 2√3
g h 3
α8 = + 1 ,
ΛΛ −√3 2√3 − N
(cid:18) c(cid:19)
g h 3
α8 = + 1 ,
ΣΣ √3 2√3 − N
(cid:18) c(cid:19)
√3g h 6
α8 = + 1 . (3.15)
ΞΞ − 2 2√3 − N
(cid:18) c(cid:19)
From these results we find that the leading order of α8 is O(N0) and that the h term
B′B c
provides a leading contribution together with the g term. This is because the expectation
values of 1λ8 σ3 are O(N ) whereas those of 1λ8σ3 are O(1). So we conclude that in
{2 }{ } c {2 }
order to get a consistent result on α8 , one should consider n-body (n 3) operators in
B′B ≥
general unless their coupling constants are suppressed. From the fitted values of D and F,
one can estimate g = 0.61 0.8 together with h = 0.02 0.1, which shows that h is
∼ − ∼ −
indeed small, less than 15% of g, but with opposite sign [10]. Therefore, one should keep in
mind the contributions from n-body (n 3) operators in the calculation of α8 . We have
≥ B′B
a similar situation when we compute the η-meson loop corrections to the baryon masses in
Section V.
6
B. One-Loop Corrections
The one-loop corrections to the baryon axial current in large N chiral perturbation the-
c
ory as shown in Fig. 1 have been discussed in Refs. [3,4,6]. Naively, these loop corrections as
they stand are not consistent with the 1/N expansion. From the meson–baryon interactions
c
(3.1), each vertex is related to an operator Xia defined as
h
Xia = g 1λaσi + 1λa σi , (3.16)
{2 } N {2 }{ }
c
with spin index i and flavor index a. This shows that the meson-baryon coupling is of order
N . Because of the presence of 1/f which scales as 1/√N , each vertex has a factor √N .
c c c
Then from Fig. 1(a), it is evident that the one-loop correction is O(Nc2) when αB′B is of
order N . Thus the loop correction dominates the tree level. However, we have to consider
c
the wave function renormalization terms (Fig. 2) in the loop calculation. When combined
with Fig. 1, this gives the commutator structure to the baryon axial current operators,
which implies that the actual order of the one-loop correction is O(Nc0) when αB′B is O(Nc).
This suppression was proved up to 2-loop order in Ref. [20] which concludes that the 2-loop
corrections are suppressed by 1/N2 as compared to the tree level values.
c
Explicitly, the one-loop correction to the baryon axial current from Fig. 1(a) is given by
the following expression:
i d4k 1 k k
Via = − µ ν (B′ XµbXiaXνb′ B), (3.17)
B′B f2 (2π)4(k v)2m2 k2 | |
Z · bb′ −
where mbb′(= mπ,mK,mη) is the meson mass in the loop. When combined with the wave
function renormalization factor Z from Fig. 2,
B
i d4k 1 k k
Z = 1+ µ ν (B XµbXνb′ B), (3.18)
B f2 (2π)4(k v)2m2 k2 | |
Z · bb′ −
this gives the renormalized baryon axial current operator in the form of
1
Xia + Ibb′[Xµb,[Xia,Xνb′]], (3.19)
2f2 µν
where
d4k 1 k k
Iab = i µ ν . (3.20)
µν − (2π)4(k v)2m2 k2
Z · ab −
Finally, we get the one-loop correction to the baryon axial current operator as
m2 m2 2 m2 m2
δXia = bb′ ln bb′ + [Xjb,[Xia,Xjb′]] bb′ ln bb′ i,bb′, (3.21a)
32π2f2 λ2 3! − 32π2f2 λ2 O
by evaluating the loopintegral using dimensional regularization with the regularization scale
λ (see, e.g., Ref. [21].), where
7
h
bb′ = g [1λb,[1λb′, 1λa]]σ + [1λb,[1λb′, 1λa]] σ , (3.21b)
Oµ { 2 2 2 µ} N { 2 2 2 }{ µ}
c
which comes from Fig. 1(b). So the one-loop correction to the baryon axial current matrix
elements are obtained as
m2 m2 m2
δαa = βa,Π Π ln Π +β˜a,Π Π , (3.22)
B′B B′B16π2f2 λ2 B′B24π2f2
where Π stands for π, K and η mesons.
The explicit results of βa,Π and β˜a,Π with g terms are given in Appendix B. From these
B′B B′B
results, we can see that the one-loop corrections are O(1/N ) at most since f2 scales like N .
c c
Furthermore, the corrections from Fig. 1(b) are of the same order as those of Fig. 1(a).
IV. BARYON MASSES AT TREE LEVEL
In this Section we investigate the baryon masses at tree level. To estimate the baryon
masses simultaneously in the 1/N expansion and the chiral expansion, we must specify the
c
relation between 1/N and the pseudo-Goldstone boson mass m . In this paper, we use
c Π
m δM O(1) where δM is the octet-decuplet mass difference. This gives m O(ε)
Π Π
∼ ∼
and 1/N O(ε), where ε stands for a small expansion parameter.2 A priori, there is
c
∼
no constraint on the relationship between m and N . In fact, the authors of Ref. [11]
Π c
used m2δM O(1). However as we shall see below, the leading order correction to the
Π ∼
degenerate baryon mass in the large N limit is proportional to m2N and we count it as
c Π c
O(ε). This is consistent, given that m O(ε2) in accordance with the chiral expansion,
Π
∼
and N O(ε−1). We will compare our results with those of Ref. [11] before calculating the
c
∼
one-loop corrections.
The matrix elements of the effective Lagrangian which contribute to the baryon mass
can be written as
˜(i)
= (B B), (4.1)
hLBi |Leff|
i
X
where ˜(i) represents that part of the Lagrangian which can give a contribution of O(εi).
Leff
Explicitly, these terms are
˜(−1) = a 1 , (4.2a)
Leff − 0{ }
a
˜(1) = 1 σj σj b m , (4.2b)
Leff −N { }{ }− 1{ }
c
α
˜(2) = 1 tr(m) 1 , (4.2c)
Leff −N { }
c
2This is consistent with the expansion of Ref. [23], where the ∆-nucleon mass difference is treated
as small parameter with the pion mass.
8
b c
˜(3) = 2 mσj σj c m2 2 m m , (4.2d)
Leff −N { }{ }− 1{ }− N { }{ }
c c
α β
˜(4) = 2 tr(m) σj σj 1 tr(m2) 1 , (4.2e)
Leff −N2 { }{ }− N { }
c c
c c c
˜(5) = 3 m2σj σj 4 mσj mσj 5 m mσj σj
Leff −N { }{ }− N { }{ }− N { }{ }{ }
c c c
d d
d m3 2 m2 m 3 m m m , (4.2f)
− 1{ }− N { }{ }− N2{ }{ }{ }
c c
up to O(ε5), where
m = B (ξ†m ξ† +ξm ξ). (4.3)
0 q q
We make use of the standard relations between B and squared pion and kaon masses,
0
m2 = 2B mˆ and m2 = B (mˆ + m ), where mˆ is the average mass of u and d quarks and
π 0 K 0 s
m the s-quark mass. The quark mass matrix m is given by
s q
m = mˆ +m , (4.4)
q s
U S
where
= diag(1,1,0), = diag(0,0,1). (4.5)
U S
Throughout this work, we assume SU(2) isospin symmetry with m = m . Then, up to
u d
O(ε5), there are 15 low energy constants that should be determined from experiments.
However, one can find that 6 terms give identical contributions to all baryon masses so
that 9 parameters remain which determine all baryon mass differences. In the following, we
discuss the baryon masses at each order of ε.
From the Lagrangian (4.2a), all octet and decuplet baryon masses are degenerate at
leading order, which gives the baryon mass operator,
(0)
M = a N , (4.6)
B 0 c
where the parameter a sets the scale as a “mass per color degree of freedom”.
0
To the next order, the correction to the mass formula reads
a
δM(1) = 1 σj σj +2B mˆN b . (4.7)
B N { }{ } 0 c 1
c
The a term gives the splitting between octet and decuplet while all states within the
1
octet and decuplet are still degenerate. Although the original form of Eq. (4.2b) includes
the operator , the resulting baryon masses do not depend on strangeness since the
{S}
expectation values of for our baryon states are of O(1) so that its contribution appears
{S}
at the next higher order. Thus, at O(ε1), we get
3
(1)
δM = a +2B mˆN b ,
8 N 1 0 c 1
c
15
(1)
δM = a +2B mˆN b , (4.8)
10 N 1 0 c 1
c
9
where M and M denote the baryon octet and decuplet masses, respectively.
8 10
At O(ε2) there are two contributions. One is from ˜2 of (4.2c) and the other is from
Leff
the remaining part of the b term of ˜1 :
1 Leff
(2)
δM = 2B (m mˆ)b +2B (m +2mˆ)α . (4.9)
B 0 s − 1{S} 0 s 1
It is clear that the α term gives the same mass shift to all baryons. The dependence of the
1
b term on strangeness results from the SU(3) flavor symmetry breaking and vanishes in the
1
flavor SU(3) limit. Therefore, up to this order, the baryon mass depends on total spin and
strangeness, but the Λ and the Σ are still degenerate.
The mass corrections at O(ε3) can be obtained as
2B 2B
δM(3) = 0mˆb σj σj + 0(m mˆ)b σj σj +N (2B mˆ)2(c +c ). (4.10)
B N 2{ }{ } N s − 2{S }{ } c 0 1 2
c c
The b term involves two operators. One of them depends on the total baryon spin and the
2
other depends on the spin of the strange quark(s). As a result, the Σ decouples from the Λ
at this order. Up to this order, we have 4 operators in the baryon mass formula, namely,
1 , σj σj , and σj σj . The c and c terms give the same contributions to all
1 2
{ } { }{ } {S} {S }{ }
baryons. The matrix elements of the operators can be evaluated using
= (N +N ),
5 6
{S}
= (N +N )2,
5 6
{S}{S}
σ σ = 2(N +N )+(N N )[(N N )+(N N )+(N N )]+4N N
j j 5 6 5 6 1 2 3 4 5 6 5 6
{S }{ } − − − −
+2(α α†α†α +α†α α α† +α α†α†α +α†α α α†),
1 2 5 6 1 2 5 6 3 4 5 6 3 4 5 6
σ σ = 2[(N +N )+(N +N )+(N +N )]+4(N N +N N +N N )
j j 1 2 3 4 5 6 1 2 3 4 5 6
{ }{ }
+[(N N )+(N N )+(N N )]2
1 2 3 4 5 6
− − −
+4(α†α α α† +α†α α α† +α α†α†α +α α†α†α +α α†α†α
1 2 3 4 1 2 5 6 1 2 3 4 1 2 5 6 3 4 5 6
+α α†α†α ),
3 4 5 6
σ σ = [(N N )+(N N )](N N )
j j 1 2 3 4 5 6
{S }{U } − − −
+2(α†α α α† +α α†α†α +α†α α α† +α α†α†α ). (4.11)
1 2 5 6 1 2 5 6 3 4 5 6 3 4 5 6
All the matrix elements needed to calculate the baryon masses are given in Table I.
The explicit expression of mass corrections at O(ε4) reads
δM(4) = (2B )2[(m2 mˆ2)c +2mˆ(m mˆ)c ]
B 0 s − 1 s − 2 {S}
α
+ 2 2B (2mˆ +m ) σj σj +β (2B )2(2mˆ2 +m2). (4.12)
N2 0 s { }{ } 1 0 s
c
The combination of c and c terms depends on the strangeness, and the α term gives the
1 2 2
next order contribution to the decuplet–octet splitting. Therefore, all of the above terms
can be absorbed into the formulas valid up to O(ε3).
Then, up to this order, we have three mass relations,
10