Table Of Content1 Feb 2008 LBL-33435
Chiral Corrections to Hyperon
Vector Form Factors
3
9 Jeffrey Anderson and Markus A. Luty
9
1
Theoretical Physics Group
n
a Lawrence Berkeley Laboratory
J
1 Cyclotron Road
2
1 Berkeley, California 94720
2
v
9
Abstract
1
2
1
0 We show that the leading SU(3)-breaking corrections to the ∆S = 1 f1 vector form factors
3 3/2
of hyperons are O(m ) and O(m ), and are expected to be 20–30% by dimensional
9 s s ∼
/ analysis. This is consistent with the Ademollo–Gatto theorem, in a sense that we explain.
h
p 3/2
We compute the O(m ) corrections and a subset of the O(m ) corrections using an
- s s
p
effective lagrangian in which the baryons are treated as heavy particles. All of these
e
h corrections are surprisingly small, 5%; combining them, we obtain 5–10% corrections.
: ∼ ∼
v
The pattern of corrections is very different than that predicted by quark models.
i
X
r
a
PACS 13.31.Ce, 14.20.Gk, 14.20.Jn
This work was supported by the Director, Office of Energy Research, Office of High
Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of
Energy under Contract DE-AC03-76SF00098.
1. Introduction
Inthispaper, weconsidertheapplicationofchiralperturbationtheory tothef vector
1
formfactorofoctetbaryonstates. Theformfactorsofthevectorcurrent areconventionally
defined by
ifabc(q2) ifabc(q2)
B J (0) B = u(p ) [fabc(q2)γ + 2 σ qν + 3 q u(p ), (1)
h a| µc | bi a (cid:20) 1 µ M +M µν M +M µ(cid:21) b
a b a b
where q p p . Our interest in the form factor f is due to the fact that it is usually
a b 1
≡ −
assumed that SU(3) breaking corrections to f are small due to the Ademollo–Gatto (AG)
1
theorem [1]. Indeed, nonrelativistic quark model and bag model calculations of SU(3)
breaking corrections to f typically give corrections of order 1% [6].
1
In this paper, we point out that the leading corrections to f due to the nonvanishing
1
3/2
strange quark mass are O(m ) and O(m ). The O(m ) terms are proportional to
s s s
m2
K 0.2, (2)
16π2f2 ∼
3/2
where f 93 MeV. The O(m ) corrections consist of terms proportional to
s
≃
m ∆ m3
K B 0.2, K 0.3, (3)
16πf2 ∼ 16πf2Λ ∼
where Λ 1 GeV is the expansion scale in chiral perturbation theory, and ∆ is an
B
∼
octet baryon mass splitting. Clearly, it is important to compute these corrections, since
dimensionalanalysisdoesnotguaranteethattheyaresmall. Forexample, theycouldaffect
the determination of D and F from semileptonic hyperon decay rates. (In the formalism
we employ, D and F are defined as couplings in an effective lagrangian which embodies
the low-energy theorems for chiral symmetry. In the SU(3) limit, we recover well-known
relations such as D+F = g , but there are SU(3)-breaking corrections to these relations
A
due to nonvanishing quark masses which can be substantial.)
We compute the O(m ) corrections using an effective lagrangian in which the baryons
s
are treated as heavy particles [2][3]. Using the values D = 0.61, F = 0.40 determined
using a recent fit to semileptonic hyperon decay [4], we find these corrections to be surpris-
ingly small, < 5%. These corrections have been computed in ref. [5], and we agree with
∼ 3/2
the results of this paper. The O(m ) contributions proportional to m ∆ have not
s K B
been previously computed. We find that these corrections are also < 5% for all decays.
∼
The terms proportional to m3 cannot be computed in terms of the lowest order chiral
K
1
lagrangian. The predicted corrections are increased significantly if the lowest-order fit
†
values of D and F are used, or if we make the approximation m 0. Our computation
π
≃
gives some indication that the corrections to f are < 10%, but we cannot exclude the
1
3/2 ∼
possibility that the remaining O(m ) corrections are 30% or more.
s
∼
The plan of this paper is as follows: In section 2, we review the effective lagrangian
formalism we use to carry out the computation. In section 3, we discuss the Ademollo–
Gatto theorem and how it is manifested in the effective lagrangian framework. The reader
eagerforthe bottomlinecanskipimmediatelytosection 4, inwhich wepresent ourresults.
Section 5 contains our conclusions.
2. The Effective Lagrangian
It has been known for some time that the low-energy theorems of chiral symmetry
breaking are equivalent to a description of the low-energy dynamics in terms of an effective
lagrangian [7]. Recently, it was realized that baryons could be simply included in an
effectivelagrangianframeworkusinganheavyparticleeffectivetheory[2][3]. Thisapproach
provides significant conceptual and calculational advantages: the non-relativistic limit is
incorporated from the start, and the Feynman rules for computing graphs are considerably
simplified.
In this section, we define the effective lagrangian and establish our notation.
2.1. Mesons
The field
ξ(x) = eiΠ(x)/f, (4)
is taken to transform under SU(3) SU(3) as
L R
×
ξ LξU = UξR , (5)
† †
7→
where this equation implicitly defines U as a function of L, R, and ξ. The meson fields are
1 π0 + 1 η π+ K+
√2 √6
1
Π = √2 π− −√12π0 +√16η K0 . (6)
K K0 2 η
− −√6
Ref. [5] gives some O(m3/2) corrections of the form m3 /(16πf2M ) where M is
† s K B B
the average octet baryon mass. These contributions are an artifact of the method of
computation used in that paper, and including them is not justified.
2
We will be interested in matrix elements of the vector currents. We therefore add to
the effective lagrangian a source term
µ µ
δ = J + J , (7)
L Vµ V Aµ A
µ µ
where J (J ) is the vector (axial vector) Noether current. The couplings of and in
V A Vµ Aµ
the effective lagrangian are then determined by demanding that they transform as gauge
fields (see eq. (12)). We therefore define the covariant derivatives
D ξ ∂ ξ i ξ, D ξ ∂ ξ i ξ . (8)
µ µ µ µ † µ † µ †
≡ − L ≡ − R
(Note that (D ξ) = D ξ .) Here,
µ † µ †
6
1 1
= ( + ), = ( ). (9)
µ µ µ µ µ µ
R 2 V A L 2 V −A
and are hermitian. The effective lagrangian is most conveniently written in terms
µ µ
V A
of
i i
V ξD ξ +ξ D ξ , A ξD ξ ξ D ξ , (10)
µ µ † † µ µ µ † † µ
≡ 2 ≡ 2 −
(cid:0) (cid:1) (cid:0) (cid:1)
which transform under local SU(3) SU(3) transformations as
L R
×
V UV U +iU∂ U , A UA U , (11)
µ µ † µ † µ µ †
7→ 7→
since the sources transform as gauge fields:
L L +iL∂ L , R R +iR∂ R . (12)
µ µ † µ † µ µ † µ †
L 7→ L R 7→ R
Note that A and V are hermitian. We can then define the covariant derivative
µ µ
A ∂ A i[V ,A ], (13)
µ ν µ ν µ ν
∇ ≡ −
which transforms under local SU(3) SU(3) transformations as
L R
×
A U A U . (14)
µ ν µ ν †
∇ 7→ ∇
The chiral symmetry is broken explicitly by the quark masses. (We neglect the effects
of electromagnetism in this paper.) We will ignore isospin breaking, so that the quark
mass matrix is taken to be
mˆ
M = mˆ . (15)
q
m
s
3
It is convenient to define
1
M ξ M ξ +h.c. UMU . (16)
† q † †
≡ 2 7→
(cid:0) (cid:1)
The simple transformation rules of the fields defined above makes it easy to write
down the effective lagrangian. For example, the leading terms can be written
= f2tr(AµA )+af3trM. (17)
0 µ
L
2.2. Baryons
We now discuss the inclusion of baryon fields as heavy particles [2][3]. The momentum
of a baryon field is written
P = M v +p, (18)
B
where M is a SU(3)-invariant baryon mass and v is a velocity. The key observation is
B
that for processes involving emission of soft pions, the relevant residual momenta p are
small if v is chosen appropriately. For a fixed v, we can then write an effective theory in
terms of baryon fields B whose momentum is the residual momentum p of the baryon [2].
The octet baryon fields B transform under SU(3) SU(3) as
L R
×
B UBU . (19)
†
7→
Explicitly, we have
1 Σ0 + 1 Λ Σ+ p
√2 √6
B = Σ 1 Σ0 + 1 Λ n . (20)
− −√2 √2
Ξ− Ξ0 −√26Λ
The lowest order terms in the effective lagrangian involving baryon fields are
= tr Biv B +2Dtr Bsµ A ,B +2F tr Bsµ[A ,B]
µ µ
L ·∇ { } (21)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
+σtr(M)tr BB +b tr B M,B +b tr B[M,B] ,
D F
{ }
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where sµ is the spin operator [3] and the covariant derivative acts on B as in eq. (13).
2.3. Power Counting
Theeffectivelagrangiandescribedabovehasawell-definedexpansionininversepowers
of Λ 1 GeV. A typical term in the lagrangian can be written schematically
∼
B nB nD A nA M nM
f2Λ2 ∇ . (22)
L ∼ (cid:18)f√Λ(cid:19) (cid:18)Λ(cid:19) (cid:18)Λ(cid:19) (cid:18)Λ (cid:19)
4
If we write
m2 afM , ∆ bM , (23)
Π ∼ q B ∼ q
thentopologicalidentitiescanbeusedtoshow thatloopcorrectionsarerelatedtotree-level
contributions by
Λ 2L af Nm/2
loop bN∆ tree, (24)
∼ (cid:18)4πf (cid:19) (cid:18)Λ (cid:19) ×
where L is the number of loops in the diagram, and N (N ) is the number of powers
m ∆
of m (∆ ) in the result of the loop diagram. This expansion is consistent provided that
Π B
Λ < 4πf, a < Λ/f < 4π, and b < 1. This appears to be satisfied in QCD [8].
∼ ∼ ∼ ∼
3. The Ademollo–Gatto Theorem
In this section, we review the Ademollo–Gatto (AG) theorem [1] and discuss how it
is realized in the effective lagrangian approach. Much of this section is quite elementary,
but we feel that the issues involved deserve a careful treatment.
Suppose that a quantum-mechanical system has a global symmetry G which is ex-
plicitly broken by perturbations whose size is controlled by a parameter λ. We assume λ
is sufficiently small so that the explicit symmetry breaking can be treated perturbatively.
It is then convenient to expand the physical states of the system in terms of states with
definite transformation properties under G:
α = c r j + cr rj , crα 0. (25)
| i α| α αi αj| i αjα ≡
Xr,j
Here, rj is a state belonging to the irreducible representation r; j labels the particular
| i
state. The state r j is the state corresponding to the physical state α in the limit
α α
| i | i
λ 0:
→
c 1, cr = O(λ), as λ 0. (26)
α→ αj →
The AG theorem applies if the symmetry breaking effect is such that it does not mix
states from the same irreducible representation, i.e.
crα = 0. (27)
αj
In this case, the AG theorem states that for any charge Q of G,
β Q α = q δ +O(λ2), (28)
α αβ
h | | i
where q is the charge of the unperturbed state:
α
Q r j = q r j . (29)
α α α α α
| i | i
5
This theorem can be applied to the f form factor, since
1
B Q B = d3x B J (x) B = u (p )u(p )fabc(~q = 0)+O(M2). (30)
h a| c| bi Z h a| 0c | bi † a b 1 q
The conditionsof thetheorem aresatisfied inthe case ofexplicit SU(3)breaking due to the
strange quark mass, since the mass matrix eq. (15) has definite isospin and hypercharge,
and thus does not mix members of the octet.
The proof of the AG theorem is by direct computation:
β Q α = c c r j Q r j + cr cs rj Q sk . (31)
h | | i α βh β β| | α αi βj αkh | | i
Xr,j Xs,k
“Mixed” terms proportional to e.g. c cr are absent by the assumption eq. (27). Demand-
α βj
ing that the physical states be normalized to unity gives c = 1 + O(λ2), and the result
α
eq. (28) follows immediately.
Usually, simple current algebra arguments such as this are spoiled by nonanalyticity
in M due to the presence of massless Nambu–Goldstone bosons in the limit M 0.
q q
→
However, we note that if we write
M = m 1+δmT , (32)
q 0 8
and consider an expansion in δm with m held fixed, there are no massless particles, and
0
we expect that physical quantities are analytic in δm. The AG theorem then guarantees
that corrections to the vector form factors are O(δm2). This is not the limit relevant
for the real world, where m ,δm m , but we will use this limit to check whether our
0 s
∼
calculations are consistent with the AG theorem.
Note that that the AG theorem is not trivially manifest in the effective lagrangian.
The lagrangian contains terms which appear to give tree-level corrections to the f form
1
factor of order δm, in violation of the AG theorem:
c c
1 2
δ = tr(M)tr Biv B + tr BMiv B +h.c.
L Λ ·∇ Λ ·∇
c (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) (33)
3
+ tr Biv BM +h.c. .
Λ ·∇
(cid:2) (cid:0) (cid:1) (cid:3)
(There are other terms which can be related to these by integration by parts.) However,
these terms also modify the kinetic term for the baryons so that there is no order δm
correction to f . To see this, we make the field redefinition
1
c c c
1 2 3
B = 1+ tr(M) B + MB + BM. (34)
′
2Λ Λ Λ
h i
6
B is a good interpolating field for the baryons provided that M does not mix octet states.
′ q
The lagrangian expressed in terms of B does not contain any terms of the form eq. (33),
′
and the AG theorem is manifest.
4. Results
The one-loop graphs contributing to the vector form factor are shown in fig. 1. We
write
1 m
fabc(0) = αc 1+ βc + K γc , (35)
1 ab(cid:18) 16π2f2 ab 16πf2 ab(cid:19)
where the well-known lowest-order results are
α4+i5 = 3,
pΛ −q2
α4+i5 = 1,
nΣ− − (36)
α4+i5 = 3,
ΛΞ− 2
q
α4+i5 = 1 .
Σ0Ξ− √2
For the O(m ) corrections, we obtain
s
β4+i5 = 2λ D2λ F(2D+3F)λ ,
pΛ 1 − 2 − 1
β4+i5 = 2λ D2λ +3F(2D F)λ ,
nΣ− 1 − 3 − 1 (37)
β4+i5 = 2λ D2λ +F(2D 3F)λ ,
ΛΞ− 1 − 2 − 1
β4+i5 = 2λ D2λ 3F(2D+F)λ ,
Σ0Ξ− 1 − 3 − 1
where
3 m2 m2 m2 m2m2 m2
λ = m2 +2m2 +m2 2 K π ln K 2 η K ln η ,
1 16(cid:18) π K η − m2 m2 m2 − m2 m2 m2 (cid:19)
K − π π η − K K
1 m2 m2 m2 m2m2 m2
λ = 9m2 +10m2 +m2 18 K π ln K 2 η K ln η , (38)
2 16(cid:18) π K η − m2 m2 m2 − m2 m2 m2 (cid:19)
K − π π η − K K
1 m2 m2 m2 m2m2 m2
λ = m2 +10m2 +9m2 2 K π ln K 18 η K ln η .
3 16(cid:18) π K η − m2 m2 m2 − m2 m2 m2 (cid:19)
K − π π η − K K
The combination of masses defined above are easily seen to satisfy the AG theorem in the
sense discussed in section 3. Our numerical results are summarized in table 1. The O(m )
s
corrections are < 5% for all decays, significantly smaller than what is expected on the
∼
basis of dimensional analysis. We are therefore led to consider the higher order corrections
to determine whether they are numerically important.
7
3/2
The O(m ) contributions proportional to m ∆ are computed from the graphs in
s K B
fig. 1. Because of the length and unilluminating nature of the resulting formulas, we will
give formulas only for the case m = 0, and simplify the results using the Gell-Mann–
π
Okubo relations
m2 = 4m2 ,
η 3 K
(39)
M = 3M +1M M .
Ξ 2 Λ 2 Σ − N
(We have checked that the full expressions satisfy the AG theorem.) We obtain
γ4+i5 = 1 (25 16√3)D2 1(39 16√3)DF + 3 (25 16√3)F2 M
pΛ −10 − −5 − 10 − n
h i
+1D(D F)M
2 − Σ
+ 2(5 4√3)D2 + 1 (83 32√3)DF 3 (25 16√3)F2 M ,
5 − 10 − −10 − Λ
h i
γ4+i5 = 1 (103 48√3)D2 1(39 16√3)DF + 1 (103 48√3)F2 M
nΣ− −30 − −5 − 10 − n
h i
+ 2(9 4√3)D2 + 1 (83 32√3)DF 1 (103 48√3)F2 M
5 − 10 − −10 − Σ
h i
1D(D 3F)M ,
−6 − Λ
(40)
γ4+i5 = 1 (25 16√3)D2 1(39 16√3)DF 3 (25 16√3)F2 M
ΛΞ− 10 − −5 − −10 − n
h i
+ 1 (15 16√3)D2 +2(11 4√3)DF + 3 (25 16√3)F2 M
−20 − 5 − 20 − Σ
h i
+ 1 (35 16√3)D2 +1(17 8√3)DF + 3 (25 16√3)F2 M ,
−20 − 5 − 20 − Λ
h i
γ4+i5 = 1 (103 48√3)D2 1(39 16√3)DF 1 (103 48√3)F2 M
Σ0Ξ− 30 − −5 − −10 − n
h i
+ 1 (113 48√3)D2 2(11 4√3)DF 1 (103 48√3)F2 M
60 − −5 − −20 − Σ
h i
+ 1 (319 144√3)D2 +1(61 24√3)DF + 3 (103 48√3)F2 M .
60 − 5 − 20 − Λ
h i
Our numerical results (including m = 0) are summarized in table 1. The main feature of
π
6
these results is that they are significantly smaller than expected from dimensional analysis.
Using theoldervaluesofD andF orneglecting thepionmassresultsinsubstantiallylarger
corrections.
5. Conclusions
We have computed chiral corrections to the f vector form factor for ∆S = 1 semilep-
1
3/2
tonic hyperon decay. We have shown that the leading corrections are O(m ) and O(m )
s s
and explained how this is consistent with the Ademollo–Gatto theorem. These corrections
are 30% according to dimensional analysis. Explicit calculation shows that the O(m )
s
∼
3/2
corrections and a computable subset of the O(m ) corrections are 5% for all decays
s
∼
8
using the values of D and F of ref. [4]. The corrections are significantly larger if the older
values of D and F are used, or if the pion mass is neglected.
These results are very different than those obtained from the non-relativistic quark
model and bag model [6]. In these models, the corrections to f are universal for all decays
1
and are 1%. The corrections we have computed are much larger and depend on the
≃ −
decay. From the point of view of the chiral expansion, the quark model results are rather
hard to understand. Since these model predictions are used in a determination of V from
us
semileptonic hyperon decay [10], this discrepancy is of more than academic interest. It
is possible that the inclusion of the effects of the decuplet [4] may reduce the apparent
descrepancy between the chiral corrections and the quark model calculations.
3/2
O(m ) O(m ) total
s s
Λ p 0.001 0.026 0.047 0.032 0.046 0.016
→ − ± ± ±
Σ n 0.065 0.011 0.060 0.041 0.13 0.05
−
→ ± ± ±
Ξ Λ 0.021 0.014 0.060 0.033 0.081 0.020
−
→ ± ± ±
Ξ Σ0 0.002 0.031 0.035 0.034 0.033 0.013
−
→ − ± ± ±
SU(3)
Table 1: f /f for ∆S = 1 hyperon decays using the best-fit values
1 1
D = 0.61, F = 0.40 of ref. [4]. The quoted errors are obtained by (somewhat
arbitrarily) assigning a 20% error to the values of D and F. Using the older
values D = 0.8, F = 0.5 [9] increases all of the corrections by 40%. We
∼
have kept m = 0; taking m = 0 increases all of the corrections, some by
π π
6
as much as 35%.
6. Acknowledgements
We would like to thank M. Suzuki for helpful discussions and R. N. Cahn for critical
reading of the manuscript. This work was supported by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of
the U.S. Department of Energy under Contract DE-AC03-76SF00098.
9
