Table Of ContentCEBAF-TH-93-02
January 1993
Delta-Isobar Magnetic Form Factor in QCD
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V.M.Belyaev
n
a
J Institute of Theoretical and Experimental Physics
1 B.Cheremushkinskaya 25,
2
117259, Moscow, USSR
1 and
v Continuous Electron Beam Accelerator Facility
7
12000 Jefferson Ave, Newport News, Virginia 23606, USA
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2
1
0
3
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h/ Abstract
p
- We consider the QCD sum rules approach for ∆-isobar magnetic
p
e form factor in the infrared region 0 ≤ Q2 < 1GeV2. The QCD sum
h rules in external variable field are used. The obtained form factor is
:
v in agreement with quark model predictions for the ∆-isobar magnetic
Xi moment.
r
a
1 Introduction
The QCD sum rule method suggested by Shifman, Vainshtein and Zakharov
(SVZ)inthepioneeringpaper[1]becomesnowauniversaltoolforcalculating
different properties of low-lying hadronic states. Using the original version of
this method, the meson [1] and baryon [2] masses were found from the sum
rules for two-point correlation functions. Using the three-point correlation
functions, hadron form factors at intermediate Q2 can be obtained [3]. Un-
fortunately, this method does not work if one tries to calculate form factors
in the infrared region 0 < Q2 < 1GeV2 due to power corrections 1/Q2n at
Q = 0. The new method - QCD sum rules in external field was suggested
in [4], and using this method nucleon magnetic moments were found [5] as
well as baryon axial couplings [6]. Then this method was formulated for a
variable external field [7] which gives a possibility to calculate form factors
at Q2 6= 0.
In [7] we have formulated a new method for calculating hadronic form
factors in the infrared region. To study a form factor at nonzero Q2, it is
necessary to introduce a variable external field. The calculation of a polar-
ization operator in this field encounters a number of difficulties as compared
with the case of a constant external field. The arising problems and methods
to avoid them were discussed in detail in the paper [7].
Let us note that in the papers [8] and [9], the pion and nucleon charge
radiiwereconsidered, using themethodssimilartoours. However, theresults
obtained in [8] are connected not with the calculation of the total form factor
F(Q2), but only the first derivative at zero momentum transfer < r2 >∼
F′(Q2)| . In [9] pion form factor was considered. The aim of the paper is
Q=0
to use the general method for calculating ∆-isobar magnetic form factor in
the infrared region.
2 Polarization Operator in Variable External
Field
To compute ∆-isobar magnetic form factor we shall consider the following
correlator in an external variable electromagnetic field:
ΠV∗(p,k) = i 2πδ(kx)eipxd4x < T{η (x),η¯ (0)} > (1)
µν µ ν V
Z
1
+∞
= dzi ei(p+kz)xd4x < T{η (x),η¯ (0)} >
µ ν V
Z−∞ Z
+∞
= dzΠV (p+kz,k)
µν
Z−∞
where
ΠV (p,k) = i eipxd4x < T{η (x),η¯ (0)} > (2)
µν µ ν V
Z
η (x) = ǫabc(uaCγ ub)uc (3)
µ µ
is the quark current with the ∆-isobar quantum numbers suggested in the
first paper in Ref.[2], u is the u-quark operator; a , b and c are the color
indeces, ǫabc is the antisymmetric tensor and C = −CT is the charge conju-
gation matrix; index V means the vacuum average in the presence of weak
external electromagnetic field that is responsible for adding to Lagrangian of
the following term
∆L = −V eikx e q¯ (x)γ q (x) = −V j (x)eikx (4)
µ f f µ f µ µ
f
X
where e is the charge of the quark with flavor f, V and k are the ampli-
f µ µ
tude and the momentum of the classical external field. This correlator was
suggested in [7].
Now let us discuss the reason why we need to introduce the δ-function in
the correlator (1). To calculate the correlator (2) atp2 ∼ −1GeV2, (p+q)2 ∼
−1GeV2, k2 = −Q2 < 0) we use operator product expansion in the presence
of external variable field (4). So we need to know nonperturbative quark
propagator in the field
<: T{qa(x),qb(0)} :> (5)
α β V
=< T{qa(x),qb(0)} > − < T{qa(x),qb(0)} >(pert.)
α β V α β V
Where :: denotes a subtraction of perturbative contribution. It is possible
to find perturbative part of this propagator in the form of expansion over the
coupling constant. To take into account nonperturbative interaction of the
quark with the external field, we expand eq.(5) over x
µ
<: T{qa(x),qb(0)} :> =<: qa(0),qb(0) :> (6)
α β V α β V
2
1
+x <: D qa(0),qb(0) :> + x x <: D D qa(0),qb(0) :> +...
µ µ α β V 2 µ ν µ ν α β V
Itisclearthatthen’thtermofexpansion(6)cangivedimensionless factor
(kx)n. Effectively it means that highest terms contribution of expansion (6)
into a polarization operator (2) is not suppressed because this factor (kx)n
corresponds to the factor (kp)/p2 ∼ 1. To kill the dangerous contributions
of the terms ∼ (kx)n we insert δ(kx) into the correlator (1). This correlator
(1) can be calculated in a form of series over 1/p2. Therefore at respectively
large −p2 this correlator can be calculated with a good accuracy using only
the first few terms in expansion (6).
The nonperturbative quark propagator in the external field has the fol-
lowing form:
δab
<: ua(x),u¯b(0) :>= e {Vˆ k2Π (k2)
α β u 12 αβ 1
1
+(σ ) k V Π (k2)+(Vx)[i <: ψ¯ψ :> + k2Π (k2)]δ (7)
ρλ αβ ρ λ 2 0 2 αβ
2
i i
+ Vˆ (kx)k2Π (k2)+ (σ ) k V (kx)Π (k2)
αβ 1 ρλ αβ ρ λ 2
2 2
1
− x ε (γ γ ) k V k2Π (k2)
µ µνρλ λ 5 αβ ν ρ 1
4
i
+ x2(σ ) k V [<: ψ¯ψ :> −iΠG(k2)−2ΠG(k2)+ik2Π (k2)]
12 ρλ αβ ρ λ 0 1 2 4
(kx)2
+(Vx)(kx)k2Π (k2)δ + (σ ) k V Π (k2)
3 αβ ρλ αβ ρ λ 4
2
(kx) 5
+ x (σ ) V [ i <: ψ¯ψ :> +ΠG(k2)+3ik2Π (k2)+iΠG(k2)
12 µ µν αβ ν 2 0 1 3 2
5 (Vx) i
− k2Π (k2)]+ x (σ ) k [ <: ψ¯ψ :> −ΠG(k2)
2 4 12 µ µν αβ ν 2 0 1
k2
−iΠG(k2)+3ik2Π (k2)− Π (k2)]
2 3 2 4
+ (terms with an odd number of γ −matrices)}+O(x3)
3
The correlators Π (k2) are defined as follows:
i
e Π (k2)(k2g −k k ) =
u 1 µν µ ν
i eikxd4x <: T{ q¯ γ q (x),u¯γ u(0)} :>
f µ f ν 0
Z f
X
e Π (k2)(k g −k g ) = (8)
u 1 µ νρ ν µρ
i eikxd4x <: T{ q¯ γ q (x),u¯σ u(0)} :>
f ρ f µν 0
Z f
X
e Π (k2)(k2(g k +g k )−2k k k ) =
u 3 µρ ν νρ µ µ ν ρ
→
i eikxd4x <: T{ e q¯ γ q (x),u¯ D D u(0)} :>
f f ρ f {µ ν} 0
Z f
X
e Π (k2)k k (g k −g k )+... =
u 4 µ ν ρr p ρp r
→
i eikxd4x <: T{ e q¯ γ q (x),u¯σ D D u(0)} :>
f f ρ f pr {µ ν} 0
Z f
X
e ΠG(k2)(k g −k g ) =
u 1 µ ρν ν ρµ
i eikxd4x <: T{ e q¯ γ q (x),g u¯Gn tnu(0)} :>
f f ρ f s µν 0
Z f
X
e ΠG(k2)(k g −k g ) =
u 2 µ ρν ν ρµ
i eikxd4x <: T{ e q¯ γ q (x),g u¯G˜n tnγ u(0)} :>
f f ρ f s µν 5 0
Z f
X
Perturbative contributions are subtracted in correlators (8). These expres-
sions were obtained in [7].
In this paper we neglect operators ΠG(k2) because their contribution into
a sum rule is small (see [7]).
3 The Sum Rules
In this Section, we obtain a sum rule for the ∆-isobar magnetic form factor.
First, we should choose a tensor structure which has a contribution from the
magnetic transition between baryon states with quantum numbers J = 3/2.
To this end, we consider the contribution of two baryons with masses m and
1
4
m into the polarization operator ΠV(p,k) (2)
2
V < 0 | η | ∆ >< ∆ | jem | ∆ >< ∆ | η¯ | 0 >
ρ µ 1 1 ρ 2 2 ν , (9)
(p2 −m2)((p+k)2 −m2)
1 2
where∆ and∆ arebaryonstateswithmasses m andm respectively. Here
1 2 1 2
we consider the case when only spinor parts of the Rarita-Shwinger fields in-
teract with a photon. In such case, thematrix element of theelectromagnetic
current has the following form:
< N | jem | N >=
1 ρ 2
v¯(1)(p)g [f (k2)γ + ϕ12(k2)σ k +ψ (k2)k ]v(2)(p+k) =
µ µν 12 ρ m1+m2 ρλ λ 12 ρ ν
v¯(1)(p)g [(f (k2)+ϕ (k2))γ +P ϕ12(k2) +ψ (k2)k ]v(2)(p+k) (10)
µ µν 12 12 ρ ρm1+m2 12 ρ ν
< 0 | η | N,JP = 3/2+ >= λv (p)
µ µ
P = p +(p+k) ,
µ µ µ
where v (p) is a Rarita-Shwinger spin-vector satisfying the Dirac equation:
µ
(pˆ−m)v (p) = 0, γ v = 0, p v = 0.
µ µ µ µ µ
Using (10) we can transform (9) to the following form:
λ λ
1 2 V [g pˆ γ pˆ G (k2)/3 (11)
(p2 −m2)((p+k)2 −m2) µ µν 1 ρ 2 M
1 2
+(other structures with γ placed at the beginning and γ at the end of them)]
µ ν
where G is the magnetic form factor.
M
It is important to note that there is no spin-1/2 baryon contribution in
the structure g pˆ γ pˆ which has the following amplitude:
µν 1 ρ 2
< 0 | η | J = 1/2 >= (Ap +Bγ )u(p)
ρ µ µ
where (pˆ−m)u(p) = 0 and Am+4B = 0.
From (11), it is obvious that the structure g pˆ γ pˆ (where p = p and
µν 1 ρ 2 1
p = p+k)containsmagnetictransitiononly, sinceG12/3 = f (k2)+ϕ (k2),
2 M 12 12
where G12 is the magnetic form factor. So, we shall further consider only the
M
structure g pˆ γ pˆ .
µν 1 ρ 2
Now let us discuss the factor 1/3 which have appeared in eq.(11). Con-
sider interaction of spin-3/2 particle with the electromagnetic field:
a(Ψ (P )g (P +P ) Ψ (P )+ibΨ (σ g /2+2g g )Ψ F (12)
µ 1 µν 1 2 ρ ν 2 µ ρλ µν µρ νλ ν ρλ
5
where 1σ = [γ γ ], Ψ is Rarita-Shwinger spin-vector field ( (Pˆ − m) =
2 ρλ ρ λ µ
0,γ Ψ = 0), F = ∂ A −∂ A . The first term of eq.(12) corresponds to
µ µ ρλ ρ λ λ ρ
the spin-independent part of electromagnetic interaction of 3-spin particle
2
and the second one describes the spin-dependent interaction. To express the
value of the magnetic moment (at Q2 = 0) let us consider the case when
A = 0,P = m,F = δ δ ǫ H , where H is magnetic field. Then we
0 0 ρλ ρi λj ijk k k
have
ibΨ (g σ ǫ /2+2ǫ )Ψ H = bΨ (g Σ +2iǫ )Ψ H (13)
m mn ij ijk mnk n k m mn k mnk n k
where Σ = diag(σ ,σ ) Now we see that the operator (Σ g + 2iǫ ) is
k k k k mn mnk
equal to 2S where S is spin operator for the Rarita-Shwinger field. So, the
k k
maximal energy of the particle in the magnetic field is equal to
E = 3bH = µH, (14)
int.
where µ, by definition, is magnetic moment or magnetic form factor at Q2 =
0. Thus, we have
µ = 3b (15)
Now let us consider the double dispersional relation for the function at
tensor structure g pˆ γ pˆ V :
µν 1 ρ 2 ρ
1 ∞ ∞ ρ(s ,s ,Q2)
f(P2,P2,Q2) = 1 2 , (16)
1 2 π2 (P2 +s )(P2 +s )
Z0 Z0 1 1 2 2
where P2 = −p2, P2 = −p2, Q2 = −k2 ≥ 0 and ρ(s ,s ,Q2) is the spectral
1 1 2 2 1 2
density. Due to reasons mentioned above, we cannot calculate ρ(p ,p ,Q2)
1 2
directly, but we can consider the double Borel transformed structure function
of correlator (1). Notice, that under replacement p → p +kz, p → p +kz
1 1 2 2
the structure pˆ γ pˆ V transforms to
1 µ 2 µ
ˆ ˆ ˆ ˆ
pˆ γ pˆ V → (pˆ +kz)V(pˆ +kz) = pˆ Vpˆ −2(Vp )pˆ z − (17)
1 µ 2 µ 1 2 1 2 1 1
2(Vp )pˆ z +(p2 +p2)zVˆ +z2(2(Vk)kˆ−Vˆk2)
2 2 1 2
and all other structures in (11) with smaller number of γ−matrices near
ˆ ˆ
g , could not be transformed into pˆ Vpˆ . So we can extract g pˆ Vpˆ
µν 1 2 µν 1 2
in the integral (1). Then the integral representation for the double Borel
6
transformed structure function under consideration is obtained from (16) by
applying the operator Oˆ:
+∞
Oˆf(P2,P2) = dzBˆ Bˆ f((P +kz)2,(P +kz)2) (18)
1 2 P2 P2 1 2
Z−∞ 1 2
where
(P2)n+1 ∂ n
Bˆ= lim −
n→∞ n! ∂P2!
P2/n = M2
M2 is the Borel parameter.
Applying the operator Oˆ to the left-hand and right-hand sides of double
dispersion relation (16) we get
1 +1/2 Q2z2
f(M2,Q2) = exp( )×
π2 M2
Z−1/2
∞ ∞ s +s (s −s )z
ds ds exp(− 1 2 + 1 2 )ρ(s ,s ,Q2) (19)
1 2 2M2 M2 1 2
Z0 Z0
where M2 = M2 = 2M2.
1 2
We shall use the sum rules (19) to calculate the nucleon form factor. The
function f(M2,Q2) is calculated using operator expansion in the external
field, and thespectral density ρ(s ,s ,Q2) issaturated by intermediate states
1 2
with quantum numbers of the current (3).
Therearetwodifferent typesofintermediatestatecontributionsinto(19).
The first is responsible for the diagonal transitions between the states with
equal masses. The second is responsible for nondiagonal transitions between
the states with different masses. In the first case the right-hand side of (19)
obviously has the form
λ2GM(Q2)e−m2/M2 +1/2eQM2z22dz (20)
3
Z−1/2
where λ2 is the square of the residue of the state with mass m into the
current η defined by formula (10), G (Q2) is the corresponding magnetic
µ M
form factor.
7
In the second case for the transition between states with masses m and
1
m we get:
2
λ1λ2G(M12)(Q2)e−m221M+m222 +1/2eQ2z2+(Mm221−m22)zdz (21)
3
Z−1/2
Now, to investigate the properties of (20) and (21) let us expand them in the
series on Q2/M2
G (Q2) 1 Q2 1 q2
λ2 M e−m2/M2(1+ + ( )2 +... ) (22)
3 12M2 5·25 M2
β1β2G1M2(Q2)e−m221M+m222 m22M−2m2 sinh(m212M−2m22)× (23)
1 2
Q2 1 M2 m2 −m2 2M4
{1+ [ − coth( 1 2)+ ]+...}
M2 4 m2 −m2 2M2 (m2 −m2)2
1 2 1 2
From (22) and (23) we see that diagonal transitions of the excited states are
exponentially suppressed compared to the ∆-isobar contribution in (22). Let
us write the non-suppressed part of the contribution from the nondiagonal
transition between the nucleon with mass m and anexcited state with mass
N
m∆∗
G∆∆∗(Q2) e−mM2∆2M2
λλ∆∗ M 3 m2 −m2 {1 +
∆∗ ∆
Q2 1 M2 m2 −m2 2M4
[ − coth( ∆∗ ∆) + ]+...} (24)
M2 4 m2 −m2 2M2 (m2 −m2 )2
∆∗ ∆ ∆∗ ∆
where m∆, m∆∗, λ and λ∆∗ are masses and residues of ∆-isobar and its
resonance ∆∗ respectively.
Expression (24) isanalogoustothecontribution fromthesingle-poleterm
appearing in QCD sum rules for correlators in constant external field (see
[4]).
It is easy to see that the function multiplied by Q2/M2 in (24) changes
from 1/12 at m2 −m2 → 0 to 1/4 at m2 −m2 → ∞. However, taking into
2 ∆ 2 ∆
account the continuum, only contributions from the states with m2 −m2 ∼
2 ∆
s −m2 (s is the continuum threshold) are to be considered. In the region
0 ∆ 0
8
s ≫ s , our model of continuum is quite correct, but when s ∼ s it is not
0 0
so. Then we see, that nonexponentially suppressed terms will be determined
by the states with m2 ∼ s . Taking m2 − m2 ≃ s − m2 ≃ 1.5GeV2,
2 0 2 ∆ 0 ∆
M2 ∼ 1GeV2, expression (24) can be written in the form
λ λ G∆M∆∗(Q2) e−mM2∆2M2 (1+ Q2 (1+ǫ)+... ) (25)
∆ ∆∗ 3 m2 −m2 12M2
∆∗ ∆
where ǫ < 0.1.
Thus, from (22) and (25) it is seen that when Q2/M2 ≤ 1, and M2 ∼
1GeV2, the right-hand side of (19) is
31λ2∆e−m2∆/M2(GM(Q2)+C(Q2)M2) +1/2eQM2z22dz (26)
Z−1/2
The accuracy of (26) is of an order of a few percents (it depends on the
numerical value of ǫ from (25)). It can be shown, that the next terms in
the expansion (23) in powers of Q2/M2 do not change the situation. So, in
the region Q2/M2 ≤ 1, M2 ∼ 1GeV2, the right side of the sum rule (19) is
indeed represented by expression (26).
Let us note that when Q2/M2 > 1, the expression (26) is invalid and we
cannot separate the contribution of the single-pole terms from the contribu-
tion of the us double-pole term ∼ G (Q2) which we are interested in. Thus,
M
our sum rule is expected to be valid in the region 0 ≤ Q2 < 1GeV2.
Here we have constructed the right-hand ”phenomenological” side of the
sum rule (19) and discussed the region of its applicability. Now, let us pass
to the calculation of the left, ”theoretical” part of our sum rule.
Using eq.(7) and dispersion integral (19) we have obtained the following
sum rule for G :
M
+1/2eQ2z2/M2dz s0ds1 s0ds2e−s21M+s22+zs1M−2s2 ρ∆(s1π,s22,Q2)
Z−1/2 Z0 Z0
2 +1/2
+ a2 dzeQ2z2/M2 (27)
3
Z−1/2
+2(2π)2a(iΠ2(Q2))M2e4QM22
3
9
