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Interacting RaritaSchwinger field and its spinparity content
A.E.Kaloshin and V.P.Lomov
Irkutsk State University,
K.Marx Str. 1, Irkutsk 664003, Russia
Abstract
We obtain in analytical form the dressed propagator of the massive RaritaSchwinger field taking
into account all spin components. The nearest analogy for dressing in the s = 1/2 RaritaSchwinger
sector is the dressing of two fermions of opposite parity with presence of the mutual transitions. The
calculation of the selfenergy contributions confirms that besides s = 3/2 component the Rarita
Schwinger field contains also two s = 1/2 components of opposite parity.
1 Introduction
The covariant description of the spin3/2 particles is usually based on the RaritaSchwinger formalism
[1] where the main object is the spinvector field # . However in addition to spin3/2 this field contains
extra spin1/2 components and it generates the main di#culties in its description [2, 3]. The problem
has a long history, we mention here only relatively recent works [4, 5, 6, 7, 8, 9] which contain discussion
of the problem and historical review.
The expression for the free propagator G #
0 is well known (see e.g. [10, 11, 12]), thus we do not
present it here. As concerned for the dressed propagator, its construction is a more complicated issue
and its total expression is unknown up to now. Thus a practical use of G # (in particular for the case of
#(1232) production) needs some approximation in its description. The standard approximation [13, 14]
consist in a dressing the spin3/2 components only while the rest components of G # can be neglected or
considered as bare. Another way to take into account the spin1/2 components is a numerical solution
of the appearing system of equations[4, 9]. In Ref. [13] it was noticed that the spin1/2 components are
necessary to reproduce the experimental data on the #(1232) production in Compton scattering. So the
correct account of the extra spin1/2 component in the # field has also a practical meaning.
Here we derive an analytical expression for the interacting RaritaSchwinger field's propagator with
accounting all spin components and discuss its properties. It turned out that the spin1/2 part of the
dressed propagator has rather compact form, and a crucial point for its deriving is the choosing of a
suitable basis. Short variant of this paper was published in [15].
2 Dressed propagator of the RaritaSchwinger field
The DysonSchwinger equation for the propagator of the RaritaSchwinger field has the following form
G # = G #
0 +G # J ## G ##
0 . (1)
Here G #
0 and G # are the free and full propagators respectively, J # is a selfenergy contribution. The
equation may be rewritten for inverse propagators as
(G -1 ) # = (G -1 0
) #
- J # . (2)
If we consider the selfenergy J # as a known value 1 , than the problem is reduced to reversing of relation
(2). It is useful to have a basis for both propagators and selfenergy for this procedure.
1) The most natural basis for the spintensor S # (p) decomposition is the #matrix one:
S # (p) =g #
s 1 + p p #
s 2 ++pp p #
s 3 +
pg #
s 4 + p # #
s 5 +
# p #
s 6 ++# #
s 7 + # # p # p #
s 8 + # ## p # p
s 9 + # # # 5 # ### p #
s 10 .
(3)
Here S # is an arbitrary spintensor depending on the momentum p, s i (p 2 ) are the Lorentz invariant
coe#cients, and # # = 1
2
[# , # # ]. Altogether there are ten independent components in the decomposition
1 That is the widely used in the resonance physics ''rainbow approximation'', see e.g. recent review[16]
334

of S # (p). It is known that the #matrix decomposition is complete and the coe#cients s i are free
of kinematical singularities and constraints. However this basis is inconvenient at multiplication and
reversing of the spintensor S # (p) because the basis elements are not orthogonal to each other. As a
result the reversing of the spintensor S # (p) leads to a system of 10 equations for the coe#cients.
2) There is another basis used in consideration of the dressed propagator [4, 9, 13] G # . It is constructed
using the following set of operators [12, 13, 17]
(P 3/2 ) # =g #
-
2
3
p p #
p 2 -
1
3 # # # + 1
3p 2
(# p #
- # # p )p,
(P 1/2
11 ) # = 1
3 # # #
-
1
3
p p #
p 2 -
1
3p 2
(# p #
- # # p )p,
(P 1/2
22 ) # = p p #
p 2 ,
(P 1/2
21 ) # = # 3
p 2
1
3p 2
(-p + #
p)pp # ,
(P 1/2
12 ) # = # 3
p 2
1
3p 2
p (-p # + # # p)p. (4)
Here P 3/2 ,P 1/2
11 ,P 1/2
22 are the projection operators while P 1/2
21 ,P 1/2
12 are nilpotent. As for their physical
meaning, it is clear that P 3/2 corresponds to spin3/2. The remaining operators should describe two
spin1/2 representations and transitions between them.
The set of operators (4) can be used to decompose the considered spintensor as following [4, 9]:
S # (p) = (S 1 + S 2
p)(P 3/2 ) # + (S 3 + S 4
p)(P 1/2
11 ) # + (S 5 + S 6
p)(P 1/2
22 ) # +
(S 7 + S 8
p)(P 1/2
21 ) # + (S 9 + S 10
p)(P 1/2
12 ) # . (5)
Let us call this basis as
pbasis. It is more convenient at multiplication since the spin3/2 components
P 3/2 have been separated from spin1/2 ones. However, the spin1/2 components as before are not
orthogonal between themselves and we come to a system of 8 equations when inverting the (2). Another
feature of decomposition (5) is existence of the poles 1/p 2 in di#erent terms. So to avoid this unphysical
singularity, we should impose some constraints on the coe#cients at zero point.
3) Let us construct the basis which is the most convenient at multiplication of spintensors. This basis
is built from the operators (4) and the projection operators #
# = # p 2

p
2 # p 2
, (6)
where we assume # p 2 to be the first branch of analytical function. Ten elements of this basis look as
P 1 =# +
P 3/2 , P 3 =# +
P 1/2
11 , P 5 =# +
P 1/2
22 , P 7 =# +
P 1/2
21 , P 9 =# +
P 1/2
12 ,
P 2 =# - P 3/2 , P 4 =# - P 1/2
11 , P 6 =# - P 1/2
22 , P 8 =# - P 1/2
21 , P 10 =# - P 1/2
12 , (7)
where tensor indices are omitted. We will call (7) as the #basis.
Decomposition of a spintensor in this basis has the following form:
S # (p) =
10
# i=1
P #
i

S i (p 2 ). (8)
The #basis has very simple multiplicative properties which are represented in the Table 1.
Now we can return to the DysonSchwinger equation (2). Let us denote the inverse dressed propagator
(G -1 ) # and free one (G -1 0 ) # by S # and S #
0 respectively. Decomposing the S # , S #
0 and J # in #basis
according to (8) we reduce the equation (2) to set of equations for the scalar coe#cients

S i (p 2 ) =
S 0i (p 2 ) +
J i (p 2 ), i = 1 . . . 10.
335

P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10
P 1 P 1 0 0 0 0 0 0 0 0 0
P 2 0 P 2 0 0 0 0 0 0 0 0
P 3 0 0 P 3 0 0 0 P 7 0 0 0
P 4 0 0 0 P 4 0 0 0 P 8 0 0
P 5 0 0 0 0 P 5 0 0 0 P 9 0
P 6 0 0 0 0 0 P 6 0 0 0 P 10
P 7 0 0 0 0 0 P 7 0 0 0 P 3
P 8 0 0 0 0 P 8 0 0 0 P 4 0
P 9 0 0 0 P 9 0 0 0 P 5 0 0
P 10 0 0 P 10 0 0 0 P 6 0 0 0
Table 1: Properties of the #basis at multiplication.
So the values
S i are defined by the bare propagator and the selfenergy and may be considered as known.
After it the reversing of the S #
0 leads to equations for the coe#cients
G i :
# 10
# i=1
P #
i
G i (p 2 ) # # 10
# k=1
P #
k
S k (p 2 ) # =
6
# i=1
P #
i , (9)
which are easy to solve due to simple multiplicative properties of P #
i :

G 1 = 1/
S 1 ,
G 2 = 1/
S 2 ,

G 3 =
S 6 /# 1 ,
G 4 =
S 5 /# 2 ,
G 5 =
S 4 /# 2 ,
G 6 =
S 3 /# 1 ,

G 7 = -
S 7 /# 1 ,
G 8 = -
S 8 /# 2 ,
G 9 = -
S 9 /# 2 ,
G 10 = -
S 10 /# 1 , (10)
where # 1 =
S 3

S 6 -
S 7

S 10 , # 2 =
S 4

S 5 -
S 8

S 9 .
The
G 1 ,
G 2 terms which describe the spin3/2 have the usual resonance form and could be obtained
from (5) as well. As for
G 3 -
G 10 coe#cients which describe the spin1/2 contributions, they have a more
complicated structure.
3 Dressing the Dirac fermions
The obtained dressed propagator of the RaritaSchwinger field has rather unusual structure, so we would
like to clarify its physical meaning before renormalization. First of all it's useful to consider the dressing
of Dirac fermions with aim to find the close analogy for RaritaSchwinger field case.
3.1 Dressing the single fermion
The dressed fermion G(p) propagator is solution of the DysonSchwinger equation
G(p) = G 0 +G#G 0 , (11)
where G 0 is the bare propagator and # is the selfenergy contribution.
Decomposition of any matrix 4 4, depending on one momentum p, has the form:
S(p) =
2
#
M=1
PM
S M , P 1 = # + , P 2 = # - , (12)
where we introduced the new notations to stress an analogy with RaritaSchwinger field.
DysonSchwinger equation in this basis takes the form:

G M =
G M
0 +
G M
# M
G M
0 , M = 1, 2, (13)
336

and its solution is:
(
G M ) -1 = (
G M
0 ) -1 -
# M . (14)
Let us look at the selfenergy contribution #(p). As an example we shall consider the dressing of
baryon resonance N # (J P = 1/2 ) due to interaction with #N system. Interaction lagrangian is of the
form
L int = g# # (x)# 5 #(x) #(x) + h.c. for N # = 1/2 + (15)
and
L int = g# # (x)#(x) #(x) + h.c. for N # = 1/2 - . (16)
Isotopical indexes are omitted since they are irrelevant here.
Positive parity baryon resonance
#(p) = ig 2 # d 4 k
(2#) 4 # 5 1

p +
k -mN
# 5 1
k 2
-m 2
#
= I A(p 2 ) +
pB(p 2 ) (17)
Let us calculate the loop discontinuity through the LandauCutkosky rule: 2
#A = -
ig 2 mN
(2#) 2 I 0 , #B = ig 2
(2#) 2 I 0
p 2 +m 2
N -m 2
#
2p 2 . (18)
Here I 0 is the base integral
I 0 = # d 4 k# # k 2
-m 2
# # # # (p + k) 2
-m 2
N # = # # p 2
- (mN +m # ) 2
# #
2
# # # # # # p 2 , m 2
N , m 2
# #
# p 2
# 2 , (19)
where # # a, b, c # = # a - b - c # 2
- 4bc.
From the parity conservation one can see that in the transition N # (1/2 + ) # N(1/2 + ) +#(0 - ) the #N
pair has the orbital momentum l = 1. But according to threshold quantummechanical theorems [19],
the imaginary part of a loop should behave as q 2l+1 at q # 0 3 , which does not correspond to (18).
Let us calculate the imaginary part of # M component according to (18)
Im # 1 = Im # A + # p 2 B # = g 2 I 0
4 # p 2 (2#) 2 # # p 2
-mN -m # ## # p 2
-mN +m # # # q 3 ,
Im # 2 = Im # A - # p 2 B # = - g 2 I 0
4 # p 2 (2#) 2 # ## p 2 +mN # 2
-m 2
# # # q 1 . (20)
One can see that the # 1 component demonstrates the proper threshold behavior.
Negative parity baryon resonance
#(p) = ig 2 # d 4 k
(2#) 4
1
k +
p -mN
1
k 2
-m 2
#
= IA(p 2 ) +
pB(p 2 ),
#A = -i
g 2 mN
(2#) 2 I 0 , #B = -ig 2
(2#) 2 I 0
p 2 +m 2
N -m 2
#
2p 2
(21)
Imaginary parts of # 1,2 now demonstrate l = 0 behavior
Im # 1 = -
g 2 I 0
4 # p 2 (2#) 2 # ## p 2 +mN # 2
-m 2
# # # q 1 ,
Im # 2 = g 2 I 0
4 # p 2 (2#) 2 ## p 2
-mN -m # ### p 2
-mN +m # # # q 3 . (22)
The considered examples show that only an ,,alive`` component # 1 , which has the pole 1 ### p 2
-m #
demonstrates the proper threshold behavior (i.e. the proper parity). Another component # 2 , which has
pole of the form 1 ## - # p 2
-m # demonstrates the opposite parity.
2 This is way to avoid the unphysical singularities: to renormalize the A, B components and then to calculate # .
3 q is the spatial momentum of #N pair in CMS.
337

3.2 Joint dressing the two fermions of opposite parity
Let us consider the nearest analogy to the RaritaSchwinger field: the joint dressing of two fermions of
opposite parity 1/2 . We will suppose that interaction conserves the parity.
Now the DysonSchwinger equation has the matrix form
G ij = # G 0 # ij
+G ik # kl # G 0 # lj , i, j, k, l = 1, 2. (23)
Every element in this equation has #matrix indexes which are not shown.
Basis will contain four operators:
P 1 = # + , P 2 = # - , P 3 = # + # 5 , P 4 = # - # 5 , (24)
where P 1,2 are projection operators and P 3,4 are the nilpotent ones. Decomposition of any #matrix,
depending on p, now is of the form (compare with (12))
S(p) =
4
#
M=1
PM
S M . (25)
This set of operators has very simple multiplicative properties (see Table 2).
P 1 P 2 P 3 P 4
P 1 P 1 0 P 3 0
P 2 0 P 2 0 P 4
P 3 0 P 3 0 P 1
P 4 P 4 0 P 2 0
Table 2: Multiplicative properties of basis (24)
The DysonSchwinger equation (23) reduces for equation on the coe#cients
G M :
# 4
#
M=1
PM
G M
## 4
#
L=1
PL
S L
# = P 1 + P 2 , (26)
where GM , SL are the matrixes 2 2.
It leads to matrix equations:
G 1 S 1 +G 3 S 4 = E 2 , G 2 S 2 +G 4 S 3 = E 2 ,
G 1 S 3 +G 3 S 2 = 0, G 4 S 1 +G 2 S 4 = 0, (27)
where E 2 is the unit matrix 2 2. Solutions:
G 1 = # S 1 - S 3 # S 2 # -1 S 4 # -1 , G 2 = # S 2 - S 4 # S 1 # -1 S 3 # -1 ,
G 3 = - # S 1 - S 3 # S 2 # -1 S 4 # -1 S 3 # S 2 # -1 , G 4 = - # S 2 - S 4 # S 1 # -1 S 3 # -1 S 4 # S 1 # -1 .
(28)
Now let us concretize these general formulae. Suppose that we have two fermions of di#erent parity,
but there is no parity violation in lagrangian. It means that the diagonal loops contain only the I and
p
matrixes while the nondiagonal loops should have # 5 . After it solution (28) takes the form
G =# +
# # #
-m 2 -E - # 2
22
# 1
0
0 -m 1 -E - # 2
11
# 2
# # # + # -
# # #
-m 2 +E - # 1
22
# 2
0
0 -m 1 +E - # 1
11
# 1
# # #+
+# + # 5
# # #
0 -
# 3
12
# 1
-
# 3
21
# 2
0
# # # + # - # 5
# # #
0 -
# 4
12
# 2
-
# 4
21
# 1
0
# # # .
(29)
338

Here # 1 = # -m 1 +E - # 2
11 ## -m 2 -E -# 2
22 # - # 3
12 # 4
21 ,
# 2 = # -m 1 -E - # 1
11 ## -m 2 +E -# 1
22 # - # 4
12 # 3
21 = # 1 # E # -E # ,
E = # p 2 , and i, j = 1, 2 numerate the dressing fermion states. The appearance of nilpotent operators in
decomposition (29) is an indication for transitions between states of di#erent parity. They are absent in
case of mixing of the same parity states.
Let us summarize our consideration of the dressing of Dirac fermions.
1) Besides simple multiplicative properties the projection operators # are very useful in another
aspect: its coe#cients exibit the definite parity. But as one can see from the loop calculations (20),
(21) the components # have di#erent parity. There is such correspondence: the parity of the field
# is the parity of ,,alive`` component # + , which has the pole 1 # (E -m). Another component # -
which has the pole 1 # (-E -m) demonstrates the opposite parity.
2) In contrast to boson case, even if the interactions conserve the parity, the loop transitions between
di#erent parity states are not zero: they are proportional to nilpotent operator # # 5 .
4 Spinparity of the RaritaSchwinger field
Comparing Tables 1 and 2, one can conclude that presence of the nilpotent operators P 7 ---P 10 in decom
position (9) is an indication for the transitions between components of di#erent parity 1/2 . To make
sure in this conclusion, we can calculate loop contributions in propagator. As an example we will take
the interaction lagrangian #N#
L int = g #N# #
(x)(g # + a# # # )#(x) # # #(x) + h.c. . (30)
Here a is some arbitrary parameter.
The oneloop selfenergy contribution is
J # (p) = -ig 2
#N# # d 4 k
(2#) 4
(g # + a# # # )k # 1
p + k -mN
k # (g ## + a# # # # ) 1
k 2
-m 2
#
. (31)
Let us calculate the discontinuity of loop contribution in
p basis (5).
#J 1 = -ig 2 I 0
mN
12s #(s, m 2
N , m 2
# ),
#J 2 = -ig 2 I 0
1
24s 2
(s +m 2
N -m 2
# )#,
#J 3 = -ig 2 I 0
mN
12s (# + 6a# - 36a 2 m 2
# s),
#J 4 = -ig 2 I 0
1
24s 2
[(s +m 2
N -m 2
# )# + 12as# + 36a 2 s(s 2
-m 2
# s - 2m 2
N s -m 2
# m 2
N +m 4
N )],
#J 5 = ig 2 I 0
mN
4s [(s -m 2
N +m 2
# ) 2 + 2a(s -m 2
N +m 2
# ) 2 + 4a 2 m 2
# s],
#J 6 = ig 2 I 0
1
8s 2
[(s +m 2
N -m 2
# )(s -m 2
N +m 2
# ) 2 + 4as(s -m 2
N +m 2
# )(s -m 2
N -m 2
# ) +
4a 2 s(s 2
-m 2
# s - 2m 2
N s -m 2
# m 2
N +m 4
N )],
#J 7 = ig 2 I 0 # 3
s
1
24s [(s -m 2
N +m 2
# )# + 4as(2s 2
-m 2
# s - 4m 2
N s + 2m 4
N -m 2
N m 2
# -m 4
# ) +
12a 2 s(s 2
-m 2
# s - 2m 2
N s -m 2
N m 2
# +m 4
N )],
#J 8 = -ig 2 I 0
# 3
s amN
6s [(s 2 + 4m 2
# s - 2m 2
N s +m 4
- 2m 2 m 2
# +m 4
# ) + 6asm 2
# ],
#J 9 = #J 7 , #J 10 = -#J 8 . (32)
Here I 0 is the base integral(19), arguments of # are shown in the first expression.
339

We saw that in case of Dirac fermions the propagator decomposition in basis of projection operators
demonstrates the definite parity. We can expect the similar property for RaritaSchwinger field in #basis.
Let us verify it by calculating the threshold behavior of imaginary part. Using (32) we have
#
J 1 = #J 1 +E#J 2 # q 3 , #
J 2 = #J 1 -E#J 2 # q 5 , #
J 3 = #J 3 +E#J 4 # q 3 ,
#
J 4 = #J 3 -E#J 4 # q, #
J 5 = #J 5 +E#J 6 # q, #
J 6 = #J 5 -E#J 6 # q 3 . (33)
Such behavior indicates that the components
J 1 ,
J 2 exhibit the spinparity 3/2 + , while the pairs of
coe#cient
J 3 ,
J 4 and
J 5 ,
J 6 correspond to 1/2 + 1/2 - contributions respectively.
5 Conclusions
Thus we obtained the simple analytical expression (10) for the interacting RaritaSchwinger field propa
gator which accounts for all spin components. To derive it we introduced the spintensor basis (7) with
very simple multiplicative properties.
The obtained dressed propagator (10) solves an algebraic part of the problem, the following step is
renormalization. Note that the investigation of dressed propagator is the alternative for more conventional
method based on equations of motion (see, i.e. Ref. [18] and references therein). The natural requirement
for the renormalization is that the spin1/2 components should remain unphysical after dressing. In other
words the denominators # 1 , # 2 should not acquire of zero in the complex energy plane. However a
problem of the renormalization needs a more careful consideration.
We found that the nearest analogy for dressing of the s = 1/2 sector is the joint dressing of two
Dirac fermions of di#erent parity. Some hint for such spinparity content may be seen from algebraical
properties of # basis (7) with presence on nilpotent operetors. Caculation of the selfenergy contributions
in case of # isobar confirms it: in the RaritaSchwinger field besides the leading s = 3/2 contribution
there are also two s = 1/2 components of di#erent parity.
We thank Organizers of Workshop for warm hospitality.
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