The electromagnetic corrections to the masses of the pseudoscalar mesons \pi and K are considered. We calculate in chiral perturbation theory the contributions which arise from resonances within a photon loop at order O(e^2 m_q). For the corrections to Dashen's theorem we find rather moderate deviations in discrepancy to the values found in the literature

Baur, R

Urech, R

CERN Document Server

ZU-TH 22/95
TTP95-31
August 29, 1995
On the Corrections to Dashen's Theorem
Robert Baur
Institut fur Theoretische Physik, Universitat Zurich
CH-8057 Zurich, Switzerland
(e-mail: baur@physik.unizh.ch)
Res Urech
Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe
D-76128 Karlsruhe, Germany
(e-mail: ru@ttpux2.physik.uni-karlsruhe.de)
Abstract
The electromagnetic corrections to the masses of the pseudoscalar mesons  and K
are considered. We calculate in chiral perturbation theory the contributions which
arise from resonances within a photon loop at order O(e
2
m
q
). For the corrections to
Dashen's theorem we nd rather moderate deviations in discrepancy to the values
found in the literature.
1 Introduction
Dashen's theorem [1] states that the squared mass dierences between the charged
pseudoscalar mesons 

;K

and their corresponding neutral partner 
0
;K
0
are
equal in the chiral limit, i.e. M
2
K
  M
2

= 0, where M
2
P
= M
2
P

 M
2
P
0
. In
recent years several groups have calculated the electromagnetic corrections to this
relation from non-vanishing quark masses. The dierent conclusions are either that
the violation is large [2, 3] or that it may be large [4, 5, 6].
The electromagnetic mass dierence of the pions M
2

has been determined in the
chiral limit using current algebra by Das et al. [7]. Ecker et al. [8] have repeated the
calculation in the framework of chiral perturbation theory (PT) [9] by resonance
exchange within a photon loop. The occurring divergences from these loops are
absorbed by introducing an electromagnetic counterterm (with a coupling constant
^
C) in the chiral lagrangian. They nd that the contribution from the loops is
numerically very close to the experimental mass dierence, and thus conclude that
the nite part of
^
C is almost zero.
In this article we proceed in an analogous manner for the case m
q
6= 0. We calculate
in PT the contributions of order O(e
2
m
q
) to the masses of the Goldstone bosons due
to resonances. The divergences are absorbed in the corresponding electromagnetic
counterterm lagrangian, associated with the couplings
^
K
i
, where i = 1; : : : ; 14. The
most general form of this lagrangian has been given in [5, 6, 10]. We nd again that
the contribution from the loops represents the measured mass dierence M
2

very
well, and the nite part of the
^
K
i
may be considered small. Using this assumption
also for the calculation of M
2
K
, we may nally read o the corrections to Dashen's
theorem from one-loop resonance exchange. We nd a (scale dependent) result that
is controversial to the values found in the literature.
In [2] the authors have calculated the Compton scattering of the pseudoscalar mesons
including the resonances and determined from this amplitude the mass dierences.
They concluded rst of all by using three low-energy relations that the one-loop
result is nite, i.e. there is no need of a counterterm lagrangian at order O(e
2
m
q
)
in order to renormalize the contributions from the resonances, secondly they found
a strong violation of Dashen's theorem. We are in disagreement with both of these
results.
The article is organized as follows. In section 2 we present the ingredients from PT
and the resonances needed for the calculation. In section 3 we give the contributions
to the masses and to Dashen's theorem and renormalize the counterterm lagrangian.
The numerical results and a short conclusion are given in section 4.
2 The Lagrangians at lowest and next-to-leading
Order
The chiral lagrangian can be expanded in derivatives of the Goldstone elds and in
the masses of the three light quarks. The power counting is established in the follow-
ing way: The Goldstone elds are of orderO(p
0
), a derivative @

, the vector and axial
2
vector currents v

; a

count as quantities of O(p) and the scalar (incorporating the
masses) and pseudoscalar currents s; p are of order O(p
2
). The eective lagrangian
starts at O(p
2
), denoted by L
2
. It is the non-linear -model lagrangian coupled to
external elds, respects chiral symmetry SU(3)
R
 SU(3)
L
, and is invariant under
P and C transformations [9],
L
2
=
F
2
0
4
hd

U
y
d

U + U
y
+ 
y
Ui
d

U = @

U   i(v

+ a

)U + iU(v

  a

)
v

= QA

+   
Q =
e
3
diag (2; 1; 1)
 = 2B
0
(s+ ip)
s = diag (m
u
;m
d
;m
s
)
F

= F
0
[1 +O(m
q
)]
B
0
=  
1
F
2
0
h0juuj0i [1 +O(m
q
)] : (1)
The brackets h  i denote the trace in avour space and U is a unitary 3 3 matrix
that incorporates the elds of the eight pseudoscalar mesons,
U = exp (i=F
0
)
 =
p
2
0
B
B
@
1
p
2

0
+
1
p
6

8

+
K
+

 
 
1
p
2

0
+
1
p
6

8
K
0
K
 
K
0
 
2
p
6

8
1
C
C
A
: (2)
Note that the photon eld A

is incorporated in the vector current v

. The corre-
sponding kinetic term has to be added to L
2
,
L

kin
=  
1
4
F

F

 
1
2
(@

A

)
2
; (3)
with F

= @

A

  @

A

and the gauge xing parameter chosen to be  = 1. In
order to maintain the usual chiral counting in L

kin
, it is convenient to count the
photon eld as a quantity of order O(p
0
), and the electromagnetic coupling e of
O(p) [5].
The lowest order couplings of the pseudoscalar mesons to the resonances are linear in
the resonance elds and start at order O(p
2
) [8, 11]. For the description of the elds
we use the antisymmetric tensor notation for the vector and axialvector mesons, e.g.
3
the vector octet has the form
V

=
0
B
B
B
@
1
p
2

0

+
1
p
6
!
8

+

K
+


 

 
1
p
2

0

+
1
p
6
!
8
K
0

K
 

K
0

 
2
p
6
!
8
1
C
C
C
A
: (4)
This method is discussed in detail in [8], we restrict ourselves on the formulae needed
for the calculations in the following section. The relevant interaction lagrangian
contains the octet elds only,
L
V
2
=
F
V
2
p
2
hV

f

+
i+
iG
V
2
p
2
hV

[u

; u

]i
L
A
2
=
F
A
2
p
2
hA

f

 
i
f


= uF

L
u
y
 u
y
F

R
u
F

R;L
= @

(v

 a

)  @

(v

 a

)  i[v

 a

; v

 a

]
u

= iu
y
d

Uu
y
= u
y
U = u
2
: (5)
The coupling constants are real and are not restricted by chiral symmetry [11],
numerical estimates are given in [8]. In the kinetic lagrangian a covariant derivative
acts on the vector and axialvector mesons,
L
R
kin
=  
1
2
hr

R

r

R

 
1
2
M
2
R
R

R

i R = V;A
r

R

= @

R

+ [ 

; R

]
 

=
1
2
n
u
y
[@

  i(v

+ a

)]u+ u[@

  i(v

  a

)]u
y
o
; (6)
where M
R
is the corresponding mass in the chiral limit. Finally we collect all the
dierent terms together into one lagrangian,
L
eff
2
= L
2
+ L
R
2
+ L

kin
+ L
R
kin
: (7)
The one-loop electromagnetic mass shifts of the pseudoscalar mesons calculated
with this lagrangian (see section 3) contain divergences that can be absorbed in a
counterterm lagrangian. In its general form, this lagrangian has one term of order
O(e
2
) and 14 terms of O(e
2
p
2
) [5, 6, 10],
L
C
2
=
^
ChQUQU
y
i
L
C
4
=
^
K
1
F
2
0
hd

U
y
d

UihQ
2
i+
^
K
2
F
2
0
hd

U
y
d

UihQUQU
y
i
4
+^
K
3
F
2
0

hd

U
y
QUihd

U
y
QUi+ hd

UQU
y
ihd

UQU
y
i

+
^
K
4
F
2
0
hd

U
y
QUihd

UQU
y
i
+
^
K
5
F
2
0
h

d

U
y
d

U + d

Ud

U
y

Q
2
i
+
^
K
6
F
2
0
hd

U
y
d

UQU
y
QU + d

Ud

U
y
QUQU
y
i
+
^
K
7
F
2
0
hU
y
+ 
y
UihQ
2
i +
^
K
8
F
2
0
hU
y
+ 
y
UihQUQU
y
i
+
^
K
9
F
2
0
h(U
y
+ 
y
U + U
y
+ U
y
)Q
2
i (8)
+
^
K
10
F
2
0
h(
y
U + U
y
)QU
y
QU + (U
y
+ U
y
)QUQU
y
i
+
^
K
11
F
2
0
h(
y
U   U
y
)QU
y
QU + (U
y
  U
y
)QUQU
y
i
+
^
K
12
F
2
0
hd

U
y
h
c
R

Q;Q
i
U + d

U
h
c
L

Q;Q
i
U
y
i
+
^
K
13
F
2
0
hc
R
QUc
L

QU
y
i +
^
K
14
F
2
0
hc
R
Qc
R

Q+ c
L
c
L

Qi
+O(p
4
; e
4
) :
with c
R;L

Q =  i [v

 a

; Q]. The three last terms contribute only to matrix ele-
ments with external elds, we are therefore left with 12 relevant counterterms. Note
that we have omitted terms which come either from the purely strong or the purely
electromagnetic sector in L
C
4
.
3 Corrections to Dashen's Theorem
Using the lagrangian given in (7) it is a straightforward process to calculate the
mass shift between the charged pseudoscalar mesons 

;K

and their corresponding
neutral partner 
0
;K
0
at the one-loop level. The relevant diagrams for the mass
of the charged pion are shown in Fig.1. Graph (a) contains the o-shell pion form
factor, (b) vanishes in dimensional regularization and (c) is called \modied seagull
graph". Graph (d) contains an a
1
-pole. The mass of the neutral pion does not get
contributions from the loops.
If we take the resonances to be in the SU(3) limit according to (6), i.e. all vector
resonances have the same mass M
V
and all axialvector resonances the mass M
A
, we
get the contributions listed below. For the graphs with the pion form factor,

p:f:
M
2

=  ie
2
Z
d
4
q
(2)
4
q
2
+ 4 + 4M
2

q
2
(q
2
+ 2)
 i
8e
2
F
V
G
V
F
2
0
Z
d
4
q
(2)
4
q
2
M
2

  
2
q
2
(q
2
+ 2)(M
2
V
  q
2
)
(9)
 i
4e
2
F
2
V
G
2
V
F
4
0
Z
d
4
q
(2)
4
q
2
[q
2
M
2

  
2
]
q
2
(q
2
+ 2)(M
2
V
  q
2
)
2
;
5
where
a)
pi+
γ
= +
ρo
b)
ρo
c)
a1
+
d)
Figure 1: One-loop contributions to the electromagnetic mass shift of 

.
where  = pq and p is the momentum of the pion. Using the relation F
V
G
V
= F
2
0
[11] we obtain

p:f:
M
2

=  ie
2
M
4
V
Z
d
4
q
(2)
4
2 + 4M
2

q
2
(q
2
+ 2)(M
2
V
  q
2
)
2
: (10)
The modied seagull graph gives

s:g:
M
2

= i
e
2
F
2
V
F
2
0
(3   )
Z
d
4
q
(2)
4
1
M
2
V
  q
2
(11)
with  = 4 d, and nally for the a
1
-pole graph, where unlike [2] we get an additional
second term,

a
1
M
2

=  i
e
2
F
2
A
F
2
0
(3   )
Z
d
4
q
(2)
4
1
M
2
A
  q
2
6
  i
e
2
F
2
A
F
2
0
Z
d
4
q
(2)
4
q
2
[M
2

+ (3   )] + (2  )
2
q
2
[M
2
A
  (q + p)
2
]
: (12)
We can now perform the appropriate integrals. The integrals, however, have diver-
gences in d = 4 dimensions proportional to O(e
2
) and to O(e
2
m
q
). We will show
that they can be absorbed in the counterterm lagrangians given in (8) . Their con-
tribution to the mass shift of the pions has been evaluated in [5, 6]. Adding this
result to the one-loop result above, we get up to and including order O(e
2
m
q
),
M
2

=  
3e
2
F
2
0
16
2
"
F
2
V
M
2
V
 
ln
M
2
V

2
+
2
3
!
  F
2
A
M
2
A
 
ln
M
2
A

2
+
2
3
!#
 
e
2
F
2
A
F
2
0
16
2
M
2

"
7
2
+
3
2
ln
M
2
A

2
+ I
1
 
M
2

M
2
A
!#
+
2e
2
16
2
M
2

"
7
2
 
3
2
ln
M
2

M
2
V
+ I
2
 
M
2

M
2
V
!#
(13)
+
2e
2
^
C
F
2
0
 
6e
2
F
2
0
(F
2
V
M
2
V
  F
2
A
M
2
A
)
+8e
2
M
2
K
^
K
8
+ 2e
2
M
2

^
R

 
3e
2
F
2
A
F
2
0
M
2


with
I
1
(z) =
Z
1
0
x ln[x  x(1  x)z] dx
I
2
(z) =
Z
1
0
(1 + x)
(
ln[x+ (1  x)
2
z] 
x
x+ (1   x)
2
z
)
dx
^
R

=  2
^
K
3
+
^
K
4
+ 2
^
K
8
+ 4
^
K
10
+ 4
^
K
11
 =

d 4
16
2

1
d  4
 
1
2
[ln 4 +  
0
(1) + 1]

: (14)
The divergences are now canceled by renormalizing the coupling constants
^
C and
^
K
i
,
^
C =
^
C() + 3(F
2
V
M
2
V
  F
2
A
M
2
A
) (15)
^
R

=
^
R

() +
3F
2
A
2F
2
0
 : (16)
Using the second Weinberg sum rule [12]
F
2
V
M
2
V
  F
2
A
M
2
A
= 0 ; (17)
7
the divergence in (15) cancels, but the divergence in (16) does not. Even if we used
an extension of this sum rule to order O(m
q
) [13],
F
2

M
2

  F
2
a
1
M
2
a
1
' F
2

M
2

(18)
and assumed F
A
= F
0
[11], the divergence would not cancel, on the contrary, it
would become larger.
We nally get the result
M
2

=  
3e
2
F
2
0
16
2
F
2
V
M
2
V
ln
M
2
V
M
2
A
 
e
2
F
2
A
F
2
0
16
2
M
2

"
7
2
+
3
2
ln
M
2
A

2
+ I
1
 
M
2

M
2
A
!#
+
2e
2
16
2
M
2

"
7
2
 
3
2
ln
M
2

M
2
V
+ I
2
 
M
2

M
2
V
!#
(19)
+
2e
2
^
C()
F
2
0
+ 8e
2
M
2
K
^
K
8
+ 2e
2
M
2

^
R

() ;
where we used (17) to simplify the rst term. In the chiral limit M
2

reduces to
the expression given in [8].
The mass dierence for the kaons is determined in an analogous way, in the contri-
bution from the loops we merely have to replace M
2

by M
2
K
. Finally the formula
for the corrections to Dashen's theorem may be read o,
M
2
K
 M
2

=  
e
2
F
2
A
F
2
0
16
2
(
M
2
K
"
7
2
+
3
2
ln
M
2
A

2
+ I
1
 
M
2
K
M
2
A
!#
 M
2

"
7
2
+
3
2
ln
M
2
A

2
+ I
1
 
M
2

M
2
A
!#)
+
2e
2
16
2
(
M
2
K
"
7
2
 
3
2
ln
M
2
K
M
2
V
+ I
2
 
M
2
K
M
2
V
!#
 M
2

"
7
2
 
3
2
ln
M
2

M
2
V
+ I
2
 
M
2

M
2
V
!#)
(20)
 2e
2
M
2
K

2
3
^
S
K
() + 4
^
K
8

+ 2e
2
M
2


2
3
^
S

 
^
R

()

;
where
^
S
;K
represent the contributions from the counterterm lagrangian to M
2
K
,
^
S

= 3
^
K
8
+
^
K
9
+
^
K
10
^
S
K
=
^
K
5
+
^
K
6
  6
^
K
8
  6
^
K
10
  6
^
K
11
^
S
K
=
^
S
K
() +
3F
2
A
2F
2
0
 :
8
4 Numerical Results and Conclusion
We put F
0
equal to the physical pion decay constant, F

= 92:4 MeV and the masses
of the mesons to M

= 135 MeV;M
K
= 495 MeV. We take F
V
= 154 MeV [8] and
M
V
= M

= 770 MeV. To eliminate the parameters of the axialvector resonances
we use Weinberg's sum rules [12],
F
2
V
  F
2
A
= F
2
0
F
2
V
M
2
V
  F
2
A
M
2
A
= 0 : (21)
Putting the numbers in (19) we get for the contribution from the loops to M
2

at
the scale points  = (0:5; 0:77; 1) GeV (see Fig.2a)
M
2

= 2M

 ( 4:9 ; 5:0 ; 5:0 ) MeV : (22)
This remains close to the experimental value of 2M

 4:6 MeV [14]. We therefore
may conclude that the (nite) contributions from the low energy constants
^
C;
^
K
i
are small, i.e.
^
C() =
^
C  0
^
K
i
()  0 : (23)
Of course, in order to get a scale independent result, the counterterms are not
allowed to vanish completely. We assume, however, that they do not shift the result
essentially, at least in the case of M
2

. This statement may be expressed in a
formal way: Consider PT without resonances, then the contributions from the
vector and axialvector mesons to electromagnetic matrix elements are absorbed in
the counterterm lagrangian (8) and the coupling constants may be written as
C =
X
R
C
R
+
^
C
K
i
=
X
R
K
R
i
+
^
K
i
R = V;A ; (24)
where the terms with the superscript R represent the contributions from the reso-
nances. Eq.(23) now means that the non-resonant part is assumed to be considerably
smaller than the resonant contributions, in some analogy to the case of the coupling
constants L
i
in the strong sector [8]. However, in contrast to the latter case we have
a scale dependence in the result (22) reminding us of an additional contribution
beside the resonances, which we assume to be small numerically.
Using this assumption we obtain for the corrections to Dashen's theorem at the
three scale points  = (0:5; 0:77; 1) GeV
M
2
K
 M
2

= (  0:48 ;  0:18 ; 0:01 ) 10
 3
(GeV)
2
; (25)
which does not agree with the values found in the literature,
M
2
K
 M
2

=
8
>
<
>
:
1:23 [2]
1:3  0:4 10
 3
(GeV)
2
[3]
0:55  0:25 [4]
(26)
9
0.5 0.6 0.7 0.8 0.9 1
4.2
4.4
4.6
4.8
5
5.2
5.4
0.5 0.6 0.7 0.8 0.9 1
-1
0
1
2
a) b)
Figure 2: The solid lines show our results, the dashed and dotted curves represent
in a) the experimental value [14], in b) the result of [2], respectively.
In [2] the authors have calculated the Compton scattering of the Goldstone bosons
within the same model that we use in the present article and have determined the
corrections to Dashen's theorem by closing the photon line. Their calculation is
nite (without counterterms) and gives a considerably large value for M
2
K
 M
2

.
The dierence to our result may be identied in (12), where we nd an additional
(singular) term that gives a large negative and scale dependent contribution. The
two results are compared in Fig.2b. Note that in [2] the physical masses for the
resonances are used in the calculation of M
2
K
, whereas we work in the SU(3) limit
throughout.
The other calculations are not strongly connected to our approach. The discrepancy
to Bijnens' result [3] remains to be claried, for a discussion of the value given in
[4] we refer to [5].
We conclude with the observation that taking into account the resonances properly
and working strictly in the SU(3) limit for the resonances leads to moderate rather
than large corrections to Dashen's theorem. But we have to keep in mind the un-
certainties due to the missing part needed for the nal and scale independent result.
10
Acknowledgements
We thank G.Ecker, J.Gasser, J.Kambor, H.Leutwyler and D.Wyler for helpful dis-
cussions.
References
[1] R.Dashen, Phys. Rev. 183 (1969) 1245
[2] J.F.Donoghue, B.R.Holstein and D.Wyler, Phys. Rev. D47 (1993) 2089
[3] J.Bijnens, Phys. Lett. B306 (1993) 343
[4] K.Maltman and D.Kotchan, Mod. Phys. Lett. A5 (1990) 2457
[5] R.Urech, Nucl. Phys. B433 (1995) 234
[6] H.Neufeld and H.Rupertsberger, Isospin Breaking in Chiral Perturbation The-
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UWThPh-1994-15, to be published in Z. Phys. C; The Electromagnetic Inter-
action in Chiral Perturbation Theory, University of Vienna preprint UWThPh-
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[7] T.Das, G.S.Guralnik, V.S.Mathur, F.E.Low and J.E.Young, Phys. Rev. Lett.
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11


On the corrections to Dashen's theorem

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