Corrections to the Weiss-Zumino term in chiral perturbation theory

Nuclear Physics B (Proc. Suppl.) 23B (1991) 295--298 North-Holland

295

CORRECTIONS TO THE WEISS-ZUMINO TERM IN C~_IRAL PER~I/RHATION %~EORY

Albert BRAMON Grup de Flsica Te6rica, Univ. Aut6noma de Barcelona, 08193 Bellaterra, Spain Johan B I ~ S Sektion Physik, Univ. M[tnchen, D-8000 Munich 2, F.R.G. Fernando COI~NET Depto. de Flsica Tedrica y del Cosmos, Univ. de Granada, 18071 Granada, Spain

All the terms needed to cancel the d i v e r g e n c e s a p p e a r i n g in o n e - l o o p c a l c u l a t i o n s involving the W e s s - Z u m i n o term are obtained. We a p p l y this result to the Pyy and PPPy vertices, fixing the finite part of the c o e f f i c i e n t s of the O(p 6) terms in the l a g r a n g i a n with vector m e s o n contributions. In all the cases the c o r r e c t i o n s improve the agreement between the t h e o re t i c a l p r e d i c t i o n s and e x p e r i m e n t a l data.

The

strong

and e l e c t r o w e a k

tions of the lightest sons at low energy described ryl masses

are

gian has an

Perturbation

limit where

into its v e c t o r

appear

In this process

8 goldstone

where

subgroup,

bosons

B(s-ip).

fields.

degrees

lagrangian

is

(2)

x°

M =

K +

K°

/

the m e s o n

momenta

in terms

and quark masses,

-1/3,

-1/3)

The quark mass matrix,

field s=M+ ....

in the This

B in terms

contribute

at higher

according

to

meD=2+Z N d ( d - 2 ) + 2 N L d

which

0920-5632/91/$03.50 © 1991 - Elsevier Science Publishers B.V.

field

is the quark

alof

and quark masses.

Loop d i a g r a m s

of

can

the i d e n t i f i c a t i o n

der in the e x p a n s i o n

the octet of p s e u d o s c a l a r

D~Z=

vector

A~ is the p h o t o n

md, ms) , is included

scalar

X=

interactions

lows to fix the constant

-v{.] and can be e x p a n d e d

with

matrix.

M=diag(mu, external

~+

external

Electromagnetic

where

exter-

fields

derivative,

contains

and Q=diag(2/3,

(i) ~,~°+ ~

M e V and X contains

The c o v a r i a n t

be i n t r o d u c e d

charge

external

The

is O(p2) :

and p s e u d o s c a l a r

L~=R~=eQA~,

of the m a t r i x

2i E = exp.-:M , ]

sons,

f=f~=132

nal scalar

3uZ+iLuZ-iZR~,

symmetry

the relevant

The chiral

in terms

containing

lagrangian

to

scale.

-f~ ~ t r ( D . E D " E ~ + X ~ t + EX ~) ,

L2 =

to be sponta-

spontaneous

and become

written

order

compared

breaking

Theo-

SU(3) L x SU(3) R chiral

neously b r o k e n

of freedom.

lowest

to be small

symmetry

set to zero the QCD lagran-

is b e l i e v e d

break i n g

the chiral

the q u a r k

symmetry w h i c h

SU(3) V.

are a s s u m m e d

me-

are c o n v e n i e n t l y

by Chiral

In the

interac-

pseudoscalar

All rights reserved.

,

(3)

or-

296

A. Bramon et al./ Chiral perturbation theory

where

pD is the order of the

gram,

N d is the number

loop dia-

O(p d) c o n t a i n e d the number

of v e r t i c e s

in the d i a g r a m

and N L is

× [~.~

of loops 2° The free coef-

ficients

of the O(p D)

grangian

can be used to absorb

gences

appearing

grams.

Gasser

terms

the d i v e r g e n c e s

involving

constructed

N~ - 4

,

( x ~ ' + ~ x i)

_N_~£~£~]

i

all

to cancel

from

the most general

_

- 4s~:p~"w"{tr([--~N

loop dia-

[ £ . £ v ( R . ~ - La~) - ( R . . - L.v)£a£Z])

in o n e - l o o p

vertices

i L ~ + ~°)~1} - ~(

2

the diver-

found

needed

appearing

D2EEt]]

- !C~(L~ + ~)

iN

in the la-

in the O(p D)

and L e u t w y l e r

the O(p ~) c o u n t e r t e r m s

diagrams

¼[£~, E D 2 E t -

of

(2) and

i ~ + ~tr(fv(RaZ - LaZ))tr(E.(EX l +

xEt -

£~£~))+

lagrangian

at O(p ~ ) 1 The W e s s - Z u m i n o has

term,

to be introduced

for anomalous tion

which

in order

processes 3'4,

at lowest

order

is O(p~), to account

The full ac-

is then given by

where

£~=ZD~Z ~,V~v=O~V~+3~V~+i[V~,V v]

Vu=R ~ and R ~ v = Z R ~ Z %. N o t i c e the terms

in Eq.

coefficient

- NcSwz ,

S =/d4xL2

(4)

that all

(6) are O(p6),

of the W e s s - Z u m i n o

not renormalized. We can now apply our result

where

Nc=3

is the number

of colors

i S w z =2_.~_6_~ [J dE ij~l~ tvE iL EjL E~L E Lt E ~L

i/

(5)

processes

Z~=Z3i Z~.

According volving

to Eq.

one-loop

In particular, the vertices

to discuss

and

(we will

appearing

in these

can be found expan-

E around

its classical

and using

the heat

expansion

regularization.

After

and a te-

one obtains

(6)

z one-loop W Z o c=

take

8=-I/3).

( N e N I - e~vc~tr ~ [DXLA. + 16r2(d - 4) k,72~'2f z ""

hag] + 4[£ x,

~ is k n o w n

EDxD.Et-

ED~R~Et D ) , D . E E 'f]

discussion

and 8.

to be a mixtu-

and singlet angle

states,

0=-19.5 °, c o r r e s p o n d i n g Consequently,

for the p h y s i c a l the singlet

considering

a nonet

way of i n t r o d u c i n g symmetry

9) to

one cannot ma-

thout

simplest

~s

8~ -20°(ref.

ke any p r e d i c t i o n

context.

part.

the

This

easily

achieved

enlarging

in Eq.

(i) with

the singlet

the

~

~ wiThe is in

can be Z matrix

part.

The o n e - l o o p d i a g r a m s c o n t r i b u t i n g to ~0 ,~,~' ~ y y ( * ) are the ones shown in Fig. i, t o g e t h e r

1

1 ,~, L ~ . + ~[£

5,6,7

D~, with mixing

sin

the m a t r i x

calculation

detailed

re of the octet in-

from L2 are O(p~).

terms

A more in refs.

The p h y s i c a l

from the W e s s - Z u m i n o

calculation

dimensional dious

for

(3) loop diagrams

and one

The d i v e r g e n t

value

full e x p r e s s i o n in

one vertex

lagrangian

ding

The

experimen-

tal data are available.

Pyy and PPPy.

can be found

for w h i c h

we are going can be found

Z~S

to some

and physical

with

so the term is

with

the ones

for w a v e - f u n c -

tion and f r e n o r m a l i z a t i o n .

The result

of the one-loop

for real pho-

tons

turns

calculation

out be finite

8

in such a

A. Bramon et al. / Chiral perturbation theory a way

that

all the effects

and n o n - a n o m a l o u s change

f into

f~8 and fnl mental

L 4 terms

f~,Ds,n1"

loops

amount

to

The v a l u e s

can be fixed

values

of the

~(rl÷y~')/8m~I'(~°÷yy)=0.62±O.05

f~s/f~=l'3 and one obtains

f~1 /fTr=l'l"

to

of

be

In this w a y

resonance

P~- m~F (~°+~Y)

(7)

We will

this

the case

is still

lous sector,

i.e.

have

is very d i f f e r e n t

~÷p+p-y.

ficient

1 gives

the

terms can be

the O(p 6) terms

with

the d i v e r g e n t

in Eq.

(6). The

lagrangian

these

in ChPT.

constants

from

coef-

finite

however, In o r d e r

and to be able

the k 2 - d e p e n d e n c e de we have

some extra

to predict

rators

the chlral

grangian

assump-

ing order

Ecker and c o l l a b o -

found that the finite

of the c o e f f i c i e n t s terms,

symmetry.

of the O(p 6) terms

lagrangian

in

of the next

0(p4),

can be related

appearing

integrated

the r e l e v a n t are fixed

out.

to

in the

la-

once

these

In p a r t i c u l a r ,

coefficients

from the ones

the r a d i a t i v e

decays

dict

the

for ~+U+~-y

contributing

to

of v e c t o r mesons,

In this way we can pre-

slope p a r a m e t e r

to be 5

1 d A ( M ~ 77)1k2=0 1.96 G e V ~. A ( M ~ 7 7 ) dk 2 = The e x p e r i m e n t a l

(8 )

part

to lead~

in the n o n ~ a n o -

is

measured

(1.9±0.4)

ment w i t h

value 12 for the slope

the

Notice

loop result

and the O(p6)

A

similar

studying we will and

through

although

the

coefficients

the p h y s i c a l

situation

result

is

is found when

PPPy vertices.

n+~+n-y.

theoretical

of p.

discuss

consider

that,

~+~+p-y

good agree-

just o b t a i n e d

prediction.

independent

FIGURE 1 diagrams for Pyy.

in the decay

GeV 2, in very

are ~-dependent,

One-loop

nonet

for v e c t o r mesons,

have been

parameter

have

the hid-

amplitu-

tions. paper,

using

are

to fix

of the ~+p+~-¥

to make

In a recent

in-

of Bando et al. II

ideally m i x e d

such as ~+~0~. part of the coefficients, free c o n s t a n t s

mesons

a contri-

and finite

to k 2. 2he first ones

introducing

found

vector

is a l l o w e d

In this case,

in Fig.

the l a g r a n g i a n

contri-

as it is the case

bution having d i v e r g e n t

removed

part

of the O(p 6) terms

approach

the c o e f f i c i e n t s

proportional

that the finite

introduced

The c o e f f i c i e n t s shell,

second d i a g r a m

that

when assuming

in the decay

here

for the anoma-

bution.

to the chiral

of the photons

to be off mass

assume

by the v e c t o r m e s o n

den s y m m e t r y (or both)

at

w i t h the e x p e r i m e n t a l

p~xP=2.4±0.4.

The s i t u a t i o n

contribution

contribution.

We

one

by the

~--m i0. In addition, they have found P that v e c t o r m e s o n s provide the d o m i n a n t

is s a t u r a t e d =2.6

in good a g r e e m e n t

are saturated

of the c o e f f i c i e n t s

the prediction.

3m~F ( ~ y )

value

lagrangian

low l y i n g

from the experi3 and pn = 3 m~F

fK/f~=l.22

malous

297

In particular,

the p r o c e s s e s

y~0~

For the first p r o c e s s

the q u a n t i t y

we

F 3~ d e f i n e d

the a m p l i t u d e

A(Tx ~ x°~) = i F 3 " ~ " V ~ A ~

'

(9)

A. Bramon et al./Chiral perturbation theory

298

w h e r e pl (p2) is the m o m e n t u m of the in.

variables.

coming

creases with respect to the tree level

mentum.

(outgoing)

~ and P0 is the ~°~mo-

At lowest order, one has F 3~=

The absolute value of F 37r in-

the e x p e r i m e n t a l value 13 F3~=12.9±0.9±

p r e d i c t i o n and the c o r r e c t i o n ranges 2 _ 2_ from a 7% at p12-(p1+p2) - 0.7m 2~ and2 P202__ 2 (p0+p2) 2= 4m~ to a 12% at p12=3.5 m~ and

0.5 GeV 3 . The agreement between the low-

PZ2=13m~'0 ~ c o r r e s p o n d i n g to the largest

9.3 GeV 3, in fairly good agreement with

est order p r e d i c t i o n and the e x p e r i m e n -

values of the v a r i a b l e s e x p e r i m e n t a l l y

tal value is worst for the n÷~+~-y decay

measured 7 .

width, where we have F(n÷~+~-y)=35 eV to Fex p C ~ ÷ ~ + ~ - Y ) = 5 3 ± 1 0 eV 14. Moreover,

the

o b s e r v e d photon energy s p e c t r u m is softer than the p r e d i c t e d one 15. The loop diagrams c o n t r i b u t i n g to y~÷ ~r°z are shown in Fig.

2, while only the

first three diagrams c o n t r i b u t e to ~+~+ ~-y. The result of the loop c a l c u l a t i o n is d i v e r g e n t in both cases/ but the di-

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Physica 96A

(1979)

3. J. Wess and B. Zumino, 37B (1971) 95.

Phys.

4. E. Witten,

B223

Nucl.

Phys.

327.

Lett.

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v e r g e n c e s can again be removed introducing the appropriate terms from Eq.

(6).

5. J. Bijnens, A. Bramon and F. Cornet, Z. Phys. C46 (1990) 599.

The finite part of the coefficients of 6. J. Bijnens, A. Bramon and F. Cornet, Phys. Rev. Lett. 61 (1988) 1453.

the O~p 6) terms in the lagrangian is again fixed from the radiative decays of v e c t o r mesons.

7. J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. 237B (1990) 488.

In this way we o b t a i n

F ( ~ ÷ ~ + ~ ' y ) = 4 7 eV at next to leading order,

thus improving the ageement with

the experimental data.

In addition,

the

loop diagrams and the O(p 6) tree level c o n t r i b u t i o n depend on the photon energy in such a way that the p r e d i c t e d energy s p e c t r u m is softer than the obtained at lowest order.

The c o r r e c t i o n to F 3'~ is

8. J.F. Donoghue, B.R. Holstein and Y.-C.R. Lin, Phys. Rev. Lett. 55 (.1985) 2766. 9. F. Gilman and R. Kauffmann, Phys. Rev. D36 (1987) 2761, A. Bramon, Phys. Lett. 51B (1974) 87. 10.G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321 C1989) 311.

complex and depends on the phase space 11. M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217, T. F u j i w a r a et al., Prog. Theor. Phys. 73 (1985) 926. ~ . R.I. D z h e l y a d i n et al., Phys. Lett. 94B (1980) 548. ~ . Y.N. Antipov et. al., (1987) 21. 14. Particle Data Group, (1988) i. FIGURE 2 One-loop diagrams for PPPy.

15. J.G. Layter et al., (1979) 2565, 1569

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