Nuclear Physics B60 (1973) 233 266. North-llolland Publishing Company
YANG-MILLS FIELDS
AND PSEUDOSCALAR MESON SCATTERING
D. I A G O L N I T Z E R , J. Z I N N - J U S T I N and J.B. Z U B E R ~
Service de Physique Thdorique, Centre d'Etudes Nucl~aires de Saclay ~$
Received 28 May 1973
Abstract: We have studied a model for pseudoscalar meson scattering (n, K, r~) based on a Lagran#an in which the mesons are coupled through massive Yang-Mills fields, identified with
the physical (o, K*, ~) vector meson resonances.
The calculation of the various two-body scattering amplitudes has been performed, in
perturbation, in the one loop approximation, and the Pad6 summation method has been applied
As a result .all the well-known d-wave resonances fo, f', A2 and K*(1420) have been generated. No exotic states have been found. The rrn and KK. s-wave phase shifts are in excellent
agreement with experimental data; in particular the strong KK threshold effect in the rrrr
scattering can be reproduced. Very broad o, K, 6 resonances are obtained. In the last section
it is shown that our model, which is not renormalizable, is a limit of a renormalizable model
containing Higgs-Kibble mesons which would lead to similar results.
1. Introduction
In a recent paper [1], it has been shown that it is possible to give a c o m p l e t e representation o f the low-energy ~rn scattering amplitude in the following way: one
starts with a Lagrangian in which the pion interaction is mediated by the p meson
considered as a massive Yang-Mills field [2]. C o m p u t i n g the 7rn scattering in the one
loop a p p r o x i m a t i o n and using the Pad6 s u m m a t i o n m e t h o d , one obtains an amplitude which reproduces all the low-energy pion scattering features: s-wave phase
shifts below 1 GeV, p-wave, and d-waves with an fo resonance in isospin zero as well
as an exotic behavior in isospin two.
It seems quite natural to exploit the same idea in order to c o m p u t e the o t h e r twobody (rr, K, r/) amplitudes, using the (p, K*, tp) as massive Yang-Mills fields. This has
been done in the present paper.
This m o d e l realizes the very attractive idea o f Sakurai [3, 4] that strong interactions are m e d i a t e d by v e c t o r mesons coupled to nearly conserved currents (I, Y, B).
:1:Attach6 de Recherches au C.N.R.S.
:1::1:Postal address: Bt' no. 2, 91190 Gif-sur-Yvette, I:rance.
D. lagolnitzer et aL, Yang-Mills fields
234
We have computed all the two body scattering amplitudes o f the (n, K, r/) system.
The Lagrangian that we used is the sum of a part invariant under local gauge transformation of the group SU(3) and of mass terms which break the SU(3) symmetry.
Actually, in order to avoid some technical difficulties, we have introduced different
masses for the vectors only through renormalization of tile one loop graphs.
One of the very nice features of the model is that tile forces generated by the exchange o f vector mesons are automatically repulsive in the exotic channels and attractive in the non-exotic ones. The p waves of the various two body (n, K, r~) amplitudes, corresponding in the exact SU(3) limit to the 8 a representation, are elementary in the model and contain poles corresponding to tile vector bosons (O, K*, ~0)
present in the Lagrangian. As is well known 14] the ratios of the widths of the vector
mesons are reasonably obtained in the model when one uses the physical masses for
the vectors and unitarizes the Born term by the K-matrix method. The [ 1, 1] Pad6
approximant, being unitary [5] will transform the double poles present in the perturbation series at the one loop approximation, into a simple pole in the second
sheet o f the amplitude [ 1]. The s- and d-waves are predictions of the model. In the
d-waves, in the non-exotic channels, one finds resonances with masses in reasonable
agreement with the various experimental 2+ resonances (fo, f', A2, K*(1420)). The
phase shifts in the exotic channels remain very small. In the s-waves one finds a set
of very broad resonances which can be identified with the not well established scalar
nonet. The scattering lengths, as in ref. [ 1 ], are in rough agreement with current algebra predictions. The phase shifts in the nn and rrK systems are in excellent agreement with the corresponding experimental results. In particular, we are able to reproduce the Kt( effect in the mr s-wave but this feature is rather sensitive to the exact
values of the parameters o f the model.
The paper is organized as follows: in sect. 2, we describe the Lagrangian and discuss some basic features o f the model. In sect. 3, we give a summary of the results
of ref. [ 1] as well as the results of a calculation in exact SU(3) symmetry in order to
show qualitative properties o f the model. Sect. 4 will be devoted to a brief analysis
of the Born terms. In sect. 5 we give all the numerical results. In sect. 6 we discuss
briefly a renormalizable 17] model using the Higgs-Kibble phenomenon [6] which we
believe, would yield very similar results and which furthermore takes into account
the co-op mixing. Sect. 7 contains our concluding remarks. The details of the calculation are given in appendix.
2. The Lagrangian
The Yang-Mills [2] Lagrangian is the sum of two parts G o and Z?B, where ~ o
is invariant under local gauge transformation of SU(3), and ./2 B is the sum of the
mass terms which breaks the symmetry.
.6?0 = --~ Tr{~uVV + ig[VU,V~]}2 + ~ Tr {BuS + ig[VU,S] }2
(2.1)
D. lagolnitzer et al., Yang-Mills fields
235
where S and V are 3 X 3 hermitian traceless matrices.
/2 o is invariant under the following infinitesimal transformations:
6 V u = OuH + ig [VU,H],
(2.2)
6S = ig [S,H] ,
where H ( x ) is also a 3 X 3 hermitian traceless matrix parametrizing the local SU(3)
gauge transformations.
~ B is of the form:
"~B = ~ ~.. [s01V~ [2 - 13i/ISi/121,
tl
(2.3)
where ~i/and 13# are real, symmetric matrices satisfying:
Otll =o~12 =o~22,
o~13 =o~23,
1311 = 1312 = /322 '
1313 = 1323 '
(2.4)
in order to preserve a SU(2) symmetry. V~ are a set of massive Yang-Mills fields and
Si! represent an octet of pseudoscalar boson fields.
At this stage arises the question of the renormalizability of such a Lagrangian.
Following the works of Feynman [8], Mandelstam [9], de Witt [ 10], Faddeev and
Popov [I1], 't Hooft and Veltman [12], Lee and Zinn-Justin [7], one knows that the
Lagrangian completely invariant under local gauge transformations, is renormalizable.
(This implies in particular that the Yang-Mills fields are massless unless the symmetry
is spontaneously broken.)
When the local gauge symmetry is broken by a mass term in the Lagrangian, one
has reasons to believe that the theory is no longer renormalizable [13].
Nevertheless it can be shown that, on the mass shell, a certain number of divergencies actually cancel [14] and the degree of divergencies of a graph G with L loops
can be bound by [15, 7]
6(G) < 21:.
(2.5)
If the local gauge invariance is only broken by a symmetric mass term for the
Yang-Mills fields, then the divergencies of the one loop graphs [13, 14] are the same
as in a renormalizable theory.
In the case of the Lagrangian (2.1), in the one loop approximation and for the
amplitudes which we compute, the same applies.
Nevertheless the non-renormalizability of the Lagrangian (2.1) will have the following consequences: the mass-term breaking the SU(3) will generate in the one
loop approximation more divergent terms (6 = 4) breaking the symmetry. In order
236
D. l a g o l n i t z e r et aL, Yang-Mills f i e l d s
to mininaize this effect we will only introduce different masses for tile vector mesons in the renormalization constants of the one-loop graphs. Nevertheless we will
have to introduce, in the s-waves, an independent subtraction constant for each
channel.
Other divergences in the vector propagator and pseudoscalar-pseudoscalar-vector
vertex can be removed by subtracting from the graphs the corresponding graphs o f
the symmetric theory [20].
So in this article we propose to consider the Lagrangian (2.1) rather as a phenomenologic',d Lagrangian and we sh',dl not try to define a consistent renormalization
scheme for the complete perturbation theory.
In sect. 6, we shall discuss a renormalizable Lagrangian which we believe to give
very similar results to the Lagrangian (2.1), and with less parameters.
3. Results for exact SU(2) and SU(3) symmetry
3.1. SU(2)
In ref. [ I ] the rrTrscattering amplitude is computed in the one-loop approximation
with the Lagranguan:
=-~(0 pv-~up ~+gpuX pv)2+~rn pP~
2-2
+ 2!t
~+~p~X~)
2
_
~m2~2
~
~
(3.1)
,
where p~ is an isospin one meson vector field which can be related to the physical
p resonance. For rnp = 0 this Lagrangian is invariant under local gauge transformation
of the group SU(2).
Let us write the Born term o f the rrTrscattering amplitude:
1=0
2G[ s-u + s - t ],
Lm2-t
p
I=1
I=2
m 2 -.u
G[2(t-u)+
s-u
[m2-s
p
m2-t
p
s-t
m-f--u]'
p
-G[S-U + s-u ],
Lm2_t
p
m2_u
(3.2)
where s, t, u are the usual Mandelstam variables, G =g2/16rr2 and the pion mass is
taken as unity.
A few remarks can be made about these expressions.
D. lagolnitzer et al., Yang-Mills fields
237
(i) One sees immediately that the fl)rces are attractive in isospm zero and one,
and repulsive in isospin two.
(ii) G and m 2 are fixed by the width I" and the mass m of the p resonance in isospin 1.
The unitarity relation being:
/r
hn t (s) = 5 [ s ~ ]
i
' It~0(s)12
(3.3)
a simple K-matrix unitarization gives:
G ~ 3
n
m21TM I - "
4)~
m'
(3.4)
(m 2 -
for 1-""-" 100 MeV this yields to G ~ 0.16 which is tile effective expansion parameter
near threshold.
(iii) The Born term vanishes at the Adler point s = t = u = 1 satisfying the Adler
self-consistency relation.
It is therefore interesting to see how current algebra constraints (Weinberg condition) are satisfied.
A simple calculation shows that the Weinberg condition is satisfied provided the
following relation holds:
g2f2
1
m2 - 3 '
(3.5)
where f~ is the pion decay constant.
With 1"= 100 MeV, m o = 760 MeV one obtains
3'n -~ 90 MeV,
(3.6)
which agrees nicely with the experimental value f,~ -~ 95 MeV.
This explains that this kind o f models are in rough agreement with the predictions
of current algebra.
It provides also a justification for subtracting the second order amplitude at the
Adler point s = t = u = 1.
The ratio o f the scattering lengths derived from the Born term is ao/a 2 = 2. The
Weinberg solution corresponds to ao/a 2 = - 3 . 5 .
(iv) Second order amplitude. As stated in sect. 3, the theory behaves in the oneloop approximation as a renormalizable one: only one subtraction is needed in order
to renormalize the scattering amplitude. This subtraction is fixed by demanding the
amplitude to vanish at the Adler point s = t = u = I. But because the p particle is unstable (rnp ~> 2m~r) another problem occurs. The second order amplitude has a double
pole near the p mass
238
D. lagolnitzer et al., Yang-Mills fields
A(r)(s,t,u)
A(s)(t
fors~-m 2.
u)
(3.7)
(s-- m 2)2
A ( s ) is complex near the pole, so one cannot impose on A ( s ) and its first derivative
that they vanish at s = m 2. In this case the most natural generalization of the usual
conditions is to set:
ReA(m2)= Re#I
OS
[s=m 2
A(s)=0.
(3.8)
By using these conditions, one keeps the values of the mass and tile width of the
resonance close to those obtained from the K-matrix approximation, computed with
the Born term.
The perturbative amplitude has a bad analytic structure near s = m 2. The p-wave
has tile following form:
s ~- m 2 ,
t(s) = ~
G + ~(s______~) G2 .
s-..m 2
(s-m2) 2
(3.9)
On the contrary the [1, 1] Pad6 approximant is unitary and behaves differently:
s~m2
'
t[l, ll(s) =
aG
s m 2 - (~(s)/~)G
(3.10)
/3(s) has a cut starting at s = 4 and t l l , l](s) has a simple pole in the second sheet of
the function. It is clear in this expression that, for G small, the conditions (3.8) imply that the pole is close to the pole o f tile K-matrix constructed from the Born
term.
(v) The results in the p-wave. As said before one finds a pole in the second sheet
of the J = I = 1 amplitude.
F o r m = 5.45 mzr and G = 0.17 one obtains:
340 ~ 760 M e V ,
I~ ~- 110 MeV.
(3.11)
Furthermore the phase shift is in good agreement with the experimental data [16].
(vi) The s-waves. The I = 0 phase shift as expected is positive and increases up to
80 ° showing a very broad resonance which can be identified with the o,
G = 0.17 ,
M o = 424 M e V ,
r o = 514 MeV.
(3.12)
The I = 2 phase shift is small and negative. Both are in good agreement below
1 GeV with the down solution o f the experimental data of ref. [16].
D. lagolnitzer et al., Yang-Mills fields"
239
Of course, in the I = 0 channel, the agreement breaks down near the KK threshold where strong inelastic effects have been observed.
The scattering lengths are:
a
O
= O. 1 8 m /r-1
a 2 =- 0.06 m ~ 1 ,
ao/a 2 . . . . 3 .
(3.13)
All these results are surprisingly similar with those obtained in the linear o-model
[ 17] which satisfies the constraints o f current algebra and PCAC.
One can speak here o f a real o - p reciprocal bootstrap.
(vii) The d-waves. In I = 0 one finds a broad resonance which can be identified
with the experimental resonance f o:
1260 MeV.
ml o .talc = 1280 MeV,
mf o exp
Pro talc = 260 MeV,
I'fo exp ~ 150 -+ 25 MeV.
(3.14)
The I = 2 phase shift is small and negative.
In conclusion one sees that in ref. [ 1] a global and consistent description o f tile
low energy 7r,r scattering amplitude has been given with two parameters fitted to the
experimental p-wave p resonance.
3.2. SU{3) symmetry {figs. 1, 2, 3)
The Lagrangian is o f the form (2.1)
= - ~ Tr{a u V v - ~ _ _ [V u, VU]} 2 + ~m 2 Tr V 2
+~ T r { a S
- ~ ig [ Vu,S]} 2 - ~/a2 T r S 2 ,
(3. i 5)
with the same conventions as in sect. 2. The calculations are very similar to those o f
the preceeding case.
Let us write down the Born term for the various (1, 8a, 8s, I 0 - 10, 27) pseudoscalar scattering amplitudes:
AI=3G[
s-u
m 2--
+ s-t ]
m 2- u
(t-u). 3 Gr s-u
A8a= 3Gm2 - s T ~ [-~-~-t
s-t ]
m2-~--u '
240
D. lagolnitzer et aL, Yang.Mills fields
I
60 (degrees)
80~| .
.
.
6°I
/
o
,
~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(i)
.
~ ( e , )
' 8~o 9ao 1~oo1~oo1~oo1~o---
-20~-
-z,O~
(271
Fig. 1. s-wave phase shifts in the case of exact SU(3) symmetry.
I 61(degrees)
te0~ ........
t..,
160t
~
18°1
Izo~
1°° t
80[- . . . . . .
[ J
0500
L-'''~600 7 0'0
_/L/
. . . . . . . . . . .
T
. . . . . . . . . .
I
EIMeV)
.
800
' 900
1 1000
, 1100
l 12
t O0
--'-
Fig. 2. p-wave phase shift in the case of exact SU(3) symmetry.
A8 s
~A1 ,
A I 0 =Ai-~ = 0 ,
A27 = --~A 1 .
13.16)
One notices immediately that, as expected, the forces are attractive in the channels ( 1, 8 a, 8s).
For the exotic channels, the amplitude vanishes in 10 and 10, and the forces are
repulsive in 27.
D. lagolnitzer et al., Yang-Mills fields
~
180~-
62(
degrees}
--~
............................
,~oF
'<°I
~
'~°~
'°°t
-'~-="--.
/
F
<','/
-/- . . . . . . . . . . . . . . . . l ......
.................
F
o, ~
_20t
241
1 .......
F
, , ~<~v,-
900 1000 1100 1200 1300 lt.O0 1500 16001700 ( 271
-
/
Fig. 3. d-wave phase shifts in the case o f exact SU(3) symmetry.
This can be compared with a simple ~,tp4 type model [ 18] in which the SU(3)
symmetry implies an 0 ( 8 ) symmetry so that all the channels have the same behaviour and (8s, 27) as well as (8a, 10, 10) are respectively degenerate.
Furthermore one can see that the Carlsonian extrapolation of 8 a and 8 s amplitudes are identical, leading for higher partial waves to an exact exchange degeneracy.
This phenomenon remains "also true for the second order amplitude. The (10, 10)
amplitude vanishes identically in the two first orders of the perturbation series.
Therefore in the case of an exact SU(3) symmetry, the Yang-Mills theory provides
us with a very attractive model.
The phase shifts are very small in the exotic channels (10, 10, 27); in the other
channels the forces are attractive generating resonances in the s- and d-waves of the
(1, 8s)channels. Furthermore the 8 a and 8 s are exactly degenerate. In the s-waves
the results are similar to those of the SU(3) X SU(3) linear o model in the limit of
exact SU(3) symmetry.
All these features are of course very encouraging, showing us that when the SU(3)
symmetry is broken by mass terms, the results will at least be in qualitative agreement with the experimental situation.
In figs. 1, 2, 3 we show the phase shifts for a particular set of parameters:
m = 1.83 rn n = 252 MeV,
M=5.9
mTr=812MeV,
G =g2/16n2 = 0.192.
D. lagolnitzer et aL, Yang-Mills fields
242
4. The Born terms
The complete set o f Born terms, as well as the second order amplitude is given in
the appendices. We will discuss here only some particular points.
4.1. The resonating p-waves (figs. 4, 5, 6)
nTr -+ nrr
I= 1
G'2(tu)+
.m2 s
s-.u
m2_~
P
rrK ~ 7rK
I =
P
F_t-. +(£! -.
(s-t) 7
m2-uJ '
P
m2?/m:l
l It-s+(la2-m2)/m2K*] + ( s - u ) }
KK~KK.
I=0
13(t-u)+3(s-u)+~(-SmiU) }
G (2m2- s 4 m 2 - t
-t '
~p
~0
(4.1)
p
where/1 is the pion mass, m the kaon mass and m ' will be used for the r~ mass, mp,
mK*, m~o are the masses of the p, K*, ~pvector mesons respectively. For the reasons
outlined in sect. 2, in the other sections we shall use in the Born terms the same mass
for the vectors mp = inK* = m~ = M. But because the vector ~p meson is extremely
sensitive to the ~o mass, we will first use first order amplitudes with different masses
in order to construct the K-matrix approximation. A short calculation [4] gives:
F
P
=4
G,
FK* = 1.4 G ,
F~o
=0.16G;
(4.2)
for G = 0.20 this yields:
r
pcalc
= 130
I-'K* calc=
talc
=
38
MeV
F p exp
MeV,
rK*exp = 5 0 +
4.5 MeV,
1-'~oexp
= 120 + 2 0 M e V ,
=
1.8
1 MeV
MeV ;
(4.3)
which is in reasonable agreement with the experimental data. When one takes into
account the (co, ~o) mixing [19] the agreement is even better, one gets:
1~ talc ~ 2.8 MeV.
D. lagolnitzer et aL, Yang-Mills fields
,8ot6_"_Ld'_g_r'_'_')
.
.
.
.
.
,,.o....-
,,op
.
.
/
.
.
.
.
.
.
.
.
243
.
p.
80 . . . . . . . . . . . . . . . . . . . . .
o Oi_
...... .,
MeV)
0
L
300
i
500
i
700
i
900
1100
-
Fig. 4. p-wave nrr phase shift obtained by K-matrix unitarization of the Born term. Experimental
data from Baton et al. [ 161.
61112 (degrees)
T
,~or . . . . . . . .
~
1~0~-
/4
120).
~ '
80
.
.
.
.
.
.
.
.
.
.
.
.
6O
~o ~
20
o
•
,.~,.
700
-,,//////~
~
,
900
/./,
,
,
1100
.
,'/,','////
,E,,~(M,V)
1300
//'/
Fig. 5. p-wave nK phase shift obtained by K-matrix unitarizadon of the Born term. Experimental
data from Mercer et al. [ 28].
244
D. lagolnitzer et al., Yang-Mills fields
~
180~
60
(degrees)
. . . . . . . . . . . . . . . . .
160
140
120
100
8O'-
'°foJ,
40
20
Ek-R(MeV}
|
o looo lolo lo2o lo3o lo~o ~ s o ~
i
I
Fig. 6. p-wave, I = 0, KK phase shift obtained by K-matrix unitarization of the Born term.
We have neglected here the (w, so) mixing but we shall return to this problem in
sect. 6.
If one gives the same value M ~- 770 MeV, to the vector boson masses, I'p and PK*
do not change very much, but clearly the so resonance becomes a bound state. The
subtraction constant of the one-loop amplitude will enable us to fit the so mass but
we expect some difficulties with the so width.
4.2. T h e s-waves
We can compute with the Born terms the nK scattering lengths. We shall compare
them to those of the rrTr system:
m l
a o ~ 87rG m~--~,
P
7rff ~
71"71"
112
a 2 ~- - 4 r i G -
;
m 2
p
a, ~
7rG I - 3 - ( m - / 2 ) 2 1 0a+m) 2 8/am]m
2 J
rn+/a
2 m2K + 2 m 2
*
'
nK ~ nK
rrG I - (/a+m)2
a~ ~ m~-'fi~
m2.
4/aml
m2 3"
(4.4)
D. lagolnitzer et al., Yang-Mills fields
245
For G ~ 0.2 the scattering lengths are:
a o ~ 0.16 m~ 1 ,
a 2 ~ 0.08 tn~ 1
.
a, -~0.13 m~ 1 ,
2
a_~ -~ - 0 . 1 3 m-~ 1
(4.5)
.
2
So we see that the nK scattering lengths are of the same order of mag~itude as
the nn scattering lengths, a[ is positive and a~ is negative and one expects the second
order to decrease somewhat la]l and to increase la~/a~l exactly as it does in the nn
system. One can therefore "also expect a broad K resonance in isospin ½.
For the other scattering lengths, no experimental data are of course available. In
our model for example the n'O ~ n~ scattering amplitude vanishes identically at this
order.
5. The results
5.1. The parameters
The masses of the n, K and r~ particles are known. The masses of the vector resonances p, K* and r/are also taken from experiment. The coupling constant G =
g2/16n2 is fixed within 20% by tile widths of vector mesons. Furthermore, as mentioned in sect. 2, in order to renormalize the theory we have to introduce an SU(3)
symmetric mass m o for the pseudoscalar octet and a mass M o for the vector octet.
A complete theory would not depend on these renormalization constants, but at
this very low order the results are sensitive to their values. We have fixed tile three
parameters G, m o and M o in order to have a reasonable p- and d-wave spectrum. Our
calculations have been made with:
G=0.2,
too= 1.83m,r~250MeV,
Mo=5.6m~r~-770MeV.
(5.1)
It remains now to fix the subtraction constants of the different amplitudes. The
crossing relations leave us with seven independent constants which can be taken as
the values at some points of seven different amplitudes. We have chosen the following amplitudes:
nn~nn,
nK ~ ~ K ,
I=0,
nK~nK,
~K ~ ~K,
I-3,-I
KK-~KI(
m7 ~ n ~ ,
(l=0orl=
lseelater)
rr0 -'- fro.
246
D. lagolnitzer et aL, Yang-Mills fields
Then the subtraction constants in the other channels are determined by the crossing
relations.
In ref. [ 1], in order to fix the nn subtraction constant, the Adler self-consistency
condition which is satisfied by the Born term is imposed on the second order amplitude. In this paper we have used the same procedure: the second order amplitude
has been subtracted at the off-mass-shell point:
s=t=u=m
2
.
?l
But this procedure cannot be applied on tile rrK ~ rrK amplitude because the 7rK
Born term does not satisfy exactly the Adler self-consistency condition. However,
as stated in ref. [ 1 ], one of the basic ideas of the model is that the contribution of
the vector meson poles dominate the s-waves at low energy. So in order to conserve
this property in tile one-loop approximation, it seems very natural to subtract from
the second order amplitude its value at threshold. In such a way, the second order
amplitude will bc small at low energy and the vector pole dominance will be maintained. Of course this statement is not very precise and so we will not be able to determine very accurately the s-wave scattering lengths.
This has been done for all the above mentioned channels, except tile KK. ~ KK
channel. For this channel, we have tried to exploit the relative arbitrariness of the
subtraction constants in order to study the strong KK threshold effect in the rrTr swave. We will come back more precisely to that point in the following.
5.2. The Padd sohttion {-
Itaving computed the various two-body scattering amplitudes in the one-loop approximation, we expand them in partial waves. For Re l > 1 we can use the Froissart-Gribov formula: the partial waves have a Carlsonian interpolation; for l = 0 and
I the partial waves do not belong to tile Carlsonian sequence and we have to modify
somewhat tile Froissart-Gribov formula in order to take into account the subtraction polyilomial. We shall now replace the perturbation series by the [ !, 1] Pad~ approximant. We will not give here any rationale for this approximation. More details
and properties can be found in ref. [5]. We recall only that tile analytic properties in
the energy variable o f the Pad6 solution are in general the same as those of the perturbation series, except that, for example, new poles can appear which are generated
by the Pad6 approximant. The diagonal Pad~ approxJmants of a unitary analytic matrix are unitary. The perturbation series amplitude satisfies exactly the crossing relations. This is no longer true for the Pad6 solution, but various tests have been made
[1,21 ] using Martin inequalities [22], B.N.R. sum rules [23] and dispersion relations,
showing that the Pad6 approximation is a unitarization procedure which preserves
the crossing properties numerically well (see also ref. [24]).
:1: Refs. [5, 18l.
247
D. lagolnitzer et al., Yang-Mills fields
In the present model, for I and isospin I fixed, the scattering amplitude is a matrix
(at most of dimension 3) in the space of the various channels. We have the choice between computing the [ 1, 1 ] Pad~ approximant of each particular matrix element, or
the Pad6 approximant of the whole matrix (called hereafter matrix P.A.). If the convergence is very good, one expects both methods to give very similar results. But for
a low order Pade approximant as the I 1, 1 ] approximant, this will not "always be true.
In this case one can give various arguments, in order to show that in general the approximant of the matrix is better. For instance only this approximant is unitary [5].
We will in general compute both, because in certain cases nevertheless the approximant of the matrix element has a better analytic structure. The (rrrr ~ 7rn) amplitude
for instance, computed from the matrix P.A., will have a spurious KI( left hand cut
produced by the coupling to the KK amplitude. Therefore, in the region of the KF,
left hand cut, it is not possible to use the matrix P.A. in order to compute the
(Trrr -~ nn) amplitude. We shall discuss this point in some details when we come to
the phase shifts. We give here the explicit form of the Pad6 approximant. The perturbation series scattering amplitude, for I and I fixed, is given by:
T = G T l + G2T2 ,
(5.2)
where T l and T 2 can be matrices in the coupled channels space. The Pade approximant is then:
T[I, I I = G T I [ T I - G T 2 ]
(5.3)
IT 1 .
We will now present the numerical results.
5. 3. The spectrum, table 1
In table 1 we show the set of resonances that we have obtained in various channels for the set of parameters:
m o = 1.83 mTr -~ 250 MeV,
M o = 5.6 mTr -~ 770 MeV,
G = 0.2.
(5.4)
All these resonances have been found as second sheet poles of the matrix P.A.
The experiment',ally well established 2 + nonet fro, f', K*(1420), A2) comes out
reasonably well. Except for the K*(1420) which is found too high in energy, the
masses are within 5% of the experimental value. The widths are in qualitative agreement with the experimental data except for the K*(1420) and the f'o which are too
narrow. Actually one observes that the K*(1420) and the f'o are nearly degenerate.
This is a reflection of the fact that these two resonances in this model, at this order,
are generated by the onset of the two-vector-threshold (at 1540 MeV). At this low
order of the perturbation expansion, the masses and the widths of the f'o and the K*
resonances are very sensitive to the vector mass M o.
No resonances are present in the exotic channels, which is a serious improvement
D. lagolnitzer et al., Yang-Millsfields
248
Table 1
Predicted resonances: due to present uncertainties, the values of experimental masses and
widths which axe reported here axe to be taken as indicative as fax as the 0 +nonet and some of
the 2+ mesons are concerned.
Mass (MeV)
Name
JP
Width (MeV)
IG
Experiment
.
.
.
.
.
.
.
.
.
.
.
.
.
Our model
.
.
.
.
.
.
.
e
0+
0+
500 to 750
460
lrN
S*
0+
0+
10+
975
- 1000
775
990
665
K
0+
~
-
0
K*
~o
111-
1+
l
i
0-
765
892
1020
fo
2+
0+
fo
2+
0+
A2
K*
2+
2+
11
5
1310
14t0
1332
1539
.
.
.
.
Experiment
.
.
.
.
.
.
.
Our model
.
> 100
675
58
20-.50
610 b
40 b
-
840
764
845
1022
125 ± 20
50
1.8
(partial width ,p---,KP,)
1270
1365
155 ± 25
1515
1536
73 ± 23
85
107
83,(130) a
52, (38) a
0, (4.5) a
165
8
143
5
(a) For 1-resonances, we have indicated the width calculated from Pad6 approximant, and, between brackets, from K-matrLx unitarization of the Born term.
(b) In t h e l = ! KK. channel, the position of the pole that we identify with the n N resonance depends strongly on the choice of subtraction parameters. Actually, when a sizable cusp effect
is obtained in the nrr s-wave, associated with the S* pole we list in this table, this ~rN pole is
located under the KK. threshold (see sect. 5).
over an o t h e r m o d e l [18] which used similar t e c h n i q u e s , but was based on a X,p4 type
Lagrangian.
As e m p h a s i z e d b e f o r e , it is difficult to i m p r o v e the results o f the p-waves with respect to those o b t a i n e d f r o m the K - m a t r i x . If one takes the same mass for the three
v e c t o r m e s o n s in the Born terms, it is possible to s u b t r a c t the s e c o n d o r d e r a m p l i t u d e
in such a way t h a t the [1, 1] Pad~ a p p r o x i m a n t s have poles at the physical masses o f
the v e c t o r mesons. The w i d t h s o f the p and K * ( 8 9 0 ) r e s o n a n c e s c o m e o u t c o r r e c t l y ,
but for the ~o r e s o n a n c e , the r e n o r m a l i z a t i o n e f f e c t s are t o o large and the ,p pole lies
m u c h t o o close to the real axis.
in the s-waves, a set o f very b r o a d o b j e c t s is f o u n d , f o r m i n g a scalar n o n e t . The
e x p e r i m e n t a l data are very imprecise e x c e p t for the rrTr and 7rK s-waves w h e r e s o m e
i n d i c a t i o n s can be f o u n d . We shall return to this p o i n t w h e n we discuss the phase
shifts.
5.4. The d-wave phase shifts
As s t a t e d b e f o r e , an attractive p r o p e r t y o f this m o d e l is the absence o f e x o t i c re-
249
19. lagolnitzer et aL, Yang-Mills fields
//
\,\
/
'\\
,/
/,1
i
Fig. 7. Argand diagram of the I --- 0 rr~rd-wave.
j
',
153/,
Fig. 8. Argand diagram of the I = 0 KK d-wave.
sonances in d-waves: the I = 2 7rn ~ n n , and I = 3 nK ~ nK phase shifts remain negative and very small (a few degrees) up to 1700 MeV. In the other channels, resonances appear and phase shifts have important structures.
5 . 4 . 1 . n n , Kg,, 7777,I = 0 c o u p l e d c h a n n e l s . The coupling between the r/r/system and
the m r and KK systems is very weak: the r f f / ~ r~r7 phase shift remains less than 1 degree and the amplitude nearly elastic up to 1800 MeV.
The coupling between the n n and KK channel is strong: figs. 7 and 8 show the
Argand diagram for the n n ~ n n , and Kg, -+ KK amplitudes, calculated from the matrix P.A.
One sees clearly the effects o f the two poles fo and f'o, on the amplitudes: the
coupling of the fo to the KK channel appears rather weak, whereas f'o is more coupled to KI( than to nTr. The Pad6 approximant of the matrix elements show less
structure: although the coupling between the fo and the KK resonance is larger, no
f'o resonance appears. The mr ~ n n amplitude behaves up to 1500 MeV like the coupled one.
5 . 4 . 2 . nK, r/K, I = ~ c o u p l e d c h a n n e l s . Here, the coupling between nK and r/K is
weak and tile r/K ~ 77K remain very small: it reaches - 3 ° at 1500 MeV.
Therefore the coupled and not coupled solutions are very similar (see fig. 9).
5 . 4 . 3 . nrT, KK, 1 = 1 c h a n n e l s . As mentioned above, the wr/~ nr/vanishes at first or-
der: therefore the only way to get a non-trivial rrr/-+ 7rr/amplitude is to calculate the
matrix Pad6 approximant. This amplitude has then a pole, which we can identify
with the A 2 resonance. This pole gives a strong structure to the KJ( and 7rr/phase
shifts, as can be shown on fig. 10; the A 2 resonance appears to be mainly coupled to
the wO channels. The physical A 2 resonance is strongly coupled to tile three pion
channel. Such a coupling does not exist in our model. We do not expect therefore
250
D. lagolnitzer et aL, Yang-Mills fields
//~\\,,\
/
1440
1190
1311
Fig. 9. Argand diagram of the I = 1 ~r'0(full
line) and Kg, (dashed line) d-waves.
Fig. 10. Argand diagram of the I = 1 nK
d-wave.
to be able to describe completely the situation in this channel. On the other hand,
the non-coupled Ki( amplitude remains very small and shows no structure.
5.5. The s-waves
In the s-waves, the experimental situation is nmch more complicated.
In the I = 0 nn channel, the well known "up-down" ambiguity seems now to be
eliminated, leaving the " d o w n " phase shift as tile only solution up to the 1 GeV region. No important 4n inelasticity is found up to this region, whereas there is some
evidence for a strong effect near the KY,, threshold [25, 27]. This effect could be explained by several dynamical mechanisms [30] but is generally interpreted as a cusp
phenomenon [25, 31]. In any case, it is associated with a second sheet pole, the socalled S* resonance, which would lie very close to the KI( threshold.
The I = 2 n n phase shift is now well known [ 16]. In the I = ~ ~'K channel, the
phase shift suffers an up-down ambiguity in the region of the K*(890) whereas the
I = ~, nK phase shift is small and negative [28, 29].
In the other channels, nearly nothing is known about the phase shifts. There are
some candidates nN, 8 for a I = 1 KK resonance at a mass of about 1 GeV.
5.5.1. Our m o d e l
We have seen above that we have several ways of chosing our subtraction constants: we fix the mr -~ mr subtraction by imposing Adler's condition. We subtract
the (~K -~ ~K) and the (~r/-~ nr/) second order amplitudes at threshold, in order to
preserve the vector dominance at low energy.
The main results are not very sensitive to the subtractions of the 7rK -~ nK,
7?K -~ ~K and W? -~ V~ amplitudes.
As far as the Ki( -~ KY-,amplitude is concerned, both isospin zero and isospin one
D. lagolnitzer et al.. Yang-Mills ]ields
t
251
50 (degrees)
/
10~
/
T l!
t" ~ . .
180
lO(]~
E}~n(MeV)
t
_~
11o~]
I
I
I
I
15o
t'
120
t
60-
>Y
-
i
30-
..'J
500
700
Enn(MeV)i
_
900
[
1100
[
J
-30
Fig. I 1. rrn s-wave / = 2 phase shift: full line. I = 0 phase shift and inelasticity calculated:
(a) from the Padd approximant of the matrix clement (dot-dashed line), (b) from the matrix
Padd approximant of rrrr, K[~ coupled channels (dashed line), (c) from the matrix Padd approximant of rr~, KK, rm coupled channels (dotted line). Data points: (.~ Baton et alo [ 16}, A and inelasticity: Protopopescu et al. 126].
amplitudes are large and it is impossible to subtract both at threshold. St) we can
choose to lower both amplitudes at threshold as much as possible, for instance by
subtracting the c o m b i n a t i o n A o + 3A 1. But we can also try to adjust the subtraction in order to reproduce the drastic effects in the coupled channels rrTr and KK,
o f isospin 0.
O f course, the results in the coupled channels KI~ and 7r~ o f isospin 1 and KK
and m7 o f isospin 0 will be very sensitive to those choices. On the contrary tile results concerning the nTr scattering at low energy or the 7rK phase shifts are unaffected.
5.5.1.1. L o w energy :rTr-amplitude. Our solution gives scattering lengths which are
slightly smaller than Weinberg's current algebra values but their ratio is the s a m e
D. lagolnitzer et al., Yang-Mills fields
252
ao=
O. 148m~r 1 ,
a 2 = - 0 . 0 4 3 rn~l ,
ao/a 2 = - 3 . 4 5 .
(5.5)
Tile phase shifts are, at low energy, in good agreement with the experimental data
(fig. I 1). At higher energies, the I = 0 phase shift reaches 70 ° without any drastic effect at the KK threshold, and the inelasticity remains small: the elasticity parameter
reaches 0.8 at about 1300 MeV. To the low energy I = 0 amplitude is associated a
very broad resonance that we identify with a o-resonance:
M o = 460 MeV,
F = 675 MeV.
These results are similar to those of ref. [ 1].
5.5.1.2. nK s-wave. The results in tile nK -+ nK s-wave show the same feature: tile
scattering lengths are somewhat too small with respect to experimental data and
Weinberg's values, but their ratio is correct:
a, = 0.127 m=!
a3 =
'
0.059 m~ 1
~
a,/a,_ = - 2 . 1 5 .
'
~
(5.6)
2
The phase shifts are in good agreement with experimental data of the "down" type
(fig. 14). Finally we find that the I = 21-amplitude has a second sheet pole which corresponds to a very broad resonance that we identify with the hypothetical K meson,
M = 665 MeV,
1-'K = 840 MeV.
At higher energies, the coupling of the r/K and nK channels does not modify significantly the 7rK -+ ~rK I = 1 phase shift (dashed line on fig. 14). The 7rK absorption
due to the r/K intermediate state is rather small: the elasticity parameter reaches 0.8
at 1300 MeV.
Let us now come to the results which depend on the choice of the subtraction of
the KK ~ KI( amplitude.
(i) If we choose to subtract the combination A o + 3A ! of isospin 0 and 1
KI( -+ KI( amplitudes, the I = 0 7rrr~ ~rn amplitude obtained from the matrix Pad6
approximant does not differ strongly from the non-coupled one: file phase shift
does not vary very rapidly in the KK threshold region, and remains at 73 ° above
1 GeV. The inelasticity is rather smail, as in tile non-coupled case.
The KK. 1 = 0 amplitude has a positive scattering length.
In the 7rr/and 1 -- 1 KI( channels, the KK ~ K~, amplitude has a negative scattering length, whereas the rrr/~ wr/phase shift has a " d o w n " feature similar to lrlr ~ 7fir
a n d r r K ~ l r K 1= I .
A very broad resonance is found with a mass of 775 MeV and a width of 610 MeV
(ii) We want now to show how we can choose the KK ~ KI( subtraction constant
in order to reproduce the strong anom',dy in the rrrt s-wave: we adapt here for the
Pad6 approximant an analysis which is usually done in terms of the K-matrix [30].
D. lagolnitzer et aL, Yang-Millsfields
253
In the vicinity of the KK. threshold, we take a two-channel lornlalism: let T 1 and
T 2 be the first and second order of the (symmetric) T-matrix:
7rTr-~ KK
KK KI~
(5.7)
,:(2 0)
and let
P2
be the phase-space matrix: with our conventions:
' ]/s-am2~
Pl = ~Tr[ - -
=n~
--4m2
02
(5.8)
As it is well known, the different determinations of Pl and ,02 define the different
sheets of the complex s-plane [30].
Now, in the physical region, along the real axis, the unitarity condition,
hn T 2 = T 1 p T l
(T 1 real symmetric),
(5.9)
tells us that the matrix T 2 - i T I pT 1 is real analytic. The same applied to the matrix
l T 2 T~ I +ip in terms of which the Padd approximant
M = T~ 1_ T(
P = T I [T l-- T2 ]-1 T l
(5.10)
reads
P= [M- ip] -1 ;
(5.11)
the matrix M plays the role of the inverse K-matrix in the usual approach. In order
to study the variation of the elements of the matrix P in the vicinity of p2 = 0, one
can now assume that M is a constant real matrix in this vicinity. One can then show
[30] that the 7rn ~ rrn amplitude P11 may have a rapid change around P2 = 0, depending on the values of the elements o f M: the amplitude then has a pole near the
KI( branching point" in sheet (1) reached from the physical region by going
through the elastic rr,r cut below the KK threshold, or in the sheet (12) reached
from the preceding one by going through the KK cut. The real part of the scattering
length of the KK ~ KI( amplitude is related to the position of the pole: if the pole
is in the first sheet, it is negative, and positive if it is in the sheet (12).
Our method is then simple: by varying the subtraction of the second-order KK.
amplitude, one subtracts some matrix from M. What we have done is to choose this
subtraction in order to get a pole of P ( t h a t is a zero of the determinant o f M - ip)
254
D. lagolnitzer et aL, Yang-Mills fields
1180~ \
/
lO O./
\
/
/
Fig. 12. Argand diagram of the I = 0 nzr s-wavecalculated from the matrix Pad~ approximant
of rrrr, KK. coupled channels.
in the vicinity of the K/~ branching point. In this way we are able to reproduce very
well tile experimental data.
The only problem comes from the fact that, by calculating coupled-channels
Pad6 approximants, we give to the nn amplitude a spurious KK left-hand cut. So we
are unable to get a coupled-channel rrTr amplitude below 950 MeV. But it is likely
that our elastic solution obtained by a simple Pad~ approximant is a good solution
at low energy, where the influence of the KF, channel is expected to be small and
that the coupled-channel solution is a better solution at higher energies, say above
the KK threshold, since in this region it has the right analytic properties and is unitary. The correct solution should be some interpolation between these two solutions.
Our best results are obtained when the pole is situated in the first sheet, corresponding to an S* "resonance"
MS, = 990 MeV,
PS* = 40 MeV.
Tile nn-phase shift (fig. I 1) is in good agreement with the experimental data of
Protopopescu et al. [26]. The nn inelasticity has a good behaviour near to KK
threshold. At a higher energies, above ! 050 MeV, our solution seems to be too elastic; then the effect o f the rt'q threshold is quite visible.
In order to correct this feature, we can try to incorporate the rt0 channel in a
three coupled channels formalism. In our model, at this order of perturbation theory, the r t r / ~ 770 amplitude has no contribution from nn intermediate states and so,
has no left-hand cut starting from s = 4 m 2 - 4 m 2._ So all our amplitudes have correct analytic properties in tile vicinity of the KK threshold, and the preceding analysis can be repeated: we can adjust our KI~ -~ KK, and r/r~ ~ ~ subtractions in order
to get a cusp phenomenon. It turns out that the results are very similar to those obtained with only rrn and Kk~ coupled channels, except that the nn inelasticity now
has a better behaviour (dotted line in fig. I !).
D. lagolnitzer et aL, Yang-Mills fields
255
<y~>
°2t!
O.l
i-tO
M(~+x-)MeV
9~o
-0.1
Fig. 13. <Y~) moment of the n + n
data from Protopopescu [ 26].
-
~
n + n --
reaction between 950 and 1030 MeV. Experimental
These results concerning inelasticity and phase shift are summarized in the Argand
diagram of fig. 12. We have also calculated the spherical harmonic moment (YI) for
the reaction n+Tr- -+ ~r+~r- , since the strong discontinuity of this moment is the most
striking effect of the cusp phenomenon. We have plotted (fig. 13) our results between 950 and 1030 MeV: we use, for calculating (Y1), s- and d-partial waves obtained by Pad~ approximant and p-partial wave obtained by K-matrix unitarization
of the Born term. As can be seen, we reproduce qualitatively the strong discontinuity
of (YI). We think that the lack of quantitative agreement does not come from our
1 = 0 s-wave but from our I = 2 s- and I = 1 p-waves: for instance, taking constant
elastic phase shifts o f - 3 0 ° for the p-wave and -25 ° for the I = 2 s-wave in the vicinity of 990 MeV would provide us with a moment (Y1) quite compatible with experimental data.
As far as the nr~ and ! = 1 KK channels are concerned we now get, with this choice o f
KK subtraction, repulsive KK. and nr/waves: KK has a strong negative (real part)
scattering length, and the wO phase shift remains at about 90 °. A pole is found in the
second sheet reached by going through KK. and rrr/cuts at M, = 570 MeV and 1-' =
75 MeV.
256
D. lagolnitzer et al., Yang-Mills fields
60 (degrees)
| |
a}
|
60
30
!
900
1100
1300
MeV
-30
[
'1[
°'
1 full line: Pad6 approximant of the matrix element;
Fig. 14. nK s-wave shift: (a) isospin 3:
dashed line: matrix Pad6 approximant of nK and r~K coupled channels, (b) isospin 3: experimental data from Mercer et al. [ 28].
5.5.1.3. Crossing properties. We have tested how well our Pad6 amplitudes are compatible with the crossing conditions, using Roskies relations [ 2 1 - 2 3 ] . We have written ,all
• the relation involving s-, p- and d-waves o f the lrTr~ nn, 7rTr-~ KK, rrK -+ 7rK,
KI~ -+ KK reactions. The crossing equations have been found to be satisfied within a
few percent: this reflects the fact that, inside the Mandelstam triangle, the second
order amplitude is much smaller than the Born term, so that the Pad6 approximant
does not differ substantially from the perturbation series which satisfies crossing exactly [5, 24].
As a conclusion, we have obtained very good results for the spectroscopy and
phase shifts of all the meson-meson channels, in s-, p- and d-waves. As expected by
studying the Born term, the p-waves have been reproduced with good agreement.
Then all the d-wave spectroscopy has been predicted without any exotic resonance.
Finally in the s-wave, we have been able to reproduce all the experimental results.
This model, of course, is not renormalizable but as we shall see, it is very easy to
construct a renormalizable model, using the Higgs Kibble mechanism, which would
yield similar results.
6. A renormalizable m o d e l *
Using a general argument given in ref. [7], one can construct a Lagrangian using
the Higgs-Kibble phenomenon, which is renormalizable, and will certainly give similar results to Lagrangian (2.1).
In order, furthermore, to introduce a ( ~ , ~o) mixing angle, necessary to explain
* ReL [33].
D. lagolnitzer et aL, Yang-Millsfields
257
the vector mesons mass spectrum [19], we will use a xnodel with broken U(3) symmetry:
O = . ~Tr{O u v v - O v V u + i g ' [ V u , w ] } 2
+ ~ Tr{auS + i g ' [ V , S l } 2 .. ~rn 2 T r S 2 - - ] ~ , I T r S 2 ] 2
M + i g ' V U M ] 10 M + - - i g ' M +V u]}
+~Tr([O
½TrAM+M
(6.1)
- ~ b(Tr MM+) 2 - ] c Tr(M+M) 2 - ~ d Tr [MM+S 21 - ~ h Tr M+M Tr S 2 ,
where V u, S and M are 3 X 3 complex matrices
V u = (VU) +,
S =S + ,
TrS = 0,
(6.2)
and
a
0
A--
a
,
0
a'
with a, a' real. g ' is related to g, the coupling constant used in tile preceding sections
by g = - g ' x ~ .
The Lagrangian is invariant under the group U(3) L × U(3) R where U(3) L is a
local gauge symmetry and U(3) R is an ordinary symmetry broken by Tr AM+M.
Vu is the set of nine gauge bosons associated with U(3) L. The transformation
laws of the different fields are:
M
--
)
U 1 MU~2
S
o
,
U1SU~I ,
(6.3)
where U 1 and U 2 are two unitary 3 X 3 matrices. The U(3) R symmetry is broken
by T r A M + M to a SU(2) symmetry.
The U(3) L symmetry will be broken spontaneously by the vacuum expectation
value o f M
{0[M]0) = F ,
with
258
D. lagolnitzer et aL, Yang.Mills fields
!
F =
0
v
0
0
0
v'
V'
and v,
real.
(6.4)
A result, provided in ref. 17] shows that the theory we obtain in this way is a
theory with nine massive Yang-Mills fields associated with a broken symmetry U(3)
and in which the nine would-be Goldstone bosons can be eliminated by a redefinition of the fields. These bosons have been used to give a third degree of freedom to
the massless gauge bosons in order to transform them into massive vector bosons.
Physically the Vu fields correspond to the (p, K*, ~0, w) resonances, S is the octet
of pseudoscalar mesons. The nine remaining components of M are the Higgs-Kibble
scalar mesons. The physical interpretation of these scalar mesons is more difficult,
but because their masses are arbitrary, they can be made very massive, or alternatively one can consider tile theory as tile regularization of the theory obtained by taking
the infinite mass limit. In ref. [32], this has been already done by considering the
linear o-model as the regularization of the non-linear o-model. In what follows we
shall "always assume that these masses are very high so that in the discussion of the
Born terms we shall be able to replace M by F.
First nevertheless let us write the equation for F in order to show that F can be
chosen in the form indicated in (6.4)
522 M:F,
= 0 -* A + b T r F 2 + c F 2 = 0 .
8M
vU=o, s=o
(6.5)
For any value of F o n e can certainly choose A to satisfy eq. (6.5).
The discussion of the quantization, renormalization and unitarity of tile theory
generated by the Lagrangian (6.1) can be found in ref. [7].
Let us now discuss the tree approximation of the theory. For this purpose we replace M by F and obtain an equivalent Lagrangian at this order
22 = - ~ Tr{~ u V - ~v V + i g ' [ V u, V v] }2 + ~g,2 T r ( V 2 F 2)
+ i1 T r { O u S + i g , [ V , S ] } 2
- :, m o2 T r S 2 - ½ d T r F 2 S 2
•
(6.6)
We have also omitted two terms (Tr $2) 2 and T r F 2 T r S 2 which play no role in the
tree approximation.
The masses of the vector are given by Tr V 2 F 2,
m 2 =m 2 =g'202
mi 2] = x~(vi2 + v2)g'2 ~
m 2 . : / g , 2 ( o 2 +v,2)
k m ~p=g
2
,2 V,2 .
(6.7)
259
D. lagolnitzer et al., Yang-Mills fields
The mixing angle 0 between ,p and co is given by
I
tg 0 - x / 2 '
0 ~- 35 ° .
(6.8)
Here a comment is necessary. In Lagrangian (6.1), in the spirit of SU(6) we have assumed that tile universal coupling to tile SU(3) and U(I) parts of U(3) are the same.
This is not necessary and we could find a better mass spectrum by introducing two
different coupling constants. But it is certainly reasonable to attribute corrections to
formula (6.7) to second order effects.
The masses of pseudoscalar mesons satisfy the Gell-Mann Okubo fornmla and are
given by:
m 2 = do 2 + m 2
71"
: }d( o2 +
(6.9)
m ,7
2 = -~d(o
,
2 + 2u,2) + m ~ .
Another SU(6) prediction gives:
d = g ' 2 ~ m 2K
2"
m n2 = m 2 , _ m p
(6.10)
Using this relation, for a typical set of parameters we get:
mp
= m w =760MeV
F
P
= 100
MeV
MeV
g,2
= 13
mK, = 900 MeV
I'K, = 31
o2
= 2.3 m 2
m ~o
= 1020 MeV
F~0
o'2/o 2 = 1.75
m IT
=
mo2
mK
= 490 MeV
m
= 570 MeV.
=
29m 2
=
2.1 MeV.
140 MeV
(6.11)
This set of values certainly provides a very nice starting point for a theory which
will very likely improve a little the results of the model studied in this paper without
nevertheless changing them too much. At the same time, for this new theory all the
problems related to renormalization can be solved. We propose to study this model
in greater detail in a forthcoming paper in order to investigate the eventual changes
that it can bring in our results.
7. C o n c l u s i o n
We summarize what we have done in this paper. We have studied a model based
D. lagolnitzer et al., Yang-Millsfields
260
on a massive Yang-Mills Lagrangian, in which the p, K* and ~0vector mesons are used
as gauge fields: these vector meson forces have been shown to generate correctly all
the d-wave resonances. In the s-waves, we have been able to reproduce tile experimental data available, such as the low-energy behaviour of rrTr and nK scattering and
the drastic S* effect at 990 MeV. This model satisfies unitarity exactly and with good
precision the crossing constraints.
It is based on a non-renormalizable Lagrangian. Therefore we have also given another Lagrangian, using the Higgs-Kibble phenomenon, which is renormalizable, takes
into account the co ¢ mixing, and would lead to numerically similar results.
We want to thank D. Levy, A. Morel and J.L. Basdevant for very interesting discussions.
Appendix A. Born terms
We list hereafter the Born terms of the amplitudes between two-pseudoscalar ( 0 - )
states and two-vector ( 1- ) or pseudoscalar states.
A.1. 0 - ( r o t ) + O-(m2) -~ O-(m3) + 0-(m4) amplitudes
Let us define the function
X1234 (s, t, u, MI2, M~)
t - u +(m~-m2)(m2-m2)/M~
+
s-u
+ 2
2
2
2 2
(ml-m3)(m2-m4)/g 2
M2-s
M~-t
X1234(s, t, u, M 2, M 2) is the sum of the diagrams of fig. 15a.
Note that the combination X1234(s, t,u,M~,M~) + X1243(s,u, t,M~,M~) does
not depend on M2: in the following, we have omitted the fourth argument of the
functions X when they enter such a combination (for instance, in the lrr/~ KK, amplitude).
The different amplitudes are expressed in table 2, in terms of these functions X,
(we omit the overall factor G).
A.2. O - ( m l ) + O--(rn2) '+ I-(M3) + 1-(M 4) amplitudes
Let us call q, q', k, k' the four momenta of the two pseudoscalar and vector mesons. We define the function:
X~rTrKK( s ' t ' u ' p ' K * ) +
(m/~rrn)
=0,
I:1.
,K*)+X
= - g Xr~Kr/K(S, u, t, K*, K*),
(wO'--,KK) =-V/~[Xnr~KK(s , t , u ,
(r/K ~ rtK)
3
XTrK,rK(S't'u'K*'o)+
= 43-X K,TK(S, u, t, K*, K*) ,
(nK--*rrK)
(,,K ~ ,~K)
_~
I~
(i)
(3
'
I = 1.
XnKK,r(LU't'K*'K*)
,K*)],
i~o
{I= I'
1
3
[1=~
'
_
(i)
'P)'
,~o
{1 = 1"
X K K K K (s'u't'
XKKKK(S,t,u,p,p),
'P)+
(~)
XKKKK(S, t,u,
(3
~"
I = 0.
XKKKK(S, u, t, , so) +
KK(S , u , t ,
I = ~.
I= ~ .
XKI~I~Iz(s, t, u, , so) +
el) (')
+
(KK ~ KK)
0.
- 3x/2- [XKK.0r~(S, t, u, , K*) + XKKr~r~(s, u, t, , K*)],
(KY. --* nn)
I=0.
0,
=2
=1.
XKKKK(S, t, u, O, so) +
XTrnKK(S'u't'o'K*)'
(~,~1
XnTr~rn(s
, u, t, p, O),
[XKKKK(S, t, u, so, p) + XKKKK(S, t, u, SO,SO)]+
(~)
Xnlrmr(S, t, u, O, O) +
(mr -~ r/~)
(KK --* KK)
0rn ~ KK)
(/171" "-'¢"71"if)
Table 2
1"
2=o~
{I
2
D. lagolnitzer et aL. Yang-Millsfields"
262
'1
~m 2-
o)
'1
"m~
b)
t~
q,rnl
q',mz/
k,M3,B
"1. k' Ml.,v
q', rn 2
q' m 2-
k', M/. ,v
• k',H4, v
Fig. 15. First order diagrams contributing to: (a) 0 - + O- --* 0- + 0 - amplitudes;
(b) O- + O- --* 1 - + 1 - amplitudes.
uv
,
,
Yl.~34(q,q,k,k,M
2
, m 2) = -
(2q-- k)U(2q'-.-k') v
"
m 2
+g~V
t
__ 1 I(q,_q)(k_k,)g~U +(q_q,)U(2k+k,)V +(q,_q)U(2k,+k)U
M 2- s t
+ (m}M 2m2) [(M2__.M~)gUV+kUk~,__k,Uk,U]}
Graphically, Y1234
u~ is the sum of the diagrams drawn in fig. 15b.
Due to the symmetry between the pseudoscalar- and file vector-meson octets:
n ,-' p, K ~ K*, 77 ,-, ¢, any amplitude 0- + 0 -~ I - + 1- is obtained from the corresponding 0 - + 0 - ~ 0 - + 0 - anaplitude by replacing each X1234(s , t, u, M~, M22)
by Y1234
uv (q, q ' , k, k ,' M~,p2) and X1243 (s, u,t,M~,M2) by Y1243
v. (q, q ,' k ,, k,
M~, ~u2).
For instance, the nK ~ p K * amplitude is
(nK-~pK*) =
()
-I
"nKpK*tqq KK , K*, n) +
()
-i
I rKK,p(qq K K, K*, K).
Appendix B. Second order calculation
We introduce now a function ~P1234,uu,(s, t) which is the sum of the contributions
of the one-loop graphs drawn in fig. 16. The function ~P1234,vu,(s, t) is defined by its
absorptive parts in the s and t channels. For the sake of simplicity, we write them in
the case of equal vector meson masses.
D. lagolnitzer et al., Yang-Millsfields
263
t
Wile r'l
"WIg
Fig. 16. Second order diagrams contributing to 0- ~ 0- ~ 0- + 0- amplitudes.
B. 1. s-channel absorptive part
lm s ~o1234,tz ,(s, t) = X(t) Y(t) lm s B1234($, t,/.t2,/a '2 , M 2, M 2)
+ [ 4 t - ~(m3+m4)-/a
3
2
2
,2 +(m 21 - # 2 )(m3-la'2)/M2+(m2-112)(m2-/a'2)/M
2
2
+ ( m l2- t i 2 ) ( m 22 - u 2 )(m 2+
3 m 42 - 2/.t,2)/2M41X Im s WI2(S, la2,M2,M2 )
+ [m 1 o m 3 , m 2 *+ m4,/l*+/a' ] X
lm s
W34(s, gt'2, M2,M2)
+ [2 + 2(m21__m2)(m2._ m4)/M2 2s + 'a(m12+m2_2 21a2)(m2 +m2_ 21.i,2)/M4 ]
X lms i (s, M2, M2)__ I13s +_~OM2] Ims l (S' M2' M2)
(M 2 - s)~
4Psqs cosO s
4 Im s I1234(s, M 2, M 2)
M 2- s
4Psq s cosO s
M2- s
4(m~- m2)(m
2
23 - m4)
2
Ims l(s, M 2 , M 2)
4Psq s cos0 s -- _
M2 s
m22) +
2sM 2
X (s+4/~2-2m 2 - 2m2)] X lms WI2($,/~2,M2,M2 )
D. lagolnitzer et al., Yang-Millsfields
264
+ [m I ~ m3 ' m2 ~ m4 '/1 ~./1,] hn s W34(s,/1,2, M 2, M 2)
4Psq s cos0 s
4q2(M 2 - s)
- lms
[(/12+ ~( s - m 2 --m 2- 2M2)) lm s 14/12(s,/12, M 2, M 2)
l(s, M 2, M2)]
X [4s - 3 m l 2- 3 m 2 +2 M 2 -2/1 2+ ( m l2- / 1 )2( m 2-/12)1M2]
2
4Psq s cos0 s
- [m I ~ m 2 , m 3 ~m4,/1"+/1' ] .
4p2(M2-s)
B.2. t-channel absorptive part
Im t ~p1234,uu,(s, t) = [X(t) Y(t) Im t B1234(s, t,/12,/1,2, M 2, M 2)
- X(t) Im t W13(t, M 2,/12,/1,2) ._ Y(t) Imt W24(t, M 2,/12,/1,2)+Im t i(t,/12,/1,2)]
+
/ (2M2+ t - m12- m2-/12_/1,2) im t W13(t, M 2,/12,/1,2)_ 2 lmt I(t,/12,/1,2)
4q2(M 2- t)
X X(t) 4Pt qt c°sOt- [X(t) lm t 14'13 - I m t I(t,/12,/1,2)] (/12 _/1,2)(m 2_ m2)/tM 2}
4Ptq t c°sOt]
where B1234(s , t, p2,/1,2, M 2, M2), W12(s, M 2,/12,/1,2), l(s, /12, /1,2) are the scalar
box, triangle and bubble diagrams and:
X(t) = 2t +M 2 - m 2 - m ~ - /12_ /1,2 +(m~_p2)(m~_/1,2)/M2 ,
Y(t) = 2 t + M 2 - m 22 - m 42_ /12 /1,2+(m2._p2)(m2_/1,2)/M2
qs,(Ps) and qt(Pt, kt) are the c.m. momenta of tile particles of masses ml, m 2,
(m3,m4) and m l , m 3 ( m 2 , m 4 and p,p'). For instance
qt = ~ [ ( t - (m I - m3 )2)(t- (ml +m3)2)/t] ~
Lastly 0 s and 0 t are the s- and t-channel c.m. scattering angles:
-
l i)
_1
0
3
3
~rK,rK,K~(S, t) +
3
(o)
(
0
(i)
9)~Ortrl(U,t,
+ _
9
~Prtrl(u't)+
g / ~o~rnKK,K~(t, u) +
(o)
~pnKnK,K~(U, t)
(~)
[~OnKK~
r ~rK(U, s) + ~OnKKn,nK(S, u)] .
~o~rnKK,K~(t, s) +
SOKK(t, u)
(~)
(_~)~onrl(s,t)+(~)SOKK(t,s,+(~)s%n(t,s)
[~OnKK.arK(t, u) + ¢TrKK,r,.K(U, t)l
[¢~rKnK,~r~r(U, t) + ~o~r~rKK,rrK(t, u)]
~ / ~OnnKK,K~r(t, s) ÷
t~
[~O~rKnK,~.r(S, t) + ~%~rKK,nK(t, s)] +
9
+ \ g ! ~OunKK,Kn(t, u) +
+
~
I)
(9)
2 ~%n(t, s) + 30 cnn( t, s) + (. - ~s).~ p n n ( u , t , ÷
(;)
()
(~)
+
3
[~OK~(U, t) + ~OK~(t , u)] .
2
I~)
[~onrrK~.,Klr(s, t) + ~OTrKrrK,KK(t, s)]
2
l
+ [¢lrTr(t, u) + ~Onlr(U, t)]
[~OK~(S,u) + ~OK~(U, s)] +
()(i)
[~OlrnKl~,~rK(S,
t) + ~OTrKnK,nlr(t, s)] +
[~OK~(S,t) + ~OK~(t , s)] +
3
+ [~OTrTr(s,u) + ~O~rlr(U,s)]
+(10 _ ~ ) × [the same terms where t .-. u] .
(KK --+ KK) = ( ~ )
(nK ~ ~rK) =
~
0
ti)
+ [3x/'6/16~
(~0/4)
+] ] [¢Tr,rK?.,Kn(S, t) + ~O~rK~rK,K~(t, s)] +
+
3
= [~olrrr(s, t) + ~Omr(t, s)]
(n~r ~ KK.) =
(ffff -'~ fiR)
Table 3
D. lagolnitzer et aL, Yang-Mills fields
266
4Psqs cosO s : t - u + ( m l - r n ~ ) ( m 2
m2)/s.
As explained in refs. [ 1, 18], we then c o m p u t e the c o n t r i b u t i o n o f these diagrams to
the partial-wave amplitudes by using the Froissart-Gribov formula.
We give in table 3 the weights o f the second-order contributions only for the main
channels: for the sake o f simplicity, we o m i t the overall G 2 factor and, for the
nn -~ nn and KK, ~ KI~ reactions the indices I, 2, 3, 4 o f the ~0 functions.
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