SLAGm-298
March 1967
THE SATURATIONOF SUPERCONVERGENCE
RELATIONS AND
CURRENTALGEBRASUM RULES FOR FORWARDAMPLITUDES*
Frederick
California
Institute
J. Gilman
of Technology,
Pasadena, California
and
Haim Harari**
Stanford
Linear
Accelerator
Center,
Stanford
University,
Stanford,
California
ABSTRACT
We use the algebra
of charges and their
Regge high energy behavior
forward
amplitudes.
of states
stants
to derive
The saturation
is self-consistent
and masses.
Saturating
time derivatives,
sum rules
of all
for
sum rules
and leads to relations
all
sum rules
strong
PCAC, and
interaction
by a finite
number
among coupling
for or-p scattering
con-
by ~,LD
= 21 BeV-'
m = 1100 MeV, ma= m , PA = 120 MeV, g
wfi
Al
p
1
in good agreement with experiment.
and A1' we predict:
(Submitted
*
**
to Phys. Rev. Letters)
Work supported in part by the U. S. Atomic Energy Commission.
Pref
pared under Contract AT(ll-l)-68
or the San Francisco Operations
Office,
U. S. Atomic Energy Commission.
On leave of absence from the Weizmann Institute,
Rehovoth,
Israel.
-l-
A new set of strong
posed by de-Alfaro,
interaction
Fubini,
sum rules
Furlan
of certain
dispersion
of the form (2):
amplitudes
./ Im A(s,t)ds
Other sum rules
writing
for
strong
unsubtracted
set of all
process
sum rules
represents
if
particular,
butions
a significant
the sum rules
In this
and analyse
forward
their
If
all
while
the coupling
(b)
always find
a consistent
we can express
parameters.
all
they lead to sets of
of the involved
(4).
particles
of such sets of equations
restricting
ourselves
and current
only to
conclusions:
for
on a given hadron x are saturated
by states
forming
of the SU(2) X SU(2) algebra
in the masses and coupling
of charges,
constants
of zero pion mass.
All
obey the usual results
additional
states
intermediate
solution.
correspond
donstants
mass,
algebra. (5)
to the sum rules
However, the uniqueness
masses and coupling
These parameters
of the chiral
to contribute
the
has a unique
to have a common mass equal to the target
constants
If we allow
In
sum rules
in the limit
are then predicted
scattering
algebra
representation
states
The com-
by the contri-
We have reached the following
an irreducible
solution,
low
information.
saturated
solutions
properties,
of pions
non-trivial
constants
t=O superconvergence
set of equations
a given
resonances,
by
satisfying
and PCAC(3) .
way for
are approximately
algebraic
the scattering
complete
amplitudes
derived
amount of new dynamical
the possible
(t=O) amplitudes.
(a)
for
in this
in the masses and coupling
paper we discuss
the
(1)
of currents
of a small number of s-channel
equations
that
may lead to super-convergent
relations
obtained
who noticed
have been previously
energy theorems based on the algebra
plete
been pro-
= 0
amplitudes
dispersion
w
and Rossetti
high energy behavior
relations
has recently
is lost
we
and
in terms of a few free
to the mixing
coefficients
of the
-2-
additional
irreducible
(c)
representations
In the particular
n(,(o and Al intermediate
case of l<-p scattering
states
which agrees very well
with
We use the following
1.
The vector
2.
yields
vector
relations
and is therefore
3.
of Q$t)
= -i[Qi,H].
elements
for
all
isospin
vector
and helicity
current
amplitudes
saI(O
expression
meson trajectory
between the t-channel
with
helicities
iso-
of the
have C!,(O) < 0 O)(9),
here only the example of rr-p scattering
results
to the full
Tb,t
of the axial
of the leading
features
can be extended
processes
of the complete
the n-p
will
scattering
) = (sl.P)(s2-P)A(s,t)
which exhibits
set of sum rules.
SU(3) X SU(3) case or to other
such as n-C or n-N* scattering
of these calculations
We write
an I=2 piece
(7) .
most of the interesting
Details
does not include
at t=O is given by the Regge theory
I=2 trajectories
We discuss
tering
(2)
(81,
two x-particles
All
SU(2) X SU(2) algebra (6),
(PCAC).
where CZ(0) is the t=O intercept
I
spin I and i!h is the difference
5.
S(t)
The high energy behavior
for fl-x scattering
obey the
satisfy:
of the divergence
are dominated by the pion pole
4.
of the set of sum rules
= 'ij'(')
a pure isoscalar
The matrix
of the
charges Q1 and QG (i=1,2,3)
of the chiral
[Di('),Q~(')I
= -$ G(t)
the inclusion
set of assumptions:
The time derivatives
where Di(t)
a solution
to the sum rules.
experiment.
and axial
equal time commutation
which contribute
be presented
amplitude
without
Our
scat-
any difficulty.
elsewhere.
in the form 0):
+ ; [(e,*P)(e,.Q)+
(E,.P)(E;Q)]
B(s,
t) +
(3)
-i- (~,~Q)(~2~$)Cl(s,t)
I- (~l+)C2(s~t)
-3-
where P = $Pl+
P2), Q = $(91+ q2), P~,~ are the momenta of the initial
pion and ql 2 and el 2 are the momenta and polarizations
j
f
Regge theory predicts
that at high energies A (I)(s,t)a
p mesons.
of the
and final
B(I) (s,t)=
sa 1 (t)-1;
The only s-channel
Jf~(s,t)K
helicity
,9(t),
We find
that
where I is the t-channel
amplitudes
and M where the subscripts
00
,W)-2;
that
do not vanish
denote the s-channel
isospin.
at t=O are Ml1
helicities
of the p's.
at t=O:
M00= c2+ (v2- ‘,A
(4)
Ml1 = c2
where:
(5)
v =- PaQ
mP
The complete
list
of t=O sum rules
Two independent
(4
includes:
sum rules 00) :
Adler-Weisberger
O" Im M$V)dV
(7)
; /
where f II= 135 MeV is the decay constant
PCAC to satisfy
convergence
ffi= flGA%/gnN.
relation(')(l'),
w
Two independent
of the charged pion,
Eqs. (5)-(8)
Im A(l)(V)dV
Eqs. (lo),
pairs
these sum rules
-not an independent
V
by inserting
v22
the commutator
p mesons moving with
is guaranteed
if CX2(0) < 0 (9).
(5) and (6) lead t o an additional
supersum rule:
Im M$(V)dV
-m
h=l,O
by
(9)
(10)
(11) are derived
of helicity
lead to the I=1 n-p
= 0
V Im Mi:)(V)dV
VP-m 2
ll
predicted
(2) and Moo
(2) :
for Ml1
sum rules
=o
(8)
V2- rnz
which is therefore-
co
/0
0
8
p,= M .
=O
II
[D',Q~]
= 0 between
The convergence
Eqs. (10),(U)
superconvergence
01)
relation
together
of
with
(12):
m
/
V
0
Im Ac2)(V)dV
= 0
(12)
-4(c)
A superconvergence
of the form (1):
relation
co
Im Bc2)(v)dv
= 0
(13)
U
Strictly
speaking,
the B amplitude
mentally
Eq. (13) is not a pure t=O sum rule,
does not contribute
determined
to Ml1 or Moo.
only by extrapolating
We have used PCAC in deriving
sum rules.
We will
set of equations
convergence
therefore
this
limit.
tion
assumption
We find,
that
the overall
that
consistency
ext
rnn = 0 even if we consider
this
in the limit
currents
links
that
these relations
connection
between the algebra
and the strong
interaction
zero,
and we consider
it
to the
have to con-
by PCAC or by vector
meson
of weak and
sum rules.
whenever the pion appears as an intermediate
is not necessarily
of the satura-
the case, we clearly
implied
taking
only the supercon-
to the extent
is really
the super-
without
they do so only because of their
If
independent
of the complete
We realize
that
dominance which are the crucial
however,
mR= 0.
however,
our sum rules
electromagnetic
two of the five
study the self-consistency
This may mean that
of currents.
all
to t=O.
and (13) can be derived
give symmetry results,
sider
B(V,O) can be experi-
(9),(12)
requires
vergence relations.
algebra
at least
only in the limit
relations
B(V,t)
since a.t t=O
Notice,
state,
as an additional
its
mass
physical
quantity.
We now proceed to discuss
and (13) as our five
(8>,(lO>,(ll>,
different
saturation
interest
while
presumably
1.
the saturation
assumptions.
the third
yields
the five
the TI and 'u intermediate
independent
The first
of the real
independent
states.
We choose Eqs. (7),
sum rules
and study three
two are mostly
of theoretical
very good agreement with
a good approximation
Assume that
problem.
experiment
and is
are saturated
by
world.
t=O sum rules
The equations
are:
-514g:m
2
m
P
(74
=- 8
77
m;)g*
m
(,‘-
(84
f2
-0
WJr
22
-42 g
=o
mg
u! wpfi
PJrX
g
PJrfl
and g
OPfl
are defined
m =m =m
w
fi
g
as in Reference
1.
g2
UPfi
04)
P
(1%)
The unique solution
x-z-4g;m
2
8
05)
f,'
mP
2
is:
-1
Eq. (15) predicts
-e
= 5, g
= 21BeV
, to be compared with
wn
g2p3rn
2.4 -I- 0.2 as determined from the p width and gwpn= (17 If: 3) BeV-'
--GT=
as calculated
from the Gell-Mann-Sharp-Wagner
model (13) for cD-+ny.
While it
it
experiment,
have obtained
saturation
is clear
is still
that
Eqs. (lb),
interesting
such a solution.
assumption
(15) do not agree very well
to understand
In order
is equivalent
algebraically
to do so we notice
to assuming that,
with
why we
that
at infinite
tum, the h=l components of p and cu are in the ($,-$) representation
SU(2) X SU(2) while
the h=O p and J[ are in (1,O) + (0,l).
In this
our
momenof
case,
, which is a generator of the algebra, connects p
%
only to u) and JT. The matrix elements of the operator D1 between particle
the axial
states
charge
at infinite
lim
p-+m
Z
If
@Dip)
= 0
I I
mutation relation
according
for
momentum satisfy
pz(a,DIi
I
B) = - $* (mi - mE)(a:
(16)
i B)
IQ4
The comB) # 0, Eq. (16) leads to m =m
B QI'
(2) implies that the operators D1 and S transform
to the (-$,$) representation
any irreducible
(14):
representation
((k,t)[D+&)*
of SU(2) X SU(2).
Consequently,
(k,$):
= 0
(17)
-6-
We conclude
that
subscripts
p and w (or
representation,
x sup)
sup)
if
general
result:
states
forming
all
states
If we saturate
matrix
elements
are then predicted
rules
for
I=2 t-channel
the I=1 sum rules
we now allow
easily
This is actually
t=O sum rules
representation
of D vanish.
of SU(2) X SU(2), we find
The masses of all
amplitudes
become trivial
intermediate
w,
of the charge algebra.
finite,
The resulting
from those of the previous
(15) while
identities
predictions
representation
equations
can be
example by adding C+terms
The solution
in form to the (0 contributions.
8 defined
by
to be the same as the mass of x and the sum
a cp meson to contribute
free parameters
a much more
for r[-x scattering
to study the case of a reducible,
constructed
identical
all
lead to the ordinary
In order
2.
Eq. (16) then leads to the prediction
p,w and fi (Eq. (14)).
an irreducible
h=O) are in the same
= 0 and (p. D K) = 0 where the
I I
(%ID/"l)
denote the helicities.
of equal masses for
that
p and JI, for
depends on two
which can be chosen as the cp mass and an
arbitrary
angle
as:
g
-=(PPfl
g
The general
solution
m =m
TX
tan 0
08)
(J-w
is:
2
(19)
P
mP
2
2
= mzcos 8 -k m sin'@
(20)
= -8 cos2e
g2
(WJr f2
fl
(22)
(21)
cp
8
2
= -ij- sin 8
(23)
f17
We immediately
see that
in the limit
g
= 0, Eqs.
CpPfl
The "badtl predictions
(Eqs.
(19),(21))
for rnn and g
removed by the q since J[ and cp never contribute
(except
for Eq. (13) which is not independent,
(18),(20)
Pflfl
give mp= mm.
have not been
to the same sum rule
in this
approximation).
-7-
From the algebraic
stood in the following
point
of view the solution
The addition
way:
the h=l w and rp to orthogonal
tions
while
mixtures
and IX are classified
pl,oo
pl only to states
connects
of cp is equivalent
to assigning
We define:
as before.
(y>
= -sin
e/(0,0)>
representation
in the ($,$)
We therefore
p1 only to (0,O):
can be under-
of the (-$,$) and (0,O) representa-
I‘Ol>= ~0se((o,o)>* sin e/($,%)> (24)
%
(ly)-(23)
+ cos e[(-$,$)>
while
D connects
find:
(27)
(26)
Eq. (26) is identical
with
to (18) and leads to (21),(22).
The angle 6 that
(16) leads to Eq. (20).
in Eq. (18) is now interpreted
Its
(0,O) representations.
will
therefore
neglect
as the mixing
experimental
was arbitrarily
introduced
angle between the (-$,$) and
value
the contribution
Eq. (27) together
is close to zero,
of the (j in the following
and we
dis-
cussion.
3.
We finally
the A,(J'=
l',ICG=
couplings
g (p
L
-gT
2
!J-
consider
If-,
the contribution
of the next JC-p resonance,
There are two independent
m = 1080 MeV).
and we choose them as the longitudinal
(P*q)Px
(psq)qv1 (q)J
q*
P2
paqf3pa1q3tehep
mAl
and polarization
coupling
> ehep ' and the transverse
p@Yp~‘P’Y
of the Al(p).
' where p(q)
The five
Alps
and e(e')
sum rules
coupling
are the momentum
give:
2
g:P, +
vAl
2
8
gT = f2
(7b)
@ >
l-r
vu)g2
wpfi-
2
gT = 0
(lob)
h’ng;nn+
v3
A
--& g; = 0
mAl
(lib)
(25)
-82
2
where
V
X
2
= -$(mx- mz).
vAl
2
vAl 2 =
--i?-- gT
mA1
mAl
The general
mw =m
P
solution
(l3b >
0
is (17) :
of these equations
gT = 0
(28)
4g:m
8 cos211r
-=2
f:
(30)
(31)
mP
Al
-2-T
m~mAl
Eq.
(31)
2
2
8
= 2 sin $
gL
fTl
= 125 MeV give cos2+ = 0.48.
+m)
and P(P
(18):
(33) we predict
in remarkable
2
(33)
P
Inserting
this
in
mA = 1100 MeV
1
agreement with
A, decays mostly
mzcos29 + m2 sin2$ =m
Al
(32)
into
Eqs. (29), (32) predict
experiment.
longitudinal
p’s
(34)
that
the
and that:
‘v;:
1
rA =
1
evidence
tional
of the predictions
for
the validity
states
(28), (30),(34),(35)
experiment
small in all
The inclusion
and can be saturated
the equations
of addifor the qq
and improves
from its
experimental
width,
is
cases P-0) .
the set of sum rules
from a Regge'-type high energy behavior,
assumptions
as strong
at the expense of adding many new parameters.
of the A2, as calculated
We have shown that
specific
agreement with
can be regarded
of our set of assumptions.
us much more freedom in solving
the agreement with
relatively
The remarkable
such as A2 or the ,Jp= 1-I-,I CG= O-- meson predicted
L=l system allows
The contribution
(35)
3nf2m3
* Al
P = 130 2 40 MeV(19).
to be compared with
experiment
sin29 = 120 MeV
for forward
the algebra
amplitudes,
of charges,
obtained
PCAC and
on the absence of I=2 components,
is self-consistent
by any finite
resonances.
number of s-channel
The
-9-
general
solution
Additional
may depend on some unknown parameters
information
is required
meters and to predict
mediate
states.
from additional
Eq. (13),
plete
It
all
successful
results
that
that
this
information
values
used in this
at t=O.
we have obtained
of the inter-
can be obtained
similar
We believe
work is strongly
angles).
these para-
constants
at t # 0 PC , or from sum rules
based on extrapolated
set of assumptions
we want to determine
the masses and coupling
is possible
sum rules
if
(mixing
that
supported
for the saturation
to
the comby the
by fl,cc, and Al.
Footnotes
1.
V. de-Alfaro,
S. Fubini,
and References
G. Furlan
and C. Rossetti,
Phys. Letters
2,
576 (1966).
2.
This will
be the case if
at high energies
cases we may have stronger
3.
The Adler-Weisberger
Rev. Letters
14
-J
relations
sum rule
A(s,t)
In some
B < -1.
of the form 1 s" Im A(s,t)ds
is an obvious
1051 (1965),
a s',
example.
W. I. Weisberger,
= 0.
S. L. Adler,
Phys.
Phys. Rev. Letters
I&
1047 (1965).
4.
infinite
The complete,
be consistently
set of all
saturated
This does not imply,
by a finite
however,
that
a small number of resonances
finite
ceedings
5.
It
of the 4
is well
to the coupling
the saturation
constant
We find
among the particle
discussion
number of states
of an irreducible
that
for
of t cannot
states.
a more restricted
S. Fubini,
Pro-
1967, to be published.
of the charge commutator
of the chiral
which are usually
referred
sum rules
algebra
leads
to as "sJ(~)
as well
as some new relations
by a wider
set of sum rules.
these relations'
masses are predicted
by
of the case of infinite
see, e.g.
representation
relations
values
approximation
Coral Gables Conference,
known that
by the states
results".
th
all
we should abandon the saturation
as a valid
by an infinite
for
number of intermediate
For a general
set of sum rules.
sets saturated
sum rules
6.
M. Gell-Mann,
Phys. Rev. 125, 1067 (1962).
7.
Eq. (2) is valid
both in the a-model and the free
the conservation
of the vector
charges implies
quark model.
that
Notice
the commutator
that
in (2)
does not have an I=1 piece.
8.
9.
The high energy behavior
process
is slightly
Letters
-'17
of helicity
more complicated.
amplitudes
See, e.g.
for
a general
scattering
T. L. Trueman, Pnys. Rev.
11% (1966).
This assumption
has already
led to a few interesting
results
and no
contradiction
with
experiment.
The only direct
is the energy dependence of c~(~r-p -+T['N*-)
cross-section
for
this
process
zero above the resonance
is consistent
with
10.
with
I > 1, for
all
Phys. Letters
11.
S. Fubini
and G. Segre,
12.
Eq. (13) was first
of this
The poorly
known
and consistent
the present
with
experimental
of
any
523 (1965)'
S.
exchange
data
system
Phys. Letters
2,
-19,
344 (1966)'
Nuovo Cimento 3,
P. H. Frampton,
Gasiorowicz
to be
641 (1966).
proposed by F. E. Low, to be published in the proth
of the 13
conference on high energy physics, Berkeley'
1966.
M. Gell-Mann,
D. Sharp and W. G. Wagner, Phys. Rev, Letters
We use I'(w --+nr) = 1.2 + 0.3,
Sakurai,
test
processes.
in Nuovo Cimento.
13.
In fact,
the absence of the t-channel
published
ceedings
at small t.
seems to be negligible
region.
V. S. Mathur and L. K. Pandit,
and D. A. Geffen,
experimental
Phys. Rev. Letters
fz/4n
8, 261 (1962).
= 2.5 5 0.4 as determined
17, 1021 (1966).
by J. J.
The quoted errors
in our
value for
14.
g
do not include the errors introduced by the model.
wfl
The operator D was used by S. Fubini, G. Furlan and C. Rossetti,
Nuovo
Cimento 2,
1171 (1965) in deriving
V. de-Alfaro,
S. Fubini,
G. Furlan
SU(~) mass formulae.
and C. Rossetti,
See also,
to be published
in
Nuovo Cimento.
15.
Our analysis
(7)’
c8)J
(lo)
of the matrix
of Q and D applies to the sum rules
5
and (11) which are derived from the commutation relations
of the charges or their
these considerations.
is a linear
elements
derivatives.
We did not derive
Eq. (13) from
However, in the pure representation
combination
of the four
other
equations
case Eq. (13a)
and therefore
adds
no new information.
16.
This approximation
was first
considered
by the authors
of Ref. 1 for
the
sum rules
(Ref.
tent
(9) and (13).
They found that
12) remarked that
is g
solution
g(JJPR=
is obtained.
have discussed
g c&In=
In the limit
objection
mass) Low’s
0.
mext= 0 (and a free
3l
is not valid,
the saturation
(12)'
however,
Low
the only consisthe inhomo-
intermediate
and a consistent
University
of Fq. (9).
and consequently
pqM= 0.
This would contradict
R. Oehme and G. Venturi,
the I=2 sum rules
0 leads to g
when we add the sum rule
pm=
genous Eqs. (7),(8).
Vu=
pion
solution
of Chicago preprint,
They did not discuss
could not obtain
any of
any relations
between
masses.
17.
The commutator
relate
rule
18.
Vu,
sum rules
(7),(8),
(10 8) and (11) give Eqs. (30)-(33)
to gT in terms of an additional
(13) fixes
this
unknown parameter.
parameter
and predicts
vu=
m2
If we use cos2$ - i and neglect + we can rewrite
“P
This relation
from a different
was found by S. Weinberg'
set of assumptions,
0 (Eqs.
gT=
20.
et al.,
Rev. Mod. Phys. a.,
P. H. Frampton and J. C. Taylor,
1 (1967).
Oxford preprint.
(28),(29)).
who derived
the same numerical
mations.
1-9. A. H. Rosenfeld
The sum
(34) as mA = @mp.
1
to be published,
but using
and
it
approxi-