REFERtNUt
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
A NEW FAMILY OF SUM RULES
FROM CURRENT ALGEBRA
D. AMATI
R. JENGO
AND
E. REMIDDI
1966
PIAZZA OBERDAN
TRIESTE
IC/66/86
INTERNATIONAL ATOMIC E N E R G Y A G E N C Y
INTERNATIONAL C E N T R E F O R T H E O R E T I C A L PHYSICS
A N E W FAMILY O F S U M R U L E S F R O M C U R R E N T A L G E B R A
D.Amati , R.Jengo
and E.Remiddi
TRIESTE
July 1966
+ Submitted to "Physics Letters".
;
'CSRN, Geneva, Switzerland, and Istituto di Fisica Teorica dell'Universita, Trieste, Italy.
AA
Xstituto di Fisica Teorica dell'Universita, Trieste, Italy.
-1In this note we propose a new family of sum rules based on
the algebra of currents.
We discuss also some of its physical implica-
tions, deferring to another paper a thorough analysis of the derivation
and the results that can be obtained on their basis.
We start from the equal time commutation relation of (vector or
axial) currents
. •
(1)
where the indices
o(,
5 » Y refer to isospin (or SU ) and parity
transformation properties. The last term in the r.h^-s. is the Schwinger
QL
(i
., , „ .
_ O '
could be an operator. We will frequently
c«.,p
indices in order to simplify the notation.
drop the
Let us consider the matrix element of the commutator of two
vector currents between two states of momenta p
and p 3 and its Fourier
transform
and let us Introduce the following kinematical notations:.
t- +A1.,
Assuming for simplicity that the states
pa.-iicles with the same mass,
Vv -
a
U«v
|pV) and
|p,^ represent spinless
can be developed a s :
^ p v + a, £ Av +• a 3 ^ *CV +
Pv ^
-2-
2
2
w h e r e a., b . , c. , d a r e s c a l a r functions of s , t , k , k ? .
we write
r
Similarly,
F,r(t)Pv+ F/
, > -
We make no use of particular properties of currents as conservation,
time reversal invariance etc. Those properties
can give rise to restric-
tions on the invariant functions in eqs.(3) and ( 4 ) , i . e . ,
and a relation among the a ,
F2 ft) - 0
b., and c. in e q . ( 3 ) .
Two methods have been substantially used in order to derive sum
rules from commutators: the use of P—> oO
(2)
persion relation approach
,
frame of r e l . (1) and the dis-
With both these methods the general sum rule
that has been derived - and which we shall call Dashen-Gell-Mann-Fubini
(DG-MF) sum rules - can be written as
4-TT
This sum rule contains more particular c a s e s that have been proposed, for
(3)
instance, the A d l e r - W e i s b e r g e r
one when the c u r r e n t s in ( l ) a r e axial
2
2
(1 )
and k
- k = t - 0 ; the Cabibbo-Radicati sum rule when both currents are
vector,
k2=
K! = k l = k ,
t - 0
and deriving (5) with respect to k
at
O.(5)
In obtaining eqs.(5) one makes the assumption that the integrals
converge.
This can be phrased in different
but equivalent ways: that there
is a limited (or convergent) set of intermediate states that saturate the sum
rule or that the amplitude satisfies an unsubtracted dispersion relation. This
could happen of course for only some of the integrals of e q s . ( 5 ) -
Using methods very similar to those that allow the derivation of
the DG-MF sum rules, one can obtain several others that involve a differvariable of integration as well as different combinations of invariant
(XK.)
functions
'
•
•
• Of these new families of sum r u l e s w e obtain, w e want to
discuss h e r e one that, in o u r point of view, is of particular i n t e r e s t , i . e . ,
4-ir J
4TT
To visualize better the meaning of these sum rules we can saythat while the DG-MF sum rules can be obtained by introducing a complete
set of states between the two currents in the matrix element
< f i l [ j ^ ( | ) , j v t ' l ) ! pi) in a P
system, our sum rules come by
using an analogous procedure in the matrix element
<ptpj[ JL(f) , j y (- j)J IQ?-
2
2
By varying s at fixed values of k , k g and t In eqs.(5) we pick
up all the contributions of intermediate states in the s and u channels in the
commutator of e q . ( l ) , i.e., all the contributions of the form
We shall discuss in a subsequent paper the relation between the different
methods of deriving sum rules. We shall show that an equivalent and simple
way to obtain all the sum rules is to start from the Fourier transform of the
matrix: elements of e q . ( l ) . In this case one obtains (CG.Bollini and J . J .
Giambiagi, Universidad de Buenos Aires, preprint) :
= F,(t) P, + F.Ct) A v + Git) fC
This equation must hold for every frame of reference and for every K.
All the sum rules can be obi;a?.ned from it, in particular the ones of eqs.(5)
using a frame of reference in which P4 —> oO , and those of eqs.{6) for
a frame in which Aj —t oO . In both cases K must be zero, while the
independence of the r . h . s . on K and K implies the independence of the
r . h . s . of eqs.(5) on k^ and k , and of eqs,(6) on w and s.
-k-
(7)
1
where, as usual, a slashed intermediate line in a diagram indicates the
summation over spins of positive energy states and not the whole Feynraan
propagator. On the other hand, when in eqs.{6) we vary v, keeping fixed
2
2
s, t and w, we sweep all the contributions in k
and k , so that we pick
X
fc
up all the states that can connect either of the currents with the vacuum .
Diagrammatically, we pick up all the contributions of the form
- X" Of course, as in the DG-MF sum rules, we implicitly accepted
that only a rapidly convergent set of states would actually contribute. This
assumption is probably more justifiable in our sum rules that in the DG-MF
ones, the reason for this being that while in ours only the states with a
definite angular momentum can contribute (J - 1 due to the fact that we are
considering vector and axial currents), in the DG-MF sum rules the intermediate states can have any angular momentum.
Increasing s f more and
more angular momenta can in principle contribute.
We shall come back
later to the problem of the spin of intermediate states.
In eqs.(5) and (6) there are two variables in the l . h . s . which
do not appear in the r . h . s .
This implies that
\ b. ( s,t, v, w)dv is a cons-
tant with respect to s and w or, in other words, that it has no singularities
in those variables.
expressed by:
If, for instance, the singularities in s of b.(s,t,v,w)
are
(9)
t-n-*; **, " i
w
;
"?»!
i'_
<
then e q s . (6) imply
(10a)
^ ^ s ^ t , ^ vX/)dv = 0
and equally
(10b)
( |o, k t , v#ow)dnT = 0
for every value
of s,t and w.
The analogous relations
(ii)
()
obtainable from (5) have been investigated by Fubini et al. It is clear that
eqs. (10) and ( l l ) a r e not really related to the algebra of commuting cur
rents but contain only the local commutativity— microcausality conditions—of
the current operators.
E q . (10a) can be represented by the diagrammatical equation
for every state
the transition
|rvi^
. The bracket of e q . ( l 2 ) contains a form factor for
<pAl jy>lvw^
evaluated at
Jc, = \>C- V
tive part of a form factor for the transition <Vv\| j ^ | p^')
tinies the absorpat
Vl - 'nr+vc/p
minus the same expression in which the absorbitive part is shifted from the
first to the second form factor.
The integral over v of such an expression
must vanish by virtue of eq.(lOa) .
F o r instance, let us fix the particles 1 and 2 to be pions s and let
2
us consider s = m T in e q . ( 1 0 a ) , i.e., take the intermediate state \my in the
-6diagram (12) to be a pion.
If we call F
(k ) the pion form factor,
eq.(lOa) reads
(13) \ [I»F r (k 1 I )F T (k:) - Fr(kl)X»F,(fc;)Jdv = 0 .
Eci-(13) is of course automatically satisfied.
2
Let us now consider s ^ m ^ , i . e . ltfi> to be an GJ-meson.
Due to the spin 1 of the t*} , we shall have an extra momentum coming
from the sum over the u> spin components, so that we obtain, besides the
analog
of eq. (13) , trivially satisfied, the condition
J
L
2
where F ^
(k ) is the form factor for the Ui^TTV transition. Let us
write for F w ] r (k ) the representation
It is easy to verify that satisfying eq.(14) for every w implies
F
F
WTT
' WIT
where
Of course, we need that the integrals converge; we shall come back later
to this problem. In a pole model, in which F _,(fe*) would be given by
(16)
- 7 -
. .
. • - . . .
E q . ( l 6 ) implies that the sum of the coupling constants vanishes, i . e . ;
X. a, 'x - 0
\
•
Tlle
quantities F J>x
• .
<j
meaning in the asymptotic expansion of F
k
of eqs.(17) have a simple
2
(It*) for k —^> oO
.Indeed
k+
Choosing the intermediate state. \*my to have higher spin J,
other extra momenta enter into the game giving relations analogous to eq. (14)
with higher powers of v . If the sum rules obtained still Converge, more
and more coefficients of the asymptotic expansion must be zero. A simple
solution is F ^ 0 ' F^1' »
^FJJ' ^
0, which implies that
F7 (I:1)—> O(r^57i)
• Again we meet the requirement that the involved in-
tegrals converge. This would be of course the case if a limited set of
2
states saturates the sum rules. In this case, indeed, ImF_._ {It ) would be
2
*
different from zero only in a finite range of k . We note, however, that
the smaller J , the weaker the requirements on the convergence of the sum
rules of eq.(lOa) . Therefore it could happen that some of those a r e meaningful even if the whole sum rule (6) is not. This reflects the already stated
fact that the basis for eq.s (10) is quite wider than that for eq.s (6) .
We have, up to now, investigated eq.(lOa). It is easy to verify
that eq.(lOb) is automatically verified. The w dependence of the sum rule
appears indeed always in polynomial form and has been exploited in obtaining
eq.s (16).
It is amusing to remark the complementary way in which the angular momentum J of intermediate states appear in our and in the DG-MF sum rules.
2
2
Indeed while the k
and k
singularities involve states with a fixed J {1 for vec-
tor and axial currents) the s-singularities involve any J . Therefore in our sum
., . ,_
.states with/.a definite , , „..
.
, .u
.,
,
,
rules the mtegralruns over/J and tne study of s-singularities allows us to reach
(one by one) any J . In the DG-MF sum rules the situation is just opposite:
the integral runs over every J in the s-channel, while the study of k
2
k singularities involve only states with a definite J .
and
-8-
Let us now consider the case in which particles 1 and 2 are nucleons .
In this case the expansion of eq . (3} is no longer complete, due to the spin of the
external particles.
Correspondingly, the sum rules assume a different and more
complicated form , because of the Lorentz transformation properties of the Dirac
spinors .
The procedure described before
can be generalized by decompos-
ing eq . s(3 ) , (k) into invariants that include
y -matrices, and then consider-
ing the matrix elements of eq. (1) in the system
helicity.
A, —7 co
between states of definite
Again, the r . h . s . of the sum rules is free from singularities in s,
so that the singularities in s of the l . h . s . must be equated to zero. This con2
dition for s
m
(i. e . } nucleon intermediate state
[f^y in the diagram
(12) implies-that some integrals involving nucleon form factors must vanish.
Let us again consider vector currents, both isovector or isoscalar.
Defining, as usual,
<Filj,U»lp,> = i u . » [ F , f . - . i
F.o:
]
we obtain
(20)
from the coefficient of
from the coefficient of
[\ T,
in the expansion of
Z\
i>
and
Ft(k|) L. F, (Oj dv = 0
from
l\ A
A v v rf.
rf.
. Other invariants give linear combinations of e q . s ( 2 0 ) ,
(21) and ( 2 2 ) . Assuming for F
we obtain
(23)
, F
e x p r e s s i o n s analogous to e q . ( l 5 ) ,
-9-
where the F
l
are defined as in eq.s(17).
The condition F*
- 0 implies that if we analize F {k ) in the
polar form
the sum over the residua of the Pauli magnetic form factor must vanish, i . e . .
Z. <A • •= 0
• "W© know that this relation is v e r y well satisfied in all
analysis of the isovectdr o r isoscalar form factors
, nearly independently
from the m a s s of the second vector r e s o n a n c e which is always needed in the fits.
The most straightforward solution of the whole system
F(
F
0)_
(0)_
F
l
~ 2 ~
2
»
seems also compatible with experimental data (7) .
As discussed previously, thefactthat.the r . h . s . of eq.s (6) is a
function that depends on less variables than the integrand in the 1 .h. s. makes
use only in a very limited way of eq . (1) . In order to say more than that, we
must use explicit properties of the current matrix element (4) which appears in
the r . h . s . of eq.s (6) .
Apart of this information can be expressed in terms of the singularities
in t of the current matrix elements.
(6) to show the
For instance, we expect both sides of eq.s
0 -pole for t - mp . The equality of the residua at
larity corresponding to any state
the t-singu-
| %•> on both sides of the matrix element of
e q . ( l ) gives rise to an equation of the form
»j
We see therefore that the problem is reduced to a simpler one in which
one considers the matrix elements of the same commutator between that
particle and the vacuum.
(xc)
Let u s r e m a r k that in our c a s e the singularities lie in the region in which we
find o u r sum r u l e s - timelike A
- while for the D G - M F sum rules a c o m bination from the spacelike A region w a s needed. I n d e e d , the t-singularities
a p p e a r in the D G - M F sum rules in a m o r e unnatural w a y .
-10-
ACKNOWLEDGEMENT
One of the authors (E . R.) wishes to express his gratitude
to professors Abdus Salam and Paolo Budini for the hospitality
extended to him at the International Centre for Theoretical Physics,
Trieste.
REFERENCES
I) S.Fubini and G.Furlan,
Physics 1,, 229
(1965).
R.F.Dashen and M.G ell-Mann, CALT-68-65 (1966;
Coral Gables Conference) .
2), S.Fubini, G.Furlan and C.Rossetti, Nuovo Cimento
A, 1171 (1965) .
S.Fubini, Nuovo Cimento l^A , 4?5-(l966)
3) W.I.Weisberger, Phys.Rev.Letters 12t »
S.L.Adler,
Phys.Rev.Letters lit , 1051
lO2
f7 (1965).
(1965).
4) N.Cabibbo and L.A.Radicati, Physics Letters ,19., 697
(1966).
5) C.Bouchiat and Ph.Meyer, preprint Orsay Th/143 (1966).
F.Buccella, G.Veneziano and R.Gatto, Nuovo Cimento
A, 1019 (1966)
6) S.Fubini and G.Segre, Nuovo Cimento (to be published).
V. de Alfaro, S .Fubini, G. Furlan and C. Rossetti, Physics
Letters 2JL , 576 (1966)
7) Cf. for instance F.M. Pipkin - Proceedings of the Oxford
Conference on Elementary Particles (1965), page 6 1 .
t
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•••-".*?'•
^<H&-