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The phenomenological current algebra vertex
OF PHYSICS
59, 129-164 (1970)
The Phenomenological
Current
Algebra
Vertex
J. SCHNITZER
HOWARD
Department of Physics,* Brandeis University, Waltham, Massachusetts 02154 and Rockefeller lJnioersity,t New York, New York 10021 AND MICHAEL
L. WISE
Department of Physics,* Brandeis Unicersity, Waltham, Massachusetts 02154 and Znstitute for Theoretical Physics, * State University of New York at Stony Brook, Stony Brook, New York 11790 Received February 9, 1970
A method is presented for constructing amplitudes which are the most general solutions to the Ward identities of current algebra. These solutions, expressed in terms of undetermined scalar functions, exhibit a wide class of amplitudes which satisfy the current algebra constraints. Further information is obtained from the model dependent equal-time commutators of the space components of the currents by means of the Bjorken-Johnson-Low limits, which provide boundary conditions for certain asymptotic expansions of the scalar functions. Simple phenomenological models are discussed within this framework, and a connection between the Gilman-Harari model and hard pion models is established.
I. INTRODUCTION
Current algebra [l] has proven to be an extremely useful tool in hadron physics, particularly as a method of generating threshold theorems and sum rules. The basic input is a postulated algebra of chiral currents involving equal-time commutators of the form [ J2(x), Jb”( y)] 8(x0 - y”) = icab, J:(x) a4(x - y). * Research supported in part by the AEC under contract AT(30-1) 3178. + Address 1969-70: Research supported in part by the Air Force Otlice of Scientific Research under Grant AF-AFOSR-69-1629. * Present Address: Research supported in part by the AEC under contract AT(30-1) 3668 B.
129 595/59/r-9
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SCHNITZER
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It is however obvious that current algebra does not provide the foundation for a complete theory of the hadrons and their interactions with weak and electromagnetic currents, although it does provide useful constraint on the hadron system, exhibited by means of Ward-Takahashi type identities [2,3]. If one examines a typical Ward identity, -iq,,
s
d4x esal’z(f 1 T{.Iap(x) Jav(0)} [ i) = k&f
1Jcv(0) 1i)
say,
one immediately observes that the relation gives no information for those terms on the left side of the equation proportional to (qlzguy - qluqlv), since they are transverse to qlu . This is also true in more general cases. Since one frequently wishes to apply current algebra to a domain in which threshold theorems are no longer applicable, new procedures must be worked out. In these cases current algebra provides constraints which may be thought of as analogous to the requirement of gauge invariance in the electrodynamics of hadrons, although the current algebra constraints involve a non-Abelian gauge group, and some of the currents are only partially conserved. This paper is addressed to the question of the construction of an explicit representation of hadron amplitudes which satisfy the Ward identities. Since these representations involve considerable arbitrariness, additional dynamical information is incorporated by making assumptions about the commutators and the commutators of the time-derivatives of the currents, as well as phenomenological assumptions as to the smoothness of the amplitude in momentum space. It has been observed that even these highly model dependent and subjective conditions are not sufficient to specify a theory, since there remains additional arbitrariness in the amplitudes. A related problem is the incorporation of the requirements of unitarity into these schemes, which certainly will provide additional information. It is the purpose of this paper to present a method for exhibiting the most general solution of the Ward identities of current algebra [4],l where a solution is a set of amplitudes which automatically satisfy these identities. A further property of such amplitudes is that they are expressed in terms of a number of scalar functions which are not subject to current algebra constraints, and hence undetermined without further assumptions. At this point there is a clear need for additional principles which we have not been able to provide in a convincing way. Hopefully one will be able to formulate some dynamical equations for these undetermined functions so as to satisfy unitarity, which is an important missing 1 Preliminary results are reported in Ref. [4].
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CURRENT ALGEBRA
ingredient in current algebra theories. As a provisional step we reexamine some model dependent results of current algebra, such as the hard pion models [3], and models defined by certain equaltime commutators of the space components of currents, so as to gain further insight into their validity and domain of applicability. Here we limit the detailed discussion to the three-point functions of currents, since the basic techniques can easily be extended to the n-point functions. We first treat the vertex of three-vector currents, and then examine the somewhat more complicated caseof two axial-vector currents and a vector current. As an interesting byproduct of this work, we are able to establish a connection between the model of Gilman and Harari [6] for the A,pr and pm- vertices, constructed from a study of sum rules, and the models for these vertices constructed from Ward identities. An essentialingredient in the Gilman-Harari model seemsto be that the divergence of the axial-vector current, t3,A”, is not the canonical pion field, which is consistent with their picture of chiral representation mixing. II. THE THREE-VECTOR
CURRENT
VERTEX
Let us begin our discussion with an analysis of the vertex of three conserved isotopic vector currents. The basic machinery for study of the current algebra vertex is a set of Ward identities, (W.I.), derived from a postulated equal-time commutation rule
W,“(x) ,vbQ)l %x0- y”) = kbc V/(x) S4(x- y) + 0.(x, y)$ 6(x0- J-O),(II. 1) where Vu”(x) is the conserved isotopic vector current with isospin a, and the Oy(x, y)$ are additional noncovariant terms or “Schwinger terms” (ST.) [6]. The nature of the (ST.), which is generally determined from an underlying field theoretical model, must be known in order to construct Ward identities. Although anomalous situations are known to occur [7], none have been discovered in the sectors we study in this paper (8). Thus in (11.1)we assumethat the S.T., OV(x, y)$, vanishes for p = 0. Let us briefly review the properties of the relevant Ward identities before proceeding to their solution. One defines [9] - ddx e-ia.r J
(0 I T{vaqx)
V,“(O)} IO> = -iS,,[Ov(q)“’
- g~Og%y],
(11.1)
together with the spectral representation (11.2)
= (g”” - mW)
d Y(q) + cY4”4”/42,
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which has the properties, Cy = 1 dm2pv(mz)/m2,
(11.3)
and Ll;;l(q)‘1y = A “(q)-l
(g”” - q”q”/qZ) + c;1quq”/q2.
We also find it convenient to use the notation gT(q)up = g” - q”q”/q2. Throughout the body of the text we assume that Cy < co, while in Appendix A we consider the modifications necessary to include models in which Cy diverges. The three-point function we wish to consider is
f d4x d4y d4z e-i*~‘“e-ia~‘Ye-‘a~‘“
(11.4)
which has the crossing property T(q1 , qz)IIYA = -T(q2
The (WI.)
satisfies by the time-ordered
, qPA
(11.5)
= --T(q1 > qPy-
product is determined
from (11.1) and
(11.4) to be [13]
%boqluT(ql 9 kJuyA = %bCM &2)vA-
s s
A &)v”l
d4x d4y e--iQl%--yo
1 T{sy%“(X,
d4x d4y e--ial.z e--ia@(o 1 T{Pw”(X,
y&
6(x0 - y”) V,“(O)} I 0)
O)!, 6(x0) V,y(y)> 1o>.
(11.6)
Although the noncovariant terms from the propagators cancel, noncovariant terms survive in (11.6) if the (S.T.) has an I = 1 component, connecting the vector current to the vacuum. One defines a covariant current correlation function (or T* product) M(q, , q2)“yA by means of the definition T(q, 3 qP
= M(q1 99P
+ ml
, qP,
(11.7)
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where the so-called “seagull” term S(q, , q2)0~~is noncovariant. As it stands, the separation given by (11.7) is not unique, since one can always add a covariant piece to PA and similarly subtract it from M wvA.To be specific we assume (or must prove in model field theories) that q,,S(q, , q2)uvAcancels the noncovariant terms in the (W.I.), [ll], [12] (II.6).2 In this case the T* product satisfies
qlLLwql >q2Y
=
LJ Y(92)“A
-
(11.8)
Ll Y(q3)“A,
as well as two analogous identities obtained from (11.8) by applying crossing. If the (ST.) is a c-number, or if the (S.T.) has no Z = 1 piece, then the T-product satisfies the same (WI.) as the T* product. A. Solution of the Ward Identity It is highly desirable to be able to construct amplitudes that automatically satisfy all the current algebra constraints, since one can then impose further requirements on the undetermined portions of the amplitudes without violating the Ward identities. It is easy to solve Eq. (11.8) for M(q, , q2)MyA,i.e., ml
2 q2YYA= $
v y(q2)VA- d y(dAi
+ gdqd””
4ql , q2)? ,
(11.9)
where A(q, , q2)p”Ais an arbitrary third-rank tensor. By computing q2”M(ql , q2)rr”A and comparing with the appropriate (WI.), one can similarly solve for (ql , q2)L’! , and substitute the result into (11.9). Finally by contracting ~T(ql)uu’ A the resulting expression for M(q, , q2)U”Awith qzA, and comparing with the relevant (W.I.), one obtains a representation for M UVlthat satisfies all three Ward identities. namely,
+ dzlP’
g&2Y”’
e-(qP’
d F&l)
d dq2)
d “(93)
R!l
7 qlL*“Y
7
(II. 10) where F(ql y q2LvA is an undetermined third-rank tensor. The factors of d.(q) in the last term are separated from FaVI\ for convenience in later considerations; this 2 For a recent account, from which earlier references may be traced, see [l 1).
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in no way restricts the generality of our representation. for F,,,,,, consistent with crossing symmetry is
The most general form
%tl , q2)wh = iLdq1 - 42M&12, 422; 4a2) + g,dq, - qlM&?12, h2; + g,dq2 - q2Lm22, 435 412) + (42 - q2)u (41 - %I” (42 - dh.fx912~ q22s 42% wheref, and
422)
(II. 11)
f2 satisfy the crossing relations, fl(4127 922; ch2) = h(92,412;
422)
and
(II. 12) h(412,
q22,
412)
=
“Mq22,
q12,
qs2)
=
f2Gh2Y
h2,
422).
The semicolon in fi reminds us that the symmetry is only in the first two variables. Thus the current algebra constraints on MUy,, can be satisfied with any choice of two scalar functions which satisfy crossing. Further restrictions on the scalar functions must come from dynamical considerations. In our construction we have used the projection operator,
which induces singularities at qi2 = 0 (i = 1,2, 3) unless fi and f2 satisfy additional boundary conditions. Since there are no zero mass hadrons, these kinematical singularities must be removed from each independent covariant amplitude. For example, the coefficient of guYqsh, V.(d
-
h2
~v(q2)l
+
(412
-
422)
h2
d dd
d y(92) d ymuqlf,
q22; 92),
(11.13)
must remain finite as qs2--+ 0. One finds similar requirements for all the invariant amplitudes of MLIVh. The removal of these kinematic singularities is facilitated by the definition of two new scalar functions. The first is
We require that f3(q12,qz; 0) be finite, and thus remove the unwanted singularity from (11.13). Notice that since
f3k12,422; fls2)= f2(q22, 412i 422)9 it retains the symmetry of fi . We also define a totally symmetric function f4(4129q22, h2) =
h(q22,
q12, 423 = hkl12Y h2,
422)
(II. 15)
(II. 16)
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CURRENT
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ALGEBRA
in terms offi and fi by the equation,
q22)fi(q12, q2?h2)+ 2 termscyclicin (qlqPqp)i +i t43i l ,q12~~4q~~~2~~ 422(412 + 42)
-1
41242Y412-
d y(q2)-1 + cyclic terms .
d y(ql)-l
432)(422 - h2)
(11.17)
f
If f, satisfies the boundary condition,
f3(q12,422;
0)
i-.f4(4123
q22> 0)
=
(11.18)
03
then Eqs. (11.13)-(11.18) insure that M,,yA contains no kinematical singularities as qi2 + 0, as can be verified by a straightforward, but tedious calculation. One can now rewrite F(q, , q2)uvAin terms of scalar amplitudes free of kinematical singularities.
ml
The result is (A
q2)Ld
=
; gdq1
-
42)*
[&Y&h27
422;
e2)
+
&1)-l
d VCP
--d
q12 _
dq2H qr]
+ cyclic terms in (CL,V, A; ql , q2 , q3)/
+ (42-
q2L
+ ; [(a2+
(41
-
q3)v
(f2(q12’
(42
422;
-
(1 . &4Gll”,
41h
422)
422Y
-f2(q22’
422;
h2)
ql”))
+
cyclic
422) 412
-
4e2
1 d v(W - d&P + (1 tt 3) + (2 ~ 3) [ - 5 (422 - 422)(412 - q27
II’
1 (11.19)
Thus Eqs. (IT.lO), (11.18) and (11.19) give the most general solution of the Ward identities free of kinematical singularities. In summary any scalar functions f, and f4, with the correct crossing properties, which are kinematical singularity free, and related by (11.18), will give a solution to the current algebra equations. Further information concerning& and f4 must come from additional dynamical considerations. The construction of phenomenological current algebra models is often discussed in terms of amputated proper functions, Pv(ql , q2)uyA defined by Mb,
9 q2Y = Ll “(qp
d
y(q2)yy’
f4 v(43Y
r&441
9 4d,,“Y
*
(11.20)
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Then rUYh is simply
+ G1&-(ql)“”g&Y’ gdq2Yml , 92h’“YA number of models involve the single-particle
approximation
[2, 3, 12, 131
where it is assumed that the p-meson dominates the two-point function. The parameter g, describes the coupling of the p-meson to the isovector current according to the normalization
where @(q, h) is the p-meson polarization vector for helicity h. Since there are no simple ways to infer the detailed structure of the vertex, an alternative is to make some plausible assumptions about the behavior of the amputated proper functions. The proper functions r,,yA have the pole structures of the external legs of the vertex removed, so that one may hope that in the low and medium energy (and mass) regions the proper amplitudes are slowly varying functions of all their variables. In this spirit one may assume that Fv(ql , qJuvh has the minimal momentum dependence consistent with the (W.I.), i.e., is at most linear in the momenta [3, 12, 131. When this assumption is combined with Eqs. (II.lO), (II. 18), (11.19), (11.21), and (11.22), one finds that
MJ121922;
h2)
=
-fkh2,
q22,
422)
=
&
P
=
-
$z
P
3
(11.24)
which is equivalent to the original hard meson model, r&l,
q2PA =
&w%2
- 41Y + g”“(q2 - 42)” + t?%l
- qd%
(11.25)
Since these models have at best a very intuitive basis, one wishes to connect them with more precise criteria. In the next section we investigate the role played by the equal-time commutators of the space components of the currents in constraining the vertex.
PHENOMENOLOGICAL
CURRENT
B. Model Dependent Commutators
137
ALGEBRA
and the Bjorken Limit
The Ward identities, and their solution only depend on the equal-time commutation relations of the time-time and time-space components of the currents; no use is made of the commutators of the space-space components or of the time-derivatives of the currents. This is reflected in the fact that the (W.I.) determine the longitudinal parts of amplitudes, but the transverse parts (Eq. 11.19, e.g.) are left unconstrained. One possible way to make use of this additional information is by means of the limit studied by Bjorken and Johnson and Low [14], which relates the high-energy behavior of time-ordered products to equal-time commutators. If one accepts the Bjorken limit as correct,3 then one can make some interesting general statements, as well as offer an interpretation of several phenomenological models. To be specific, consider %bcT(%
, q2
; mp3)uy
=
-
=-ilim
d4xe-iq1+(0 1 T{ V,ll(x) I/by(O)} 1pc(q3 ; CT))
’
J
ho2 ____-
q32)
Q32’mp2d/(277)” 2q,”
g;kbcT(ql
dq3 ; 0)
, qduvA
026)
The heuristic limit of Bjorken [14] implies the formal asymptotic expansion
---f - 5
1 d4x e-iql’e(O
1 iva”(x),
. . + (4:“)2 ~
J
v,“(O)]
d4x e-igl’~(O 1 [a,V,“(x),
s(x”)
~b”(o)l
1 k-&3
s(xo)
; O);
1 P&3
; O>i
-t
‘*.
.
(11.27)
In order to explore the consequences of this “folk theorem” for our general solution, we must first substitute Eqs. (11.7) and (11.10) into (11.26), with the resuIt [41 m,
, q2 ; %2)u” = -i +
go r”q2” lim Q&mp2 d(279 2q,o I[ 422 ml
3 92
; %v”
-
WI1
3 q2
g”w 412 ; %“)““(?
w12y%A 412422 I
493
; a) (11.28)
3 Although this may not be correct in perturbation theory, it still may be valid in the real world. We consider it a reasonable working hypothesis.
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where we have defined ml,
q2 ; %2P
=
*J+t
pG7l)
4q2)
EA(qs; a)F(qx, q&//g,
gT(qP'gT(qB)YY'
0 and 8%
42
2
; %2Y”
=
,p;
* g;2(%2
-
432)
ml
3 &PA
4q3
; 4.
(11.29)
I) Since the Bjorken limit has only been studied carefully for the time-ordered product of two operators, we must accept the loss of information inherent in the mass-shell limit of (11.28). It is straightforward to apply the Bjorken limit to (11.28), keeping all terms to 0(l/q,02) in the asymptotic expansion, with the result [4]
Toi BiTie -
~-d(243
+ limPTq12,422;q‘9°011,
(11.30a)
-1 EJ + 0 qlo
(11.30b)
ig, I[ 2q,o
(--$-)I + lim[F”jl/,
ig, -- -1. ES+ - 1 q3v + 0 (--$)I B5 d(2~7)~ 2q,’ [ qlo hlo)2
+ lim[P”]/,
(11.30~)
and
k
Tij 7
--= d/(2%93 2q,o
-
I[
1 (q2V - qlV) + 0 (+)I (q10)2
+ lim[P
- S”j]/. (11.3Od)
The reader should have no difficulty with the abbreviated notation in Eqs. (11.30b)d). Observe that the first bracketed terms in Eqs. (11.30) are consistent with the vacuum-one p-meson matrix elements of the equal-time commutators of the algebra of fields model, [15] (A.F.), which have the commutation rules
84(x - y), Vao(x>,VbG)l Go - u”) = kd, VcJqx)64(x- y) + G,,S~jCya~ [VaYx), VtxY)l %x0 - v”> = 0,
(11.3la) (11.3lb)
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and
P,~,W, ~AY)I &x0 - y”) = &be[V,i(x) aiY(x - y) + a”(V,j(x) 64(x - y))] -- ko2 2 eaefcbehV+(x) V,j(x) a4(.x - y) + c numbers. 4
(II.3 fc)
This is in fact a general result. Namely that part of any current algebra amplitude which is completely determined by the Ward identities [in the sense of (11.10) or (11.28) in our case), has the same Bjorken limit as the minimally coupled, massive Yang-Mills theory, i.e., it has the same limit as the algebra of fields [4]. The implications of this observation for the study of radiative corrections and other applications have been discussed elsewhere [ 161. Let us now examine the properties of the model dependent terms and their limit li$F~’ 1
- SW”].
Firstly note that we have assumed in (11.30) that So” = 0 = Sue, since the absence of a S.T. in the time-time commutators implies [II] that these seagulls may be dropped without affecting the Ward identities. One should recall that if the divergence of the seagull is equal to the (S.T.), we have
and where
s
d4x e-i*l’“(O
/ 0,(x,
O&, Sy18(xo) / pe(q3)) =
--gp___ ~cLdJ(q~ , qJ. Y/(2 Try 2q,o
(II.321
Secondly, the conjecture that Condition
1. (11.33a)
implies that l$m[P ,
- S”j] + 0
for all components. From the conservation one obtains the useful result that if
equations q,,Fup
lim - Foe -+ 0 Bj (41’)” ’
(IL33b) 0 and q2”Fuv = 0,
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then lim 41iFio B5 (qlO)n+l ’
lim
q~Foi (qlo)“+1 ’
B5
Let us examine the conditions models of interest.
and
lim 41i42jF~’ --f 0 . B, (ql”)“+”
(11.34)
required to reproduce a number of commutator
Condition 2: Equation (11.1). Necessary and sufficient conditions are lim(q,O) Foe -+ 0, B5
(11.35) and liBm(qlo) Pi0 -+ +q&P, , while a sufficient (but not necessary) condition
is that
lim[F6j - sij] + 0 4
(11.36)
as a result of the conservation equations, (11.34). Since (11.36) is already included in (11.33b), Condition 1 implies Condition 2, and insures that we reproduce the input commutators. Of course, if we have no Z = 1 Schwinger terms, then Eq. (IL33b) becomes a condition on the components of FuY directly. Condition 3: Nonminimal Algebra of Fields, Equations (11.3la-b). A nonminimal algebra of fields model is an algebra of fields model in which one permits nonminimal couplings. In this case Equation (11.31~) no longer holds. A necessary and sufficient condition is that li$ql”)
Fii -+ 0.
(11.37)
We have assumed that the (S.T.) is a c-number, as is the case in such models. Condition 4: Algebra of Fields, Equations (II.3 1) [ 151. The necessary and sufficient condition is that lim(q,0)2 I;ii --t 0. 4
(11.38)
One may investigate the requirements of other models along the same lines. Unfortunately we must report that the free quark model does not seem to be understood within our framework, which we conjecture may be related to the
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failure of the Jacobi identity for the quark model,4 however we have not been able to make this more precise. C. The Bjorken Limit of the Scalar Amplitudes We have presented a discussion of the constraints imposed by the Bjorken limit on the tensor amplitude, F(q, , q2; mP2) uy. We now analyze these same constraints in terms of the scalar amplitudes f3 and f4 . Towards this purpose let us write F(q, , q2 ; m2P = (q12 - mo2)F1 F2(ql , q2)yY- (422 - m2)-l F2(q2 , qlP - (a2 - m,“)-’ (422 - m,“)-’ F3(ql , q2)uy
+ d f&J d dqff [m?fkh2, 422;mP”>I;,Gh 9q4)~y + q2Y3(q12, mA 422)F2(ql yq2Yv- 412f3(422, mp2;a21F2(q2, qlYy Cm04- 412422) + [f4h2, qe2, m,2) - (q12 _ m,2)(q22 _ mge)f3(q12)G m?) +
(q12
(q12m,2- 424) _ q22)(q22 _ ,p,)f3W
m,"; h2)
(422m,2- q14)
nb2;c?l")]F3‘3(q1 7S)"'l9
-
(q12
_ mp2) f&l:,
q22)(q12
(11.39)
where Fdq,
3 Q2)LIy
=
~T(qlP’
&472)yY’
~A’&*“4q,
-
4212
2
F2h >S2)“”= &4ll)‘“~’ gdq2)VYl “‘&t44q3- du, 2
(11.40)
and
It is now a straightforward matter to obtain the expansion of (11.39) and (11.40) in the Bjorken region, which translates into statements about the asymptotic behavior off3 and f4 . Let us summarize the results for the case of a c-number (S.T.). Conditions 1 and 2: lirnBj Fii + 0. (Sl”) d r&l> Ll YG72)f3(q12~
a See
K.
Johnson
%Y Bi’
0,
(41o)2d&d ~~(q~)f~(91~,mp2;422)~j' (410)2d &J d &2)f3(q22T m:; q12)7
0, 0,
and
F. Low,
Ref.
[14].
422;
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and (41°) d r&l) d Y(92)f4(q12, q229mp2) -y+
0.
(11.41)
0,
(11.42)
Condition 3: limBi (qIo) FFij -+ 0. (Q0)2 d &l)
d Y(92)f3(!7129 422; m,2> --Jp
and similarly for the other three functions. Condition 4: Algebra of Fields: lirnBj (q10)2Fii + 0. k3lo)3 d v(q1) d (41o)4 4 &)
Y(q2)f3(q12,
422;
mp2) y+
0,
d &2)f3(q12,
m2; 422) 7
- 1,
(41o)4 J &h) d &2)f3(q22y
m,2; q12) 3
- 1,
(11.43)
and (qlo)3 d &)
d dq2) f4(q12, 922, mP”> 7
0.
Notice the consistency with Eq. (11.24), where one assumes
f3 = -f4 = const and
Thus the original hard pion model [3] has the same equal-time commutators minimal algebra of fields [15]. D. Phenomenological
as the
Models [2, 3, 12, 131
It is clear from our work in the previous section, that the Bjorken limit and the space-space commutators do not give a unique determination of the scalar amplitudes. Further dynamical requirements must be incorporated to complete the construction. On the other hand, if one is presented with functionsf, andf, which satisfy the boundary conditions of (II. 24), one can see what space-space commutator model is realized by the application of the Bjorken-Johnson-Low limit. That is to say, the scalar amplitudes f3 and f4 provide a realization of the space-space commutators. In the absence of deeper dynamical criteria, various phenomenological models for these amplitudes have been constructed. We now apply our general framework to these models to increase our understanding of the assumptions involved.
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this section we limit ourselves for A v(q),
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ALGEBRA
to models with
the single-particle
We further assume, thatf, andf, are at most quadratic in momenta. The restriction of the discussion to models forfZ and f4 which are quadratic in momenta is dictated in part by simplicity, and in part by the hope that these structure functions do not vary too rapidly with momenta. We are not able to prove that they do not with these methods. The most general quadratic functions of momentaj, consistent with crossing and (11.24) are (11.44)
and
where of course OL,/3, and y are constants. One constructs FuY from (11.39), and takes the appropriate Bjorken limit, i.e., qlo --f im with q1 fixed. Because of the conservation equation (11.34) we only need examine
- 2yq,~q,V f (P + Y)(t%,i + qli) 4720 + q30ciqJ- E0(q2jqsi+ qliq3j))]
+ (q;o)2 K
1tw - +&i)47a term proportional
to p and ~1
One can now systematically investigate the effects of various requirements on this model. Condition: Foe, Foi, Fj” -By 0. This is satisfied for all choices of 01,p and y. Condition: Equation (II. 1). The absence of a Schwinger term in the time-time commutator is automatic. 6 If we considered models for f3 and f4 which are finite polynomials in momenta of degree greater than two, we conjecture that this would not alter our basic results. Rather it would lead to equal-time commutators involving a greater number of derivatives of the currents on the right side of the commutators. Although we have constructed several illustrative examples, we do not have a definitive proof of the conjecture.
144
SCHNITZER AND WISE
There is an Z = 1 Schwinger term in the time-space commutator (/3 + y), which corresponds to the matrix element of
proportional
to
[VaO(x),V?JWlQ+) = ~~abc[~cwW) + 2gp4(/3 + y)(ajV,“(O) - iW,j(O)) 8%3*(x)].
(11.46)
Of course this implies the existence of a seagull, Sii, to cancel the constant term in (11.45). The l/q,O term of (11.45) corresponds to the vacuum-one p meson matrix element of [V,“(x), V,i(O)] S(x0) = iq&{pm,2g~v~(o) + $yaw,a’(o) + W,j(O))
+ (y - /I) a”a~vcyo)} P(x)
+ 4ig,4w
(O)) lw(x)},
aw> (11.47)
which also has a Schwinger term proportional to (/I + r). Notice that no choice of /I and y will reproduce the commutators of the free-quark model. Condition: No Z = 1 Schwinger terms. This can be achieved if (/3 + y) = 0. Condition: Nonminimal Algebra of Fields.
The requirement
is /I = y = 0, which implies-&
and f4 are constants.
Condition : Algebra of Fields.
The commutators In this case
of this model are reproduced if 01= -l/g:,
and /I = y = 0. (11.48)
Once again, we see consistency with Equations (11.24) and (11.48), which brings us back to the original hard pion model. It is possible to consider models forf3 and& with finite polynomials in momenta of degree greater than two, thus increasing the number of parameters, allowing considerable leeway. For example we are able to show that there are several different models which can be described as modified algebra of fields. However, we conjecture, but have not been able to prove, that Eq. (11.48) is the only polynomial model which will realize the commutators of the minimal algebra of fields.
III.
THE VERTEX OF Two AXIAL
CURRENTS AND A VECTOR
CURRENT
In Section II we have devoted a great deal of attention to the study of the vertex of three vector currents. We now consider the vertex of two axial currents and a
PHENOMENOLOGICAL
CURRENT
145
ALGEBRA
vector current, with emphasis on those features which are special to this sector. The method of attack, and flavor of the discussion is essentially the same as that of Section II, however there are some additional questions which occur. Since the philosophy is identical to that of the analysis of three-vector currents, we will tend to omit details when they are the obvious generalizations of statements made in Section II. Let us provide the additional definitions for the discussion of the Ward identities of this sector. The equal time commutators we require are for the axial-vector of isospin a, &U(X), are L‘4z”w, AbU(Y)l ~(xO - v”) = k3bC~CW 6*(x - v> + OA(X, u& %x0 - v”), (IILla) @x0 - y”) = ic abAw
[Vao(x), &w(y)]
8*(x - Y> + 0,(x,
YXb tx” - YO);
(IILlb) and
L&O(x),VbU(Y)l%x0 - Y”>= kdcU(X) 6*(x - Y>+ 0, Y(X,V>Zb Q0 - v”> (III. lc) where O&, Y),“, , %4x, v>S , and OAy(x, v)$ are the Schwinger terms which may occur in the commutation relations. We assume the absence of (S.T.) in the timetime commutators, and in the [VO, 8A] and [AO, aA] equal-time commutators. The divergence of the axial vector current defines an interpolating field for the pion, %4z”W
but we do not assume that this field has canonical commutation define the pion propagator I Throughout
dax e-iG’X(0 I W,&“tx)
(111.2)
= m,2Edtx>~
&.AYO)~ I O> = -iL
relations. Let us
A(q).
(111.3)
this work we make the approximation6 (111.4)
which makes a considerable simplification in our equations without altering the logic behind our discussion [17]. We also must consider the two-point functions
s
d*x e-i’=
A;(O)} 10) = a,, -f& Ancq),
B See [17] for a discussion of the modifications necessary when the pion propagator assumed to be completely dominated by the pole. 595/59/I-10
(111.5) is no longer
146
SCHNITZER
AND
WISE
and s
d4x e-i*‘E(O 1 T{&“(x) = -&a
A;(O)) 10)
44(q)“’ + ;;q:;2 27
(111.6)
+ g”“g”(Fn2 - G)],
where A,,(q)“’ has properties which are completely analogous to those of dy(q)uy described in Eqs. (11.2) and (11.3). There are three time-ordered products to be defined, icabcW(q, , qZ)PYh= hbc
W(ql
, q2)yA
=
-i’l’ze-“Qa’Y(O 1 T{&“(x) s d4x d4y e 1
d4x
d4y
A<(y) V:(O)} [ 0),
(111.7)
&-iql’lG~-iq~-‘(O
and iEab.$W(ql , q2)A = 1 d4x d4y e-iq1’xe-iq2’v(0 I z?b‘LYx) UtYY)
KWN I o>, (111.9)
with the symmetry properties
wtq1 , q2F
=
-
WC72
9 qlPh
and
Wta
, q2Y
=
-
W(q2
, q1Y.
The Ward identities satisfied by W cVA,Wvh, and WA are very similar in appearance to (11.6), with the exception of extra terms due to the nonconservation of the axial vector current. For example,
-
-
Eabc
Ll.(q&A - LlA(q2p - 5
d,(qd + g”gAo(C, - Fn2 - W]
d4x d4y e-iql’~e-iqa’y(O I T@‘“o,(x .’ y)tb 6(x0 - y”) VcA(0)
+ ~““0.4”(& o>tc%x0)&YY)> I 0).
(III. 10)
There are two consistency conditions among the Ward identities, namely qluq3AWtql , q2)PYhand qzvq3~Wql 9 q2)uh should be independent of the order in
PHENOMENOLOGICAL
CURRENT
147
ALGEBRA
which the Ward identities are constructed, so that qlJq3A W@vA] = q3JqluWGLaA] The conditions among the Z = 1 S.T. terms are cabcq3W(CA - F,,2 -
C,)
= qsh J d4x d4y e-iq1’zee-i*2’y(0 / T{SkAO,,(x,
O):, S(x’) AbY(
- qlu j d4x d4y e- iql’se--i*z’y(O j T{POvA(O, + S”“h4(0,
/ 0)
x)j,, @x0) A,“(y) (111.11)
YXb %YO) 4zY-4~ I O?,
and qzv / d4x d4y e-‘q1’b-in2’Y(0 * dax
=
43A
d4y
1 T{iY7’OAV(y, O)& S(y’) a,A,ll(x)j
e-iol.xe-iq,.v
(0 I mA~h(O,
i
/ 0)
YXb &Y”) 3AYX))
I o>,
WI3
which gives a relations among the (S.T.) that appear in the [A, V] commutators. The reader will recognize the relation, CA-Fm2=
(111.13)
Cv,
as Weinberg’s first spectral function sum rule [9]. A suficient condition for the validity of (111.13) is that there be no Z = 1 Schwinger terms in the equal-time commutators [A”, P] and [VO, A”], which is a considerably weaker condition than frequently stated. Throughout the remainder of the paper, we assume (111.13)to be true. With this assumption the remaining three independent identities are [3], %zbeqlLL W(q1 2 &PA =
-zEabcw(ql
_
- dax
> q2)“A
ddy
+
E,bc
Akh)“A
+
%’
on(%)
-
d.(~,)vA]
e-inl’xe-iQ2’v co
J q3A
[
1 T@j”“,(x,
db
%x0
-
y”>
v:(o))
1 O>,
(111.14)
wq1 3 q2Y
= [~akd”” - ~A(q2)uYl+ 5 k1%“4&J
- 92%?2”4%)1~
(III. 15)
and !72vwq1 2 4P
= -wq1
>92 - 3
4441).
(111.16)
A covariant correlation function N UvLcan be defined in analogy to (11.7) Wz, 3qP
= N(q, 2 qP
+ SAq, 2 qzJuyA,
(III. 17)
148
SCHNITZER
AND
WISE
however it is evident from (111.14) and (111.16) that uI”* and WA are already covariant. We again require that the gradient of the seagull cancels the Schwinger term in (III. 14), so that qluSA(q, , q2)pyAequals the last term on the right of (III. 14). From (111.15) one also has the property that (111.1s)
q,sA(ql , q2PA = 0. These properties ensure that N Uvhsatisfies (111.14) with the noncovariant omitted.
term
A. Solution of the Ward Identities The construction of solutions to the W.Z. proceeds in complete analogy with the method of Section II, so we only state the results. We find
Well 3q2Y= 3 kL(41) - 4kt)l + 4q1) 4q2) d “(43)g&3Yh’ (41- q2k mq12, lla? 432),(111.19) N(q, , qp where
id (4J q3’ WI1 9 q2Y - -y+ [T
= 3
+ 4a)
Uq2)
d dq3)
dq2)YV
ml1 7CkP = &Am?12; c&i? q32) + (41 -
-
(42’
g&3Y”’
q2h
q2 ;yq ml
+ yq (111.20)
3 q2Y’,
(41 - 93)” ff2kh2;
422i
932), (III-21)
and
R&PI + &-
+ g$
+(
(ql”,~A(q2)V’”
q223%2
4f42' &q
+
;
-
441)
[
* 43
(42
[;
-
Gzl
%)A
-
42Y
43v43A
422
4&3)
44(q2)
-
-
$
z3413Yh’
&(%I
w4q1) )
cl3
w.4
4(q3)Ah’
A&2)
+$4r(q1)
C‘4K)
42%” ---2
422432
+
-
-
- ~“k?tYl
W‘442)‘”
q12mlr2 - (F z $)I
Cd
arkl)
(41
g&3P
-
-
q2k
4q2))
mz12>
422;
qsa)]
&2)vy’
c;=:31 +4-h) 4kz2) A&)g&.P' Md"" g&W '%9qzLvr\~ 2 x
mq1
> q2h
-
(111.22)
PHENOMENOLOGICAL
CURRENT
149
ALGEBRA
where
G(q, 9c&A = gdq, - qzJ/\G(q,2, q12;qs2)+ g,,(q, - qJv Gh2> qz2,qs2) + g&z, - qdrr G.(q,2,q12,qs2) (111.23) + (43 - qJrr.(41 - q& (q2 - ql)A G(q12vqz2;qs2). The W.Z. are thus expressed in terms of six scalar functions with crossing symmetries
a712~ 422; 432) =
fMq22, 412i 432),
(111.24)
G(q12, qs2; 4x2) = G,(q,2, q12; 43%
and G(q12, qz2; qs2) = G,(q,2, q12; qs2). We now must insure that these solutions contain no kinematical singularities of the l/q2 variety. This is easily accomplished in (III.19), where we define a new function ~2Gz12,
422;
432)
=
2
~az12~
422;
432)
-
a7(ql)-1 - 4(&11 a2 - 422
4443)-1
; ’
with the property K2(q12, q22; 0) finite, so that
wzl >q2Y 4&l) =
(42
-
41)”
[
-
q12 _
4q2)
q22
-
+ 43%h2- 422) 47(41) 4&t)
432
4zl)
4(42)
O&3)
K2(912,
422;
43211
(111.25)
4 dq3) K2(412> 422; 433)
is free of spurious singularities. In order to insure that NYA is free of unwanted singularities, we define two new functions, ff3k12, q22, 432) = 5
1; fe12,
422P432) + (a2 -
922) ff2G712Y422Y4a2)
(111.26) and Kk12,
q22, 432) = &
f - ; m312, 422, 432) + (422 + 42 - 412) ff2(412Y422v437
+ id d.4(q2)-l 4q2)
K2(q12, 922; 932) - &
If
dA(q2)-l d &z3Y 1, (111.27)
150
SXNITZER
AND
WISE
which remain finite as any of the q2 + 0. These substitutions, inserted in (111.20), guarantee that N(q, , q2)yAis manifestly free of l/q2 singularities. Finally, we define two new functions to remove the kinematical singularities from N(q, , q2)uyA, (111.22), i.e.,
G&12, qg2; qa2) = $ [Gh2,qz?432) - ( d”(q’;: ; ffq2)-’ ) dv(W] (111.28)
qs2)- G(q12vqz2,qs2) G(q12,qz2;qs2)= 3 [G&l’, qz2;qs2)+ Gh2, q12,&212 - q22)
1
&29)
which are finite as qi2 -+ 0. There are two additional boundary conditions to be satisfied by the scalar functions, which are most conveniently expressed as
Cd - 4:) Gdd, O,d) + Vd,(qJ1 - &(G-ll
C;;’
+ iF,,2m, CAIHI(O, q12, q:) = 0
(111.30)
and G2C4
qz2,qs2)- WA 42’;qa2)+ 2h2 - qs2)GO, qs2;qs2) - 2im,aFv2C;;‘H2(0,
q2a, qz) = 0.
(111.31)
This can be rewritten in terms of K2 , H3 , H, , G, to G, , but (111.31) is the most compact way of expressing the result. Thus the definitions (111.28) to (111.30), together with (111.31) guarantee that Wlrvh = NU” + PA”’ has no l/q2 singularities. In summary, we have given a complete solution to the Ward identities of two axial currents and a vector current in terms of six undetermined, kinematical singularity free scalar amplitudes. Again the need for further dynamical requirements to determine the vertex is transparent. B. Model Dependent Commutators
and the Bjorken Limit
The solution of the Ward identity for this sector involves six undetermined scalar functions, which are essentially unconstrained, except for some weak boundary conditions. In the absence of an underlying theory, we study the phenomenological requirements of various model dependent commutators by making use of an heuristic asymptotic expansion due to Bjorken and Johnson-Low. (In order to simplify matters, we assume the absence of I = 1 S.T., so that seagull
PHENOMENOLOGICAL
CURRENT
ALGEBRA
151
terms may be omitted. More general cases are easily treated.) To this end one considers the following seven matrix elements: --i
I
* d4x e-iql’“(O
j T{A:(x)
A;(O)}
j pc(q3)>
1 412922
ww?1*
+ G,(q, 9q2P (111.32) where
--i
d4x e-inl’2(0 ] T(aA,(x)
s
=
g,%bc - _==_5 - 5 xQ2n)” 2q3” [
-
t’( 27T)32q,o s
K&l
1pc(q3)> 9 q2) + fJAq1 7 42)“] ;
g&abo
S-
-i
AbY(
d*y e-iqa’Y(&(-ql)~
mq1
(111.35)
3 et);
T{&“(y)
(IIT.34)
V,A(O)} 10>
(111.36) --i
s
d4y e+‘Z’“(&(-ql) -‘%bcgA
l_l_
= d/(2793 2q,o
IT{%,(y)
ffA(93 , %Y,
V,YO)) 1O> (111.37)
152
SCHNITZER
AND
WISE
where
&(q2, q3Y= Slf;;A* 4q2) d&3) gkI3P 4--Q)* --i
d”y e-i’E’w(‘(17,(-ql)
s
I T{&“(y)
ml2 9dLY ;
(111.38)
V$(O)} 10)
aFnma2 1 432 43”43’ 42’43%2 . I [ 422432
q3
1
1
42w 422
= 2/(27r)” 2q10 TQ
(111.39) and -i
d4y e-“‘“‘Y(17~(-ql)l
I
kd5mn2
T{BA,( y) VcA(0)} I 0) (42
-
41Y
= 1/(27~)~ 2qlo
[
,q3””
=
$&
a Uq2) 77
Ll Y(q3)
gdq3Y”
5 43Ya
=
*,~~~
e 4Aq2) 77
d&3)
4s2(41
mv2 -
q22
+
K&I3
(III.40)
7 q3Y],
where H,(q2
gdq3YA’
wq1
9 42h
and K&2
-
q2k
&4I3Y”’
K2(412,
422;
432).
(111.41)
One may verify that these seven matrix elements satisfy all the relevant Ward identities. It should be observed that each of these matrix elements is divided into a model independent or canonical term, and a model dependent term undetermined by the W.I. The canonical terms by themselves provide a realization of the matrix elements of the minimal algebra of fields, as can be verified by the Bjorken prescription. If the model dependent terms vanish sufficiently rapidly in the Bjorken limit, then the entire matrix element is consistent with the A.F. To be precise consider Condition
1: Minimal A.F.; [Ji , Ji] = 0; [&P, Y] = A.F. This implies
BJI Kh 3q2)-+ 0 BJI aof%, 3e)i - 0 BJI (q~“M4q, 3a) + 0 BJI (q~0)2G,(q, >cd + 0,
(111.42)
PHENOMENOLOGICAL
CURRENT
153
ALGEBRA
BJ, (q~“ML(q~ >qdi -+ 0
(111.43)
BJ, ba”)“Gm-Asz 3qP - 0, and
BJ, (q2°K,(qz >qs>’- 0
(III.44)
BJ, (q2°)2Kr(qs3qz)i’ -+ 0, where we have defined the following Bjorken limits: BJ1 =
lim
,
q32=mp2
~,bim.qfixed
and lim BJzm = Q12=m,42( m,3 * q2bim,q,fixed
(111.45)
It is a straightforward exercise to transcribe these conditions to the corresponding constraints for the scalar functions. Another important class of models is given by Condition commutator
2: Nonminimal A.F.; models requires
[Ji , Jj] = 0; [Ji , Jj] f A.F.
This class of
BJI Kk, >q2) - 0, BJI f&h 3q2)i -+ 0, BJI ffoh 342 + 0, BJI hO)G,(q, 3qP BJ, (qz”)Gt& 9qP BJ2 H&s 3eY BJ, K,(q, , qdi
- 0, -+ 0, - 0, - 0,
(111.46)
BJ, (qz”VZAqz3qP - 0. C. Hard Pion Models In order to obtain a simple realization of these conditions, discussion to the simple particle approximation, i.e.,
we limit the ensuing
and Further, we approximate each of the scalar functions by simple polynomials in the momenta. The simplest nontrivial model, consistent with Condition 2, is one in
154
SCHNITZER
AND
which KS and Gl are quadratic in momenta; ta; and G, is constant,’ i.e., &(q12,
qz2;
432)
=
k2,
+
(a2
+
Hdq12,
qz2,
qs;“)
=
hn
+
q12h12
qz2 +
-
qz2h,,
qa2) +
WISE
G, , HI , and H, are quark kzz
in momen-
7
qs2h14
+ Kq22 - 4a212 - 4141 k, + 412(422 + $x2- 412)h, 9 Hdq?>q2>qa2)= ha + q12h,,+ a2h,, + qs2h24 (111.47) + K412- G2j2+ 422(422 - &712- 2q331h2, , G(q12, qz2; 43”) = gn + qz2g,, 3 G(q12, qz2, qs2) = g2, + q12g,, + qz2g,, + qs2g2, + K412 - h2j2 + (422 - %12 - 2%x2) 4221 g25 ,
and G&la, qz2; qs2) = g, . This model will be free of kinematical satisfied. For example, one obtains
singularities
G,(q,‘, q22; 0) = -~I~C;~C,-~
provided Eqs. (11.25-31) are = g,, .
All but five of the parameters in (111.47) are determined in this manner, with the result &(q12, qz2; qs2) = k2, + (a2 + qs2 - qs2) k,, 3
Hh2, qz2,qs2)= m 2& ;,q;
[ 1 - $ + qs2(T 1 qs2 - q12) CA ( 2Fn2 m2
+i q Gh2,
1
1
- -m, ) 2Frr2 Cv m, 11
h2 + qs2- aa) k,l
+ Kq22 - h212 - 4141Al, + 912(422 + 4S2- 412) h,, , (III 48) h15 - ‘+ k,, - g23], c qz2; qs2) = c Y ; A ‘, A2 + qB2 [ 3iFvr2mn2 A
H2(q12, qz2, qs2)=
4m,2F;21CAC,
c* - T) cv + yiF2m 2 qs2k,, (m, A
+ ‘2 (a2 + qz2+ qs2)h, + ; q12h, 3 -1 G(q12, qz2, qs2) = 2cyc~2 -Ii 7 See footnote 5.
F
(7
CA
Cv Fw4mr4 k + mpz 1 + CA2 21
(a2 - qs2) h,, + qz2g,, 3
PHENOMENOLOGICAL
CURRENT
155
ALGEBRA
and G(e2,
qz2; qs2) =
The leading terms of the asymptotic expansion of the matrix elements may be calculated: BJI K,h
, cd -
2
2F,4mnampZk22 __ (26 * q1 + E0430)+ o(q:--q[, 1 91° E0 - (q;o)2 (111.49a)
BJ1 H,(q, , q2)j ---t -
gA22;mn4 G%,
+ hl,)P~5930+ e0(91- 921 + O(dq (III.49b)
BJ1 G,(q, , q2)ij + -$$
(g,, + i Fnz72h15 ) (ciq3j - dq3”) + O(q;-‘),
BJ, HA(q1 , q2)j -+ -gfqF;F4
( P,,
2
-
4iFr2mT2 CA
BJ, GA(q2 , qp
--f -~~o~2
+ hnJP+O + c0(q2- q#l
k22[ql)
1[g,, -
y
h,, + w
k,,] Eiqlj
A
+ (g23 - F + (i T
, d5 -
(111.49d)
+ wi
2
BJ, Kk,
(111.49~)
A
1
h15) Pqli h,,) g”jr”qlO/ + wrh
(111.49e)
4g2202mr4 k,,(q,jq,O) + O(q;-a),
(IIL49f)
and BJa K,Cqt , qa)ij + ‘8
!2gij [2(qlo)2 h,, + mn2h16 + i
2m,2F,,2 ($5” - Es”)]
2
+ 4q,W
[h,, + i y
kt2] 1 + O(q;-‘).
From Eqs. (111.49) one deduces the following commutators.
s
d4X e-‘Q1v5<0 I PA44 = ,,&*$f$-
AD>1
C4
&2F,2m,2(2h,, 3
matrix
(111.49g)
elements of equal-time
I po(q31s)) + ho&‘~5g,0 + cO(ql - q&,
(111.50)
156
SCHNITZER
AND
WISE
2Eabcgo (F,4m,8m,2) kz2e0, = 1/(277)3 2q30
s d”v
e-iq”‘“(&t-qdl
[aAb(.d,
-%bcgA
= 2/(277)” 2q,o -
4iFi;w2
~ci(o>l
(III.5 1)
a(J”)
kD2F,“mw4) EWI5
+
1 0)
hJWq,O
+
Gob2
-
qd9
k2&%j)],
(111.52)
and
s
d”v
e-iqa’*(&(-ql)l
=
iaAb(y)?
~,i(O)l &YO)I 0)
-4iEabcFnmn2 (F,,2m,2g,2) k,2(qIjqIo), 1/&y 2q,o
(111.53)
as well as the standard commutators of the algebra of fields. The commutators for the time-derivatives of the currents are similarly deduced. Equations (111.50-53) are consistent with the commutator algebra [&d,(X), &d,(o)] 8(X0) = -4iE,bcFr4m,8m~k2zV~(X) [a&(X), &j(o)] +O) =
-%bc(2&
s4(x),
(111.54)
+ hd gA2F,r2G2
* [2(CPV,i(x) - W>(x))
- V?(x) aq P(x),
(III.55)
and
= -4ic,,,
g,2;2mT6 k2,[Fr2aiA,0(x) - m;2C@80t3,A,u(x)] A
- E,bcgv2F,2m,,2(2&, + h,,)[2(a”&‘(x) - (A,‘(x) + m;“a”auAi,lr(x)) aj] S4(x).
S”(x)
- a’&‘(x)) (111.56)
Since the right sides of Eqs. (111.54-56) are not zero, it is evident that aA is not proportional to the canonical pion field. [Recall that V”(x) and Ai are assumed to be proportional to canonical fields in the algebra of fields].
PHENOMENOLOGICAL
CURRENT
ALGEBRA
157
As a special case consider Condition that
1: Minimal
A.F. with aA Canonical.
Equations
(111.42-44) imply
k,, = h,, = hl, = g,, = 0. Further, from the last of Eqs. (111.49) one obtains iL2 = g2, which is Weinberg’s second spectral function sum rule [9] in the single-particle approximation. With the constants so chosen, one obtains the original hard pion model [3]. It appears that the only model, involving the polynomial approximation for the scalar functions, consistent with the minimal algebra of fields equal-time commutators is the original hard pion model. D. The Gilman-Harari
Model [5]
Gilman and Harari made a study of the saturation of current algebra sum rules and superconvergence relations for forward scattering, by applying the hypothesis that these sum rules were saturated with single-particle states. In their study of the A, , p, T system they found (111.57) I g, I = 0, and
where these coupling parameters are defined in Appendix B. Heretofore, it has not been possible to connect their results with those of the Ward identity approach to current algebra. However, we are now able to make the connection. We refer back to the model described by Eq. (111.48) to construct the p and A, decay parameters, i.e.,
gLlns= 3
[y
+ 6hm,F,J4 k2,],
gT = $ [(Z + S) - 8i(m,2m,4F,4) h,,], n and a = $ [(3 + 6) - 8(m,m,FJ4 w
k,,],
(111.58)
158
SCHNITZER AND WISE
where we have made the numerical estimates, rnA2 N 2mp2, g,2 N gA2, go2 N 2mp2FT2, neglected terms of order mv2/mp2, and defined the parameter 6 as in the hard pion model. This model is easily seen to be compatible with Eqs. (111.57), with the choice of parameters 6=2
6 = -2
hmDF~)4 k2, = Q i(mn2m04F,,3 h,, = 3
(m,m,FJ4 or
k2, = - 8
(111.59)
h,, = 0
as well as two other sets involving large values for 6. The important point is that the consistency of the two approaches requires a,,Ap not to be a canonical field, as is evident from a comparison of (111.59) with (III.54-56). This is in accord with the representation mixing of chiral multiplets found by Gilman and Harari. In this connection, a modified hard pion model has been discussed by Brown and West and Horowitz and Roy [18] where they consider a nonminimal algebra of fields in which a,A”(x) is still the canonical pion field. That model corresponds to the choice of parameters
kzz= 0, W,, + 4,) = 0, hl, # 0, which adds a single parameter, h,, to the original hard pion model, so as to reduce gr without severely changing F@ -+ 7~) and P(A, -+ pr). It is now clear why they could not make contact with the picture of Gilman and Harari, as it is essential that k,, # 0 (aA not canonical) for the compatibility of the two approaches.
IV. CRITIQUE We have succeeded in solving the Ward identites for current algebra vertices in terms of undetermined scalar functions. In addition we have discussed the constraints provided by equal-time commutator models, for the general case as well as for specific phenomenological models of interest. As a byproduct of this work we have been able to establish a connection between the model of Gilman and Harari (constructed from current algebra sum rules), and the Ward identity methods; with the conclusion that a,Au should not be the canonical pion field if the two methods are to be related. We also wish to comment on the problems of this formulation of current algebra.
PHENOMENOLOGICAL
CURRENT ALGEBRA
159
1. The quark model does not have a simple characterization in terms of our separation of matrix elements into a canonical term and a model dependent term. 2. A related difficulty is that all the models we discuss satisfy the Jacobi identity, while it is known that the currents of the free quark model violate the Jacobi identity. It is not clear what modifications are necessary to discuss hard pion models valid for the quark model, since all known hard pion models satisfy the Jacobi identity in the Bjorken limit. 3. There is no unique connection between models valid at low energies, and those useful in the Bjorken limit. If we abandon polynomial models for the undetermined scalar functions, then there exists a large number of examples exhibiting drastically different polynomial behavior in the two regions. In fairness to many of the applications of the Bjorken limit, principally to calculations of radiative corrections, we should say that the hard pion models do give a correct realization of the various algebra of fields models, since they do have the correct equal-time commutators (defined by the Bjorken-Johnson-Low limit), which controls the divergent parts of the relevant integrals [4, 61. In these cases only the$nite parts of the radiative corrections may be somewhat ambiguous, but the values of the finite parts are usually of subordinate interest in a discussion of the convergence of radiative corrections. 4. Doubt has been raised as to the validity of the Bjorken limit in perturbation theory [9]. It may be that this limit is, in fact, valid in the real world, in which case all of our analysis would hold. If not, then that part of our analysis which pertains to the asymptotic boundary conditions imposed by equal-time commutators of the We [Jo(4, -WI 4-8 (as well as our general solution) would probably not be affected. Further, no anomalies have as yet been discovered for models of currents based on an algebra of fields, so that part of our work would also probably be unaffected.
APPENDIX
A.
REGULARIZED
WARD
IDENTITIES
Throughout the main text we have assumed that the formal Ward identities are valid, and well defined, as in theories based on underlying vector (and axial-vector) meson Lagrangians, such as the algebra of fields. In the event that the real world must be described by a theory with an underlying fermion field, such as the free quark model, the quantities d dq)gv and Tfiva would not be well defined. Fortunately in the case of the free quark model, the formal Ward identities retain their form except for the replacement of dv(q)uv and M uyAby the corresponding regularized quantities [8]. In that case all our phenomenological results are easily recovered. Consider the case in which s (py(m2)/m6) < co, but Cy + co, as in the case in
I60
SCHNITZER AND WISE
the free quark model. The covariant regularized propagator for the vector currents is
= gk4“”
P~~V(P)R + bvguV.
(A. 1)
The Ward identity satisfied by the regularized covariant three-point function is [8]
which is identical in form to the Ward identities given in the text, the solution of which is obtained from (II. 10) with the substitution
d&d-+&b)
and G+b,
where
(A-3) k&)
= bv + q2~v(q)it.
As before, one is led to define scalar functions free of kinematical singularities. However we must consider one new feature, i.e., the function DAq) will contain zeroes which in general should not be present in the physical amplitude, ml1 3 qz)!F. Therefore, these zeroes must be cancelled by corresponding poles in the scalar functions [20]. Once the kinematical zeroes of D,(q) are so removed, one is led back to the theory described in the text. Identical considerations are valid for the AA V sector discussed in Section III.
APPENDIX
B.
ELECTROMAGNETIC
FORM FACTORS AND MESON DECAYS
In the body of the text we solved a set of Ward identities in terms of a number of undetermined scalar functions. For completeness we relate these results to the electromagnetic form factors and decay amplitudes of the p, A, , and rr mesons. The electromagnetic interaction of the p-meson is
~cY0) I P.hD
(B.1)
PHENOMENOLOGICAL
CURRENT
161
ALGEBRA
where M(q, , qJuYh is given by (11.10). This is easily reexpressed in terms of the scalar functions, i.e.,
Gdq2)l ~c”(O)I Pa(41))
-~~abe~~(qJ 4k)* Ig,,cql + q2y 11+ tgp2Ay(t)f3(mo2, m,2;f)l =
2(27r)3 dq1Oq2O
- %wlv + gd2Jg,Zm,2A &>f3(m,2, t; m,2) - 2q2,9&+ q2Y go24 At>[L(m,2, mp2, t>+ f3b,2,m,2;f> - f3h2,t; mD2) -* -
m,”
( &f&Y
+ A v(t)-' m,2 ($
(f3@,2, m:;t) -f3(mp2, t; m,2))
x; mo2)- gf3(t, Ay(x)-l)
m,? 4 x=mp2 A ,(m 2)--l A y(m,2)-1 - A v(t)-l m,2 L.- t ( rnp2 - t
x=mp= -
iii9 03.2)
where t = (ql - q2)2. The coefficients of the three tensors are the charge, magnetic moment, and electric quadrupole moments form factors, respectively. Similarly, from (111.22) one obtains the electromagnetic vertex of the &-meson,
-4d!A” = 2(24” -
-
__ dq1fJq20
4m4q1)
2(243
dqd
A v(qd
Gfq,
3 -q2PA
/
, a12=a22=m42
&l2)
2/m=
4d*
*
I
g% + q2Y[l + a2tA 14) GhA2, ma2;t)l
- 2&W’ + g”hq2u) g2AAt) G2(mA2, m.2; t) - 2q2%‘(q1+ q2Yga2A~0) [Gh2,
m..t2;t>
+ k (& G,(x, mA2;t) - -$ G2(mA2,x t))
11, em”2
(B.3)
where once again the coefficients of the tensors are the charge, magnetic moment, and electric quadrupole form factors, respectively. 595/59/I-11
162
SCHNITZER
The pion electromagnetic
=-
AND
WISE
magnetic form-factor
is obtained from
i(m,” - s12Xm,2 - 422) - 4dA(q1 p Fw2mr4do 2qlo2q20
lim ala.a+nn
,
92)
ica’c(ql + q2)A (1 - fA “(t) F,,2m,4K2(m,2, m,2; t)} = 1/(27T)” 2q,o2q,o iEC%-dql+ qa)” F,(t). = 1/(2?r)f3 2q102q20
(B-4)
The decay p -+ nn is described by the effective interaction ~Lw, = &*(41
-
=
lim
q2)
- 433) @x2 - q12Xmn2 - q22)(m,2 - q22)NA(+ql
ala-aza=m,Z;+tt2~2
,
+q2z) En(q3j
gZr2mr4
= -gdnc2Fn2mp4K2(mT2, ma;mo2Ml - q2)- c4q3.
(B.5)
The effective matrix element for the decay A, --t p” is “4%A&Xl =
%A(q2>
%‘(d*
i
gT
rA
+ [gL + ( mA2 +z;y2 = a12=mnB. x
-
-
m*2
) gT]
w2"/
4d* & 8pI;,Wr2
%A(q2)
lim
Pz
aga=m~e.a8s=m
crnn2
4mA2m,2 - (mA2 + rn> - rn,“)” 4mA2 I
[
h2)(mA2
-
q22)(mo2
-
h2)
N”%h
, -q2)
so that 8g, gT
=
4mA2m,’ +
mA2(mA2
gAFT2m,2F42
-
(mA2 -
+
mn2)
m,,’ H4(mm2,
-
mD”)” mA2,
[(m,e
+
mA2
-
m,2)
K&x2,
mA2,
m?)
03.6)
%a)l
and 16gPg~F~2m~zmA2mg2 gL = 4mA2m,2 - cmA2 + mpa _ m,2)2 + (3mA2 + y”
I Hdm=2,
- m,z) H4(m,2, mA2, m2)].
mA2,
mo?
05.7)
PHENOMENOLOGICAL
163
CURRENT ALGEBRA
The decay widths are I$
gzmn3(mo2- 4m,2)3/2 12m02 47r
---f 7777)= -
VW
and F(A, -+ p-r) = g
2
I 3m,ze2
1
-k 3m 3 8L2
- m,2)2 -
r,
I Y
(B-9)
4m,2m,2].
ACKNOWLEDGMENT One of us (HJS) wishes to thank the physics group at Rockefeller University for their warm hospitality during the academic year 196991970, and Giuliano Preparata for a reading of the manuscript. REFERENCES 1. M. GELL-MANN, Physics 1 (1964), 63. 2. S. ADLER AND R. DASHEN, “Current Algebras and Applications to Particle Physics,” W. A. Benjamin Inc., New York, 1968; S. WEINBERG, Znt. Conf: High-Energy Phys. Proc. 14th Vienna (1968); S. GASIORO~ICZ AND D. A. GEFFIN, Rev. Mod. Phys. 41 (1969), 531; H. J. SCHNITZER, “Proceedings of the 1969 Erice Summer Institute,” Erice (Trapani), Sicily. 3. H. J. SCHNITZER AND S. WEINBERG, Phys. Rev. 164 (1967) 1828; I. S. GERSTEIB AND H. J. SCHNITZER, Phwvs.Rev. 170 (1968) 1638. 4. H. J. SCHNITZER AND M. L. WISE, Phys. Rev. Left. 21 (1968), 475; M. L. WISE, Thesis, Brandeis University, 1969. 5. F. GILMAN AND H. HARARI, PhJx Rev. 165 (1969), 1803; S. WEINBERG, Phy.~. Rec. 177 (1969), 2604. 6. J. SCHWINGER, Phys. Rev. Lett. 3 (1959), 96; T. GOTO AND T. IMAMURA, Prog. Theoret. Phvs. 14 (1955), 396. 7. S. L. ADLER, Phys. Rev. 177 (1969), 2426. 8. K. WILSON, Phys. Rev. 181 (1969), 1909; I. S. GERSTEIN AND R. JACKIW, Phys. Rev. 181 (1969), 1955. 9. S. WEINBERG, Phys. Rev. Lett. 18 (1967), 507. 10. R. P. FEYNMAN, “Proceedings of the 1967 International Conference on Particles and Fields,” Rochester, N. Y., 1967, Interscience Publishers, Inc., New York, 1967. 11. D. J. GRCAYS AND R. JACKIW, Nucl. Phys. B 14 (1969), 269. 12. S. WEINBERG, Phys. Rev. Len. 18 (1967), 188; J. SCH~INGER, Phys. Lett. B 24 (1967), 473; J. WESS AND B. ZUMINO, Phys. Rev. 163 (1967), 1727. 13. R. ARNOW~TT, M. F. FRIEDMAN, AND P. NATH, Phys. Rev. Lett. 19 (1967), 812; S. OKUBO, R. E. MARSHAK, AND V. S. MATHUR, Phys. Rev. Lett. 19 (1967), 407.
164
SCHNITZER
AND
WISE
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