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Unitarity corrections to kaon-pion scattering from current algebra
Nuclear Physics BI09 (1976) 357 -372 ,~) North-Holland Publishing Company
UNITARITY CORRECTIONS TO KAON-PION SCATTERING FROM CURRENT ALGEBRA * J. S~i BORGES lnstituto de F{sica, Unirersidade Federal do Rio de Janeiro, Rio de Janeiro, Bra.:il
Received 9 February 1976 A method is pre~nted to unitarize the hard-meson current algebra kaon-pioJt amplitude derived by Ford. Elastic unitarity is satisfied in the lowest-order correction to current algebra. The amplitude contain.~three real parameters and does not have any arbitrary function for simulating left-hand discontinuities. Once the isospin-half P-x~ave is given in the low-energy region, this method leads to a sharp determination of tht: S-wave phase shifts. Our solution favors the "down" type for the I = ~ S-wave phase shi "ts.
1. Introduction The pion-pion and heaw-target-pion scattering amplitudes compatible wi:h tile requirements of the partially conserved axial-vector hypothesis [1] together with SU(2) X SU(2) current algebra were obtained by Weinberg [2] using tlle soft-pion technique. The soft-meson method gives a linear approximation that can be interpreted as tile truncation of a complete polynomial expansion of the amplitude in a singularity free region in the space of all the variables. Deviations are certain to exist, but their evaluation is muclt more model d.'pendent. One way for obtaining corrections is, for example, tlte hard-meson met rod based on the Ward- Takahashi identities technique [3]. Hard-meson amplitudes for tile on-mass-shell particles have been obtained by Schnitzer [4] describing pion-pion scattering and by Ford [5] for kaon-pion ~cattering. These amplitudes explicitly satisfy all the constraints of the current al,~ebra and are expressed in terms of one-particle reduced tunctioJIs. Using the soft-meson estimates for the two- and three-point functions, we [6] have shown that, near the symmetric point s = t = t¢ = m~, the Schnitzer result is equal to the Weinberg result plus a "tree structure" and a "seagull term". Furthermore, when form-factor and propagator discontinuities across tile physical * Research supported in part by HNEP. ('NPq and CEPI.~(, of Univcrsidade Federal do Rio de Janeiro. 357
J. Sd Borges /Kaon-pion scattering
358
cut were evaluated in lowest order correction, it turned out that, near threshold, tile contributions of the non-polynomial part to partial waves were small compared with Weinberg's result. This way, we have obtained a crossing symmetric correction to Weinberg's expansion that satisfies, only approximately, the requirements of two-body elastic unitarity. The present analysis will parallel our study of pion-pion scattering [6]. In this paper, using vertex and propagator estimations, we can reobtain a Weinberg-like result [71 from the hard-meson amplitude. The main point in this derivation is the fact that since current algebra gives real amplitudes, the nearby partial-wave discontinuities across tile right-hand cut are small compared with tile current algebra result. These discontinuities arc known in the first order. In this way, we construct the structure functions entering the amplitude from lhcir first-order discontinuities by the dispersion relation technique. Working with three parameters, only in the seagull term, we determine a crossing symnretric correction to the kaon-pion low-energy scattering amplitude that does not contain any arbitrary function to simulate partial-wave left-hand discontinuities. In this scheme two-body unitarity constraints in partial-wave amplitudes can be satisfied only approximately. We fix the tw,, r,arameters entering the total isospin-half P-wave by fitting the experimental 18] phase shift near 90 °. The remaining parameter is adjusted and we obtain S-wave phase shifts. Our Si/2 phase shift is found to be of thc " d o w n " type above the K* meson mass and the $3/2 phase shift is repulsive and small, which is in agreement with other predictions [91 . in sect. 2 we give the basic formalism for describing the kaon-pion system and, using soft-meson estimates, we reproduce the linear current-algebra amplitude [71 from the exact representation derived by Ford [5]. In sect. 3. a model for the form-factor and propagator discontinuities regarding elastic unitarity constraints is constructed and we discuss the seagull term. In this section we also present ttae first-order corrected amplitudes and numerical results. The detailed structure of the different terms entering the amplitude is given in the appendix.
2. Low-energy kaon-pion scattering amplitude 2.1. Basic"formalism Here we compile the formulas pertaining to Kzr scattering. For the elastic scattering K,~lr~ ~ K7 q,rr{L/,,, we define tile invariant anaplitudes T ~(s, t, u) by
(K'rq,Tr~_.p,JTlKqrr~p) = X~.'l,{6CcaT+ + ~[r a, T q r } X k ,
(2.1)
J. Sd Borges /Kaon-pion scattering
359
where )~- are the isospinors, s : - ( q +p)2, t = - (q + q')2 and u = - ( q +p')2. Other symbols have their usual meaning [10]. We define partial-wave amplitudes Tll(s ) for total isospin 1 in the s-chanr~el by I 1 = ~, 3 ,
Tt(s, t, u) = ~ ( 2 / + 1)Tll(s)Pl(cos Os), 1= 0
(2.2)
where
Ti/2(s, t, u) = T+(s, t, u) + 2 T - (s, t, u ) , T3/2(s, t, u) = T+ (s, t, u) . T - (s, t, u) , 2St= Is - (m K +mTr) 2 ] [ s .
(m K-- m r r ) 2 ] ( c O S 0 s - l ) ,
with 0s being the c.m. scattering angle. For elastic scattering we get, in particular below the first inelastic threshold, 1
Im Tll(S) = ~6-~ PK"(S) I Tu(s) 12'
(2.3)
which may be solved giving 16rr i611(s) TII (s) = - - - - e sin 8tt(s ) ,oK,(s ) where 811($)
(2.4)
the real phase shifts and,
are
1
pKrr(S) = s[S - (m K + mrr) 2 I I/2 [ S - (m K - m , ) 2 ] 1/2 ,
(2.5)
is the phase-space factor, for s>~ (m K + m~,) 2.
2.2. Current algebra exact representation for the Kn system The starting point in our derivation is Ford's exact kaon-pion scattering amplitude constrained by current algebra 151. Particle names will be used to label the quantum numbers of the two-, three- and four-point functions. Without assuming any representation to which the symt:~etrybreaking belongs, it reads "~ 2
c~: F2 c. ~K'0'o;~,~/(0) F2K " c,, =- Ta~v's (q' P' q')
,aK(0)~ A6,,,(O), " ~4i/Im ~"K" (q' I,, q ') c~ A c~, .¢c~ (q p. q,) F2
k~-77 ~,~'t6
'
360
J.
[-CKA CAI
L-F~7 -F~K*r' XA
Sd Borges / Kaon-pion scattering
. ~,~KAA I K*, q'l'
I
",~~,a,,.¢~"tq, p) + ----(;%(q21"KF.,., ,' P)
FC KACA~
';V m . ~ K A A ~ K *
- ,
~
]
rl .
,
(q+p)kF K l.r q p i tx,~o,,r,(q,p')+2~.~7/.r~((q
,
+--I-L-J'~[sef~'~,', [(q- P)(q'- p')S~¢'" +(q2 ._ p2)(q'2_ ,2 -2 41"Kf' ~
P')r
p'2)¢KGr/~d( q +P)l
,,
t 1~ XAhh'(q
+I))
p'
'
•
2f- K
LI., 2 v P
1
--*
(0,FKrt, ¢ , , ~ , I .. -F-~A Kyl"(0~Arr " ~rn"" lm,g(q'p)An'h '(q , + P)+72~-F~(r'r'Ta)h'n '(`
.
ze~,ao,,,,.(v,r")+
~
r;
,1
_
t
K--Tt 3
F2 I%
.,, tq. q ~ ,,~,~q+q'~ + 2 ~ { r~. r~h, , %
%h'(q +q )
P~
(2.6)
x
A;(o)~-~.,,~, "o" r /,,,,,,~r),r,')a,~,t,,(p+Y) "~°" + ; ~l{ T ~ L?h',,'%,
,
where c~,/3, etc. are indices of SU(3) which generators are the antisymmetric matrices 7~.. As demonstrated in ref. 131 there is the following relationship between the decay constants of mesons into leptons and the symmetry-breaking parameters e: ei T/~ = (Faro 2 / x / Z b , where re(Z) is the mass (wave function renormalization) matrix for the meson multiplet. It is convenient to label the e by the name of the meson appearing on the right-hand side of that relation. Thus, for example, e~ = t'~rn2/x/Z,r. In formula (2.6), A ~ ( q ) is the spin-one current propagator while the conventional meson-field propagators Ag(q), the functions Di/(q ) and I)i/¢q).. are written in terms
J. Sd Borges / Kaon-pion scattering
361
of the spectral function pO(m2) as d (m2) pp;(m 2 )
=
fd(m2)p°.(m2) fd(m2)pB.(m 2 )
,/(q)
=-
(2.7)
,T(;,T; q i-
st) that
bq(q) : D,.c (q)
,
"~" ' ") ")
{ll.
2 + (p'- q '}k (q~eKaf(o)-p~2~2~(o))] + ~do,~<)d~K~K(O)+~,~(O))
'~K (0)
&,(0)
~)< (0)
~,,(0! -
+ crossed terms} !
where ]'aCe and da~e are the SU(3) Clebsch--Gordan coefficients introduced ')y Gell-Mann I I 1 ]. In order to obtain tile current algebra soft-meson result [7] from T cA, we approximate spin-zero meson-field propagators by saturating the spectral functi ms in (2.7) with the kaon and pion particle masses, I
-=
,~.{p) = ~
~-
l
?II~ +
p-~
~Z;,
J. Sd Borges / Kaon-pion scattering
362
and we consider F K = 1;'~ = F [7], and Z K = Z~r = 1, CA . 1 3 2 .._ 3 2 Ta3.},6(s, t, u) = 3~- ~ {fc~&]'c.rs(t -- u) + da~jcdevs(s + t + u --- ~m K im~r ) + crossed t e r m s } , SO
T+Ca(s, t, u} = 4 ~l { s + . + 2t - 2m2K - 2,n 2) 1
T-CA(s,t, U ) = ~ . 2 (S - It).
(2.8)
Note that we could correct this result by a p o l y n o m i a l if we had introduced, generically, different FK, F,r, Z K and Zrr * In order to analyse tile remaining terms of eq. {2.6), we use tile results for the three-point functions,
CKACAt
,u v,,KAA1K* -
K*'
F K -t~- q p l au, 3v, e'z'tq'p)Ac'rer(q+p)+
~F@n
..
K*,
.f~'(q
P)r'A"rer(q+P)
x Z / ~ x / - ~ F Ka~,e,r, n K . . tq, p ) A eK" 1 fa~e [ ( q . . . P} r S _~+_FTZKqr FK =VLKVZ*r , r r ( q + p ) + b~,F - .z~KZTrp F . r]
C2A, , A ' A ' o . p,)Af,~r(p , 1 . +P ) + ~,.2]~8~'(p " p')T'APc,r~T(P+P ') F 2 pVp op3v, 6o, e,r,(p, 21"n
..o
=Z~P¢6,c','(P,P')~'~
tP+P )+]~6~(P
(
)
P')" I +21F ~ ,
(2.9)
and a similar result for the KAKAO vertex. The next step is to relate the proper vertex parts and two-point functions to conventional form factors and propagators. The K13 form factors j"~_ and the electromagnetic form factors j . v are defined as
* A ratio FK/I,'Tr >~ 1.3 seems to be rather unlikely [121. Ilere the Z's are being assumed finite phenomenological parameters [ 5 J.
J. Sd Borges /Kaon-pion scattering
363
X D~,c(q+p) +]~,,ac [(P " q)rf+(s) + ( q + p ) r f (s)] , Znl ~6~.,r,(p,p )Ac, ~ (p • p) = (p' p)~J~'(t),
(2.10)
and a similar definition for the kaon electromagnetic forln factor .l~.) "0 and]'", the scalar form factors upto a constant, are connected with vertex functions by
d2K ~ + r2~.y, (q, p)~,, +q+t,) +~-FK-~-. Z,rV~°,(p.p')a°,+(p +p') +
(T,.
d~./~ (tt
T~)~..,(¢)+. =
•
Ill
~" ¢ I
and a similar connection for tile sigma-kaon-kaon formtactorjK; the relatiu~ between J. and ] . 0 % ¢ j ,.0 ~,(.,;)=
[(q2
p2)j+(s)+s J. tS)le,~e "
where
J:?i3c ='/~3c
- 2t.'K [."~ {Ta' T#)+~.'(0),,'
(2.12)
will also be used m the following. The relation between the total amplitude T ,,./$,7,5 KrrKrr, for on-mass-shell particles, and /14KrrKrr• is the proper function --o~,7,~ KnKn ,2 , KrrKn Ta~,7,5 (q,p,q , )= - 1 ,,2Kl'r:'~la~,Ta (q,p,q').
(2.13)
When the definitions (2.10), (2. l 1) and (2.13) are introduced in (2.6), we obl ain T Kr~Kn(q, p, q') =
T('A(q, p. q') + T KTrKn(q,p. q ' ) .
where 2
T"IJ`7,5 (q'P'q )
2
1"~ F~ ~ , 5
'
{
FK (
+ (q +Ph\A ,~vds) ++K ~"(s) Dn(s) "~0 (s'~ °'~""
F~Z~,,,)
I . ~ _.p2)CKe~*(s)) 1 -%,_~/~to~+(q-
J. S6 Borges / Kaon-pion scattering
364
p'2)eKe~[)K(s)} +f2~(s)~,(s)j,v~(s -o ,, :o ) + C) ,.~
+ (q2_ p2)(q'2
(q- q')(P- P ) l ~ l ~ a 6 1
S]+
4F2 F21r
2F~
d~'~Edc#6
]'~(t)Txl°(t)fg(t)
(2.14)
Next we analize formula (2.14) and show that 7" partial waves at threshold are small when compared with the current algebra amplitude T CA projections.
2.3. Solt-meson estimates The Ward identities analysis, together with the hypothesis of single-particle dominance of all propagators, lead us to the following assumptions for x ,'-' (rn K + rnrr)2 :
f+(x) ~- l + f~+l)(x) , f ~ ( x ) ~- l + f ~ ( ' ) ( x )
f__(x) ~_t(_l)(x),
}°(x) ___f0~'~(0,), f2(:,) ,-, 1 + t ~ ( % 0 ,
, A K , ( x ) ~-- C,¢, +8~2(~) ,
AK(X) ,-, ~ ( ~ ) ,
,%(X)" ~(J)(X).,
%(x) ~- cp + ~')(x) ,
(2.1S)
MI functions denoted with a superscript (1) are of order (m 2 + rn2)/X 2, where X is of the order of magnitude of the vector meson mass, in a vector-meson dominance approximation. In addition, we assume that the partial-wave contributions of the seagull term, near threshold, are small and almost real [4,13],
,~l(X)'-,tl/)(x),
o(
Im ,~'1(x) "--, \ - - X ~ -
.... /.
¢2.1 6)
When the definitions (2.1 5) and (2.16) are introduced in (2.14), we obtain
:~K"K'~(s, ,, u)~-- T(l)K"~"(s, t, u)+ O(M4/X4) , where T(1) KrrKrr.,. (l) (a, o t, U) a[3"~6 ~S, t, it) + ta.#.yb
{
J'o',,,,)]
J. Sd Borges / Kaon-pion scattering
+ 4;
j,'o3e]~t,~e[(t
u)6(KI,)(s) + (q2 __ p2)(q'2 _ p'2)%eKd(l)(s)]
+t-(l) Cs~ 1 - - - ~ r ( t ) J 0aBe"
: (~( 1)ee' " 0e'~',b (S) K
+ 2~.:Sa.reJe3~(s u)
+(s
o
u)}
"~(1)(t) +f~(l)(t) + 2 F- - 6 ( 0
((l)o(t~
+da.r~de~K
365
1
f.(l)o(t ~
' " ~-~l)(t%~"
"" "
(2.17)
o
We can interpret the result of hard-meson calculation as follows: near the softmeson point or the Adler point, T(1)(s, t, u) is small compared with the well-known soft-meson amplitudes T('A(s, t, u) explicitly shown in (2.8). In (2.17) we have assumed, for the sake of simplicity: F K = FTr = F. In sect. 3, we will develop a model for the first-order corrected amplitude T (l).
3. Un~tarization
3.1. First-order unitarity corrections Now we must construct a model for the form factors, for the propagators and for the seagull term involved in the first-order corrected amplitude T (I). First we observe that, as the limiting amplitude is real, the following discontinuities for the partial .Javes T(1) are expected: Im T[/)(s) = ~6-~pK,r(s)r]l 1 (-A2(s).
(3.1)
Let us start analysing the form factors; we intend to make progress by application of the constraints of elastic unitarity. The two-particle unitarity equation sat isfied by form factors is Im F(s) = 1 ~ p(s) T*(s)F(s) . Having in mind (3.1), the electromagnetic and 1~_ form factors are obtained fiom Im.fl~(1)(x)
= 3-~PKK
(,.)
J. $6 Borges / Kaon-pion scattering
366
1 1
unf~+~)(x) -'- i U~ °K,Cx) r1~2 (x) "
(3.27
By assuming equal phases for D ~' and A • , we can write from (2.7), lm
....
a"(s)
s
thus from (2.1 2);
" ~(s)
......
s---
On the other hand, the isospin-~ and 3 projections I"1/2 and )"3/2 of the scalar structure function ~(1) ~0~t3e and the sigma form factors j.o are evaluated from lmf(:9)(x) - =
1 ,%,,(x) ro~2(x), R,~ l
, Krr
Iml'~:](~) = i ~;, PK. (x) ro 3/2 ~ ) ' 1 wTr lm.r~ l)Oix) = ~-z~ &,,(x) Too ix),
im&(Z~O(x) = ~5~ I OKK(x) TOO ~:K(x).
(3.4)
Now we analyse the propagators. We will extend the model for the 6 o discontinuity (eq. (2.15) in ref. [6]) to the spin-one propagators (60 and 6K* ) and that for 6 o (eq(3.67 m r e f . [4] or eq. (2.14) in ref. [6]) to the scalar propagators 6s (60 and 6~), respectively. In this way, we assume
._Llm 5(~)(x) = ~ 1 pnn(X) Tll,,,, ( x ) , 172 I im~2(.,,:)= 1 ~(.l)(x) = t~' l)(x) 7"- Wa(x). In the model under consideration the 8K, discontinuity is linear in do not contain any pole. Therefore. it is reasonable to assume hn d4 1)(s) = - ~- lm 6K,(S) ,
(3.s) s
whereas d,~ (3.6)
J. S6 Borges / Kaon-phm scattering
367
which makes it possible to obtain (3.1) for ,r(l) -11/2' The first-order correction to the form factors and propagators are obtained by writing twice-subtracted dispersion relations using, accordingly, the S- and P-wave projections of the expression (2.8) and the phase-space factor in (2.5). The zxpressions of the current algebra amplitude 7 "'r~r and T KK and the definition of phasespace factors P~rrr and PKK can be found in the literature [10]. We fix subtraction constants to be zero at s = 0; that is a particular choice of free parameters o t h e model " Finally we analyse the seagull ternr in (2..I 7). This term comes from the product of all momenta by the one-particle reduced irreducible function constructec with four axial-vector torrents. In tire hard-meson method, the "smoothness" hypothesis is implemented by approximating the primitive functions by lowest-order polynomial in the momenta compatible with Ward identities. Therefore, in the present rlodel, we have three real parameters for the seagull term, as can be seen in the appendix. When the expressions of form factors, propagators and seagull term are i~ troduced in (2.17), the partial-wave first-order corrected amplitudes takes on the folltJwing form satisfying (3.1):
T[Il )(s) = T(u'A2(s)G(s) + "rt1(s) + lPtl(s) .
(3.7)
In this expression
G(s)= 1
2
P K'r(X) dx K+mzr)2 X2(X - S)
while Tll are known functions with only left-hand discontinuities and.the frec parameters of tile model come in the seagull polynomial which is included in IP:I. In the appendix G is given and it is shown how to compute the 7"'s and the IP's.
3. 2. Numerical results Having computed the first-order correction to the form factors and to to the propagators, and projected seagull term, we find that the resulting amplitudes are
Tll(S ) = TCA(s){I + Oll(S)}, d)ll(S) = T~I1)(s)/T~IA(s).
(3.8)
Over the whole elastic interval the unitarity relation on the partial waves (~:.3) becomes I1+¢11(s)12=I
.
~' A less restrictive choice of these parameters does not change our numerical results.
J. Sd Borges / Kaon-pion scattering
368 d
0.4
Z° ~/2 0.2
. z9 I/2
~,~o
7~o
760
MK~(M e V ) -0.2
-0.4
-0.~ Fig. 1. A graph OfZll(S ) = (11 for S - a n d P-waves.
+
Oil(S)(
1 )/( [ l + Oil(S)( + I ) versus M K n over the elastic interval
is8
~2d
9d.
860
96o K1T
,ooo Moss
,,oo
(MeV)
Fig. 2. The isospin-half P-wave phase shifts.
J. Sd Borges / Kaon-pion scattering
369
8° "2
6d
30
960
lObO
Krl"
-3d
ll6o
M a f-s ( M e V )
,~ o o
fo%
I"ig. 3. The 1 = ~ and I = 3 S-wave phase shifts.
The S- and P-wave phase shifts can be calculated directly from tile partial-wave amplitudes by the definition (2.4)*,
A(s)PK"(s) 167r tg 611(s ) = I + Re-~II~)
½, '
I =
~,
1 = 0.1 .
(3.9)
Our m e t h o d consists in fixing the two real parameters of the I = ½ P-way(. defined in (3.8), by fitting the experimental [8] phase shifts near 90 ° , with the definition (3.9). This done, the other parameter is fixed in order to fit approximately the I = S-wave [8]. We obtain, for ~J01/2 above the K* meson mass, a " d o w n " type ~olution and it is impossible, in our model, to reproduce " u p " solution. The seagull parameters, defined in the appendix, corresponding to the figures, are ~Jl =
0.0531,
~J2 = 0 ' 0 8 3 7 '
~3 = 0 . 0 1 8 8 .
The corrections e l, to Weinberg's S-wave scattering length a w, calculated from
* As elastic unitarity is not exactly .satisfied, the definition of phase shift is quite arbitrary.
J. Sd Borges / Kaon-pion scattering
370
(2.6), a I = a)V( l + el), are el/2 = 0.035,
e3/2 = 0.350.
Therefore, the correction to the current algebra a~/2 -- a3/2 W is about 14%, which is bigger than other theoretical predictions [12I.
4. Conclusion Estimating vertex and two-point functions, we have exhibited the connection between the soft-meson Kn scattering amplitude and its exact representation derived by Ford. This way, we have defined a first-order correction to tire current algebra result. Performing this unitarization program of current algebra, we have constructed a model for the low-energy kaon-pion scattering. The current-algebra amplitudes are real and the computation of the first-order correction to current algebra leads to amplitudes that are unitary up to this order. Furthermore, as the isospin-~ P-wave current-algebra amplitude vanishes, its first-order correction is real, in our model. The full amplitude is exactly crossing symmetric and we do not have any arbitrary function to simulate tile left-hand discontinuities of the partial-wave amplitudes. Despite the lack of exact unitarity, this method, already in lowest-order approximation, takes to a sharp determination of the S-wave phase shifts once the I -- ~ Pwave is given in the low-energy region. Among our principal results are the corrections to Weinberg's scattering lengths and the S- and P-wave phase shifts. The isospin-~ Swave phase shifts are found to be of the " d o w n " type and the isospin-~ S- and P-wave results are in agreement with other predictions [9]. Going to higher-order correction would presumably improve the unitari, ty properties of the amplitudes, while preserving exact crossing symmetry. Finally we feel that these ideas, already used for obtaining n n phase shifts, should be applicable to other process for which the Ward identities of chiral current algebra play a role.
Appendix We can express the seagull amplitudes t +, and t - , respectively symmetric and antisymmetric in the exchange o f s by u, in terms of three real parameters/~1, ~2 and ~3 as
t+(1)(s, t, u) = ~ l ( t - - 2 , n 2 ) ( t - 2m2) + ~2[(s ~ M2) 2 + (u - M2)21, t-(l)(s,t,u)=~3[(s-
M2) 2 ( u
M2)2].
371
J. Sd Borges ] Kaon-pion scattering The total amplitudes of the model, with defined isospin in the s-channel are
7.(l,& l/2W.t. t , ) t= +{l) + _"~t (l) + - - - 3-
32t :4
,__( 32F 4
1 . 5tt.
96F4
+ -~,l~(S
)
, 2M 2
-
(t
) hK.(S ) tt + _ns44_ "' ,
(
,
u ] 1/2(u)
+
6
(M 2
u)h3/2(u)
u)hp(t) ,
T(1). 3/2(S, t, u) = l +( 1)
t (1) + _81F4(M2 .- s)h3/2(s)
m4) hl/2(u)
4864
u ] ~
+
2
14 (M 2 -. u) h3/2(u) -- 41,~1 ~ (s
tt)hp(t),
M2 hK,(S ) = ~
h i/2(s) =
- 23I 2+
1 (5s
G(s)+
~ m4
25• 2 ,_ 3 7 4 ) (;(S) + ~-~2 ( 7 M4.
s)G(s)
ttp(t ) = (2m K- t)g K(t)
M2=,,4+ m2
----
647r 2 '
1) +
tTl4
•
h3/2(s )= 2(M 2
~
s
3 M2 647r 2 '
5/4
16rr 2 m 4 t
967r2'
m2 _-4
'"2. •
From tile expression (2.5), we have defined S2
a(~)= ~
f (mK+mrr)2
PKrr(X)
dx-----
X2(X S)
resulting in
16n 2 G(s) = - p KTr(S) In
S -- M 2 + SPKTr(S) 2 mKl?lrt
m 2~ ] m K \m 2
-s
}
n nz. + 1 + iTrPK~(S)
372
J. Sd Borges [ Kaon-pion scattering
t t e r e , g K is the equal mass limit of G (in this case the kaon mass). The r and IP c o m p o n e n t s of the partial waves in (3.7) can be obtained introducing the above formulas in (2.2). This done, the second-degree polynomial P I 1/2, contains only two independent c o m b i n a t i o n s of the three parameters of the model.
References [ 1 ] M. GelI-Mann and M. Levy, Nuovo Cimento 16 (1960) 705; • Y. Nambu, Phys. Rev. 4 (1960) 380. [2] S. Weinberg, Phys. Rev. Letters 17 (1966) 616. 13] I.S. Gerstein and H.J. Schnitzer, Phys. Rev. 170 (1968) 1638; 1.S. Gerstein, J.H. Schnitzer and S. Weinberg, Phys. Rev. 175 (1968) 1873; I.S. Gerstein and l-l.J. Schnitzer, Phys. Rev. 175 (1968) 1876. [4] tt.J. Schnitzer, Phys. Rev. D2 (1970) 1621. [5] P. Ford, Nucl. Phys. B57 (1975) 25. [6] J. S~ Borges, Nucl. Phys. B51 (1973) 189. [7] R.W. Griffith, Phys. Rev. 176 (1968) 1705. [81 R. Mercer et al., Nucl. Phys. B32 (1971) 381. [9] C. Lovelace, Proc. Conf. on rrn and Krr interactions (Argonne National Laboratory, 1969) p. 562. [10] J.L. Petersen, Phys. Reports 2 (1971) no. 3. [1 l ] M. Gell-Mann, Physics 1 (1964) 63. [12] D.H. Boal, R.H. Graham and B. Weisman, Phys. Rev. DI2 (1975) 1472. [13] P. Ford and H.J. Schnitzer, Phys. Rev. D4 (1971) 2402.
UNITARITY CORRECTIONS TO KAON-PION SCATTERING FROM CURRENT ALGEBRA * J. S~i BORGES lnstituto de F{sica, Unirersidade Federal do Rio de Janeiro, Rio de Janeiro, Bra.:il
Received 9 February 1976 A method is pre~nted to unitarize the hard-meson current algebra kaon-pioJt amplitude derived by Ford. Elastic unitarity is satisfied in the lowest-order correction to current algebra. The amplitude contain.~three real parameters and does not have any arbitrary function for simulating left-hand discontinuities. Once the isospin-half P-x~ave is given in the low-energy region, this method leads to a sharp determination of tht: S-wave phase shifts. Our solution favors the "down" type for the I = ~ S-wave phase shi "ts.
1. Introduction The pion-pion and heaw-target-pion scattering amplitudes compatible wi:h tile requirements of the partially conserved axial-vector hypothesis [1] together with SU(2) X SU(2) current algebra were obtained by Weinberg [2] using tlle soft-pion technique. The soft-meson method gives a linear approximation that can be interpreted as tile truncation of a complete polynomial expansion of the amplitude in a singularity free region in the space of all the variables. Deviations are certain to exist, but their evaluation is muclt more model d.'pendent. One way for obtaining corrections is, for example, tlte hard-meson met rod based on the Ward- Takahashi identities technique [3]. Hard-meson amplitudes for tile on-mass-shell particles have been obtained by Schnitzer [4] describing pion-pion scattering and by Ford [5] for kaon-pion ~cattering. These amplitudes explicitly satisfy all the constraints of the current al,~ebra and are expressed in terms of one-particle reduced tunctioJIs. Using the soft-meson estimates for the two- and three-point functions, we [6] have shown that, near the symmetric point s = t = t¢ = m~, the Schnitzer result is equal to the Weinberg result plus a "tree structure" and a "seagull term". Furthermore, when form-factor and propagator discontinuities across tile physical * Research supported in part by HNEP. ('NPq and CEPI.~(, of Univcrsidade Federal do Rio de Janeiro. 357
J. Sd Borges /Kaon-pion scattering
358
cut were evaluated in lowest order correction, it turned out that, near threshold, tile contributions of the non-polynomial part to partial waves were small compared with Weinberg's result. This way, we have obtained a crossing symmetric correction to Weinberg's expansion that satisfies, only approximately, the requirements of two-body elastic unitarity. The present analysis will parallel our study of pion-pion scattering [6]. In this paper, using vertex and propagator estimations, we can reobtain a Weinberg-like result [71 from the hard-meson amplitude. The main point in this derivation is the fact that since current algebra gives real amplitudes, the nearby partial-wave discontinuities across tile right-hand cut are small compared with tile current algebra result. These discontinuities arc known in the first order. In this way, we construct the structure functions entering the amplitude from lhcir first-order discontinuities by the dispersion relation technique. Working with three parameters, only in the seagull term, we determine a crossing symnretric correction to the kaon-pion low-energy scattering amplitude that does not contain any arbitrary function to simulate partial-wave left-hand discontinuities. In this scheme two-body unitarity constraints in partial-wave amplitudes can be satisfied only approximately. We fix the tw,, r,arameters entering the total isospin-half P-wave by fitting the experimental 18] phase shift near 90 °. The remaining parameter is adjusted and we obtain S-wave phase shifts. Our Si/2 phase shift is found to be of thc " d o w n " type above the K* meson mass and the $3/2 phase shift is repulsive and small, which is in agreement with other predictions [91 . in sect. 2 we give the basic formalism for describing the kaon-pion system and, using soft-meson estimates, we reproduce the linear current-algebra amplitude [71 from the exact representation derived by Ford [5]. In sect. 3. a model for the form-factor and propagator discontinuities regarding elastic unitarity constraints is constructed and we discuss the seagull term. In this section we also present ttae first-order corrected amplitudes and numerical results. The detailed structure of the different terms entering the amplitude is given in the appendix.
2. Low-energy kaon-pion scattering amplitude 2.1. Basic"formalism Here we compile the formulas pertaining to Kzr scattering. For the elastic scattering K,~lr~ ~ K7 q,rr{L/,,, we define tile invariant anaplitudes T ~(s, t, u) by
(K'rq,Tr~_.p,JTlKqrr~p) = X~.'l,{6CcaT+ + ~[r a, T q r } X k ,
(2.1)
J. Sd Borges /Kaon-pion scattering
359
where )~- are the isospinors, s : - ( q +p)2, t = - (q + q')2 and u = - ( q +p')2. Other symbols have their usual meaning [10]. We define partial-wave amplitudes Tll(s ) for total isospin 1 in the s-chanr~el by I 1 = ~, 3 ,
Tt(s, t, u) = ~ ( 2 / + 1)Tll(s)Pl(cos Os), 1= 0
(2.2)
where
Ti/2(s, t, u) = T+(s, t, u) + 2 T - (s, t, u ) , T3/2(s, t, u) = T+ (s, t, u) . T - (s, t, u) , 2St= Is - (m K +mTr) 2 ] [ s .
(m K-- m r r ) 2 ] ( c O S 0 s - l ) ,
with 0s being the c.m. scattering angle. For elastic scattering we get, in particular below the first inelastic threshold, 1
Im Tll(S) = ~6-~ PK"(S) I Tu(s) 12'
(2.3)
which may be solved giving 16rr i611(s) TII (s) = - - - - e sin 8tt(s ) ,oK,(s ) where 811($)
(2.4)
the real phase shifts and,
are
1
pKrr(S) = s[S - (m K + mrr) 2 I I/2 [ S - (m K - m , ) 2 ] 1/2 ,
(2.5)
is the phase-space factor, for s>~ (m K + m~,) 2.
2.2. Current algebra exact representation for the Kn system The starting point in our derivation is Ford's exact kaon-pion scattering amplitude constrained by current algebra 151. Particle names will be used to label the quantum numbers of the two-, three- and four-point functions. Without assuming any representation to which the symt:~etrybreaking belongs, it reads "~ 2
c~: F2 c. ~K'0'o;~,~/(0) F2K " c,, =- Ta~v's (q' P' q')
,aK(0)~ A6,,,(O), " ~4i/Im ~"K" (q' I,, q ') c~ A c~, .¢c~ (q p. q,) F2
k~-77 ~,~'t6
'
360
J.
[-CKA CAI
L-F~7 -F~K*r' XA
Sd Borges / Kaon-pion scattering
. ~,~KAA I K*, q'l'
I
",~~,a,,.¢~"tq, p) + ----(;%(q21"KF.,., ,' P)
FC KACA~
';V m . ~ K A A ~ K *
- ,
~
]
rl .
,
(q+p)kF K l.r q p i tx,~o,,r,(q,p')+2~.~7/.r~((q
,
+--I-L-J'~[sef~'~,', [(q- P)(q'- p')S~¢'" +(q2 ._ p2)(q'2_ ,2 -2 41"Kf' ~
P')r
p'2)¢KGr/~d( q +P)l
,,
t 1~ XAhh'(q
+I))
p'
'
•
2f- K
LI., 2 v P
1
--*
(0,FKrt, ¢ , , ~ , I .. -F-~A Kyl"(0~Arr " ~rn"" lm,g(q'p)An'h '(q , + P)+72~-F~(r'r'Ta)h'n '(`
.
ze~,ao,,,,.(v,r")+
~
r;
,1
_
t
K--Tt 3
F2 I%
.,, tq. q ~ ,,~,~q+q'~ + 2 ~ { r~. r~h, , %
%h'(q +q )
P~
(2.6)
x
A;(o)~-~.,,~, "o" r /,,,,,,~r),r,')a,~,t,,(p+Y) "~°" + ; ~l{ T ~ L?h',,'%,
,
where c~,/3, etc. are indices of SU(3) which generators are the antisymmetric matrices 7~.. As demonstrated in ref. 131 there is the following relationship between the decay constants of mesons into leptons and the symmetry-breaking parameters e: ei T/~ = (Faro 2 / x / Z b , where re(Z) is the mass (wave function renormalization) matrix for the meson multiplet. It is convenient to label the e by the name of the meson appearing on the right-hand side of that relation. Thus, for example, e~ = t'~rn2/x/Z,r. In formula (2.6), A ~ ( q ) is the spin-one current propagator while the conventional meson-field propagators Ag(q), the functions Di/(q ) and I)i/¢q).. are written in terms
J. Sd Borges / Kaon-pion scattering
361
of the spectral function pO(m2) as d (m2) pp;(m 2 )
=
fd(m2)p°.(m2) fd(m2)pB.(m 2 )
,/(q)
=-
(2.7)
,T(;,T; q i-
st) that
bq(q) : D,.c (q)
,
"~" ' ") ")
{ll.
2 + (p'- q '}k (q~eKaf(o)-p~2~2~(o))] + ~do,~<)d~K~K(O)+~,~(O))
'~K (0)
&,(0)
~)< (0)
~,,(0! -
+ crossed terms} !
where ]'aCe and da~e are the SU(3) Clebsch--Gordan coefficients introduced ')y Gell-Mann I I 1 ]. In order to obtain tile current algebra soft-meson result [7] from T cA, we approximate spin-zero meson-field propagators by saturating the spectral functi ms in (2.7) with the kaon and pion particle masses, I
-=
,~.{p) = ~
~-
l
?II~ +
p-~
~Z;,
J. Sd Borges / Kaon-pion scattering
362
and we consider F K = 1;'~ = F [7], and Z K = Z~r = 1, CA . 1 3 2 .._ 3 2 Ta3.},6(s, t, u) = 3~- ~ {fc~&]'c.rs(t -- u) + da~jcdevs(s + t + u --- ~m K im~r ) + crossed t e r m s } , SO
T+Ca(s, t, u} = 4 ~l { s + . + 2t - 2m2K - 2,n 2) 1
T-CA(s,t, U ) = ~ . 2 (S - It).
(2.8)
Note that we could correct this result by a p o l y n o m i a l if we had introduced, generically, different FK, F,r, Z K and Zrr * In order to analyse tile remaining terms of eq. {2.6), we use tile results for the three-point functions,
CKACAt
,u v,,KAA1K* -
K*'
F K -t~- q p l au, 3v, e'z'tq'p)Ac'rer(q+p)+
~F@n
..
K*,
.f~'(q
P)r'A"rer(q+P)
x Z / ~ x / - ~ F Ka~,e,r, n K . . tq, p ) A eK" 1 fa~e [ ( q . . . P} r S _~+_FTZKqr FK =VLKVZ*r , r r ( q + p ) + b~,F - .z~KZTrp F . r]
C2A, , A ' A ' o . p,)Af,~r(p , 1 . +P ) + ~,.2]~8~'(p " p')T'APc,r~T(P+P ') F 2 pVp op3v, 6o, e,r,(p, 21"n
..o
=Z~P¢6,c','(P,P')~'~
tP+P )+]~6~(P
(
)
P')" I +21F ~ ,
(2.9)
and a similar result for the KAKAO vertex. The next step is to relate the proper vertex parts and two-point functions to conventional form factors and propagators. The K13 form factors j"~_ and the electromagnetic form factors j . v are defined as
* A ratio FK/I,'Tr >~ 1.3 seems to be rather unlikely [121. Ilere the Z's are being assumed finite phenomenological parameters [ 5 J.
J. Sd Borges /Kaon-pion scattering
363
X D~,c(q+p) +]~,,ac [(P " q)rf+(s) + ( q + p ) r f (s)] , Znl ~6~.,r,(p,p )Ac, ~ (p • p) = (p' p)~J~'(t),
(2.10)
and a similar definition for the kaon electromagnetic forln factor .l~.) "0 and]'", the scalar form factors upto a constant, are connected with vertex functions by
d2K ~ + r2~.y, (q, p)~,, +q+t,) +~-FK-~-. Z,rV~°,(p.p')a°,+(p +p') +
(T,.
d~./~ (tt
T~)~..,(¢)+. =
•
Ill
~" ¢ I
and a similar connection for tile sigma-kaon-kaon formtactorjK; the relatiu~ between J. and ] . 0 % ¢ j ,.0 ~,(.,;)=
[(q2
p2)j+(s)+s J. tS)le,~e "
where
J:?i3c ='/~3c
- 2t.'K [."~ {Ta' T#)+~.'(0),,'
(2.12)
will also be used m the following. The relation between the total amplitude T ,,./$,7,5 KrrKrr, for on-mass-shell particles, and /14KrrKrr• is the proper function --o~,7,~ KnKn ,2 , KrrKn Ta~,7,5 (q,p,q , )= - 1 ,,2Kl'r:'~la~,Ta (q,p,q').
(2.13)
When the definitions (2.10), (2. l 1) and (2.13) are introduced in (2.6), we obl ain T Kr~Kn(q, p, q') =
T('A(q, p. q') + T KTrKn(q,p. q ' ) .
where 2
T"IJ`7,5 (q'P'q )
2
1"~ F~ ~ , 5
'
{
FK (
+ (q +Ph\A ,~vds) ++K ~"(s) Dn(s) "~0 (s'~ °'~""
F~Z~,,,)
I . ~ _.p2)CKe~*(s)) 1 -%,_~/~to~+(q-
J. S6 Borges / Kaon-pion scattering
364
p'2)eKe~[)K(s)} +f2~(s)~,(s)j,v~(s -o ,, :o ) + C) ,.~
+ (q2_ p2)(q'2
(q- q')(P- P ) l ~ l ~ a 6 1
S]+
4F2 F21r
2F~
d~'~Edc#6
]'~(t)Txl°(t)fg(t)
(2.14)
Next we analize formula (2.14) and show that 7" partial waves at threshold are small when compared with the current algebra amplitude T CA projections.
2.3. Solt-meson estimates The Ward identities analysis, together with the hypothesis of single-particle dominance of all propagators, lead us to the following assumptions for x ,'-' (rn K + rnrr)2 :
f+(x) ~- l + f~+l)(x) , f ~ ( x ) ~- l + f ~ ( ' ) ( x )
f__(x) ~_t(_l)(x),
}°(x) ___f0~'~(0,), f2(:,) ,-, 1 + t ~ ( % 0 ,
, A K , ( x ) ~-- C,¢, +8~2(~) ,
AK(X) ,-, ~ ( ~ ) ,
,%(X)" ~(J)(X).,
%(x) ~- cp + ~')(x) ,
(2.1S)
MI functions denoted with a superscript (1) are of order (m 2 + rn2)/X 2, where X is of the order of magnitude of the vector meson mass, in a vector-meson dominance approximation. In addition, we assume that the partial-wave contributions of the seagull term, near threshold, are small and almost real [4,13],
,~l(X)'-,tl/)(x),
o(
Im ,~'1(x) "--, \ - - X ~ -
.... /.
¢2.1 6)
When the definitions (2.1 5) and (2.16) are introduced in (2.14), we obtain
:~K"K'~(s, ,, u)~-- T(l)K"~"(s, t, u)+ O(M4/X4) , where T(1) KrrKrr.,. (l) (a, o t, U) a[3"~6 ~S, t, it) + ta.#.yb
{
J'o',,,,)]
J. Sd Borges / Kaon-pion scattering
+ 4;
j,'o3e]~t,~e[(t
u)6(KI,)(s) + (q2 __ p2)(q'2 _ p'2)%eKd(l)(s)]
+t-(l) Cs~ 1 - - - ~ r ( t ) J 0aBe"
: (~( 1)ee' " 0e'~',b (S) K
+ 2~.:Sa.reJe3~(s u)
+(s
o
u)}
"~(1)(t) +f~(l)(t) + 2 F- - 6 ( 0
((l)o(t~
+da.r~de~K
365
1
f.(l)o(t ~
' " ~-~l)(t%~"
"" "
(2.17)
o
We can interpret the result of hard-meson calculation as follows: near the softmeson point or the Adler point, T(1)(s, t, u) is small compared with the well-known soft-meson amplitudes T('A(s, t, u) explicitly shown in (2.8). In (2.17) we have assumed, for the sake of simplicity: F K = FTr = F. In sect. 3, we will develop a model for the first-order corrected amplitude T (l).
3. Un~tarization
3.1. First-order unitarity corrections Now we must construct a model for the form factors, for the propagators and for the seagull term involved in the first-order corrected amplitude T (I). First we observe that, as the limiting amplitude is real, the following discontinuities for the partial .Javes T(1) are expected: Im T[/)(s) = ~6-~pK,r(s)r]l 1 (-A2(s).
(3.1)
Let us start analysing the form factors; we intend to make progress by application of the constraints of elastic unitarity. The two-particle unitarity equation sat isfied by form factors is Im F(s) = 1 ~ p(s) T*(s)F(s) . Having in mind (3.1), the electromagnetic and 1~_ form factors are obtained fiom Im.fl~(1)(x)
= 3-~PKK
(,.)
J. $6 Borges / Kaon-pion scattering
366
1 1
unf~+~)(x) -'- i U~ °K,Cx) r1~2 (x) "
(3.27
By assuming equal phases for D ~' and A • , we can write from (2.7), lm
....
a"(s)
s
thus from (2.1 2);
" ~(s)
......
s---
On the other hand, the isospin-~ and 3 projections I"1/2 and )"3/2 of the scalar structure function ~(1) ~0~t3e and the sigma form factors j.o are evaluated from lmf(:9)(x) - =
1 ,%,,(x) ro~2(x), R,~ l
, Krr
Iml'~:](~) = i ~;, PK. (x) ro 3/2 ~ ) ' 1 wTr lm.r~ l)Oix) = ~-z~ &,,(x) Too ix),
im&(Z~O(x) = ~5~ I OKK(x) TOO ~:K(x).
(3.4)
Now we analyse the propagators. We will extend the model for the 6 o discontinuity (eq. (2.15) in ref. [6]) to the spin-one propagators (60 and 6K* ) and that for 6 o (eq(3.67 m r e f . [4] or eq. (2.14) in ref. [6]) to the scalar propagators 6s (60 and 6~), respectively. In this way, we assume
._Llm 5(~)(x) = ~ 1 pnn(X) Tll,,,, ( x ) , 172 I im~2(.,,:)= 1 ~(.l)(x) = t~' l)(x) 7"- Wa(x). In the model under consideration the 8K, discontinuity is linear in do not contain any pole. Therefore. it is reasonable to assume hn d4 1)(s) = - ~- lm 6K,(S) ,
(3.s) s
whereas d,~ (3.6)
J. S6 Borges / Kaon-phm scattering
367
which makes it possible to obtain (3.1) for ,r(l) -11/2' The first-order correction to the form factors and propagators are obtained by writing twice-subtracted dispersion relations using, accordingly, the S- and P-wave projections of the expression (2.8) and the phase-space factor in (2.5). The zxpressions of the current algebra amplitude 7 "'r~r and T KK and the definition of phasespace factors P~rrr and PKK can be found in the literature [10]. We fix subtraction constants to be zero at s = 0; that is a particular choice of free parameters o t h e model " Finally we analyse the seagull ternr in (2..I 7). This term comes from the product of all momenta by the one-particle reduced irreducible function constructec with four axial-vector torrents. In tire hard-meson method, the "smoothness" hypothesis is implemented by approximating the primitive functions by lowest-order polynomial in the momenta compatible with Ward identities. Therefore, in the present rlodel, we have three real parameters for the seagull term, as can be seen in the appendix. When the expressions of form factors, propagators and seagull term are i~ troduced in (2.17), the partial-wave first-order corrected amplitudes takes on the folltJwing form satisfying (3.1):
T[Il )(s) = T(u'A2(s)G(s) + "rt1(s) + lPtl(s) .
(3.7)
In this expression
G(s)= 1
2
P K'r(X) dx K+mzr)2 X2(X - S)
while Tll are known functions with only left-hand discontinuities and.the frec parameters of tile model come in the seagull polynomial which is included in IP:I. In the appendix G is given and it is shown how to compute the 7"'s and the IP's.
3. 2. Numerical results Having computed the first-order correction to the form factors and to to the propagators, and projected seagull term, we find that the resulting amplitudes are
Tll(S ) = TCA(s){I + Oll(S)}, d)ll(S) = T~I1)(s)/T~IA(s).
(3.8)
Over the whole elastic interval the unitarity relation on the partial waves (~:.3) becomes I1+¢11(s)12=I
.
~' A less restrictive choice of these parameters does not change our numerical results.
J. Sd Borges / Kaon-pion scattering
368 d
0.4
Z° ~/2 0.2
. z9 I/2
~,~o
7~o
760
MK~(M e V ) -0.2
-0.4
-0.~ Fig. 1. A graph OfZll(S ) = (11 for S - a n d P-waves.
+
Oil(S)(
1 )/( [ l + Oil(S)( + I ) versus M K n over the elastic interval
is8
~2d
9d.
860
96o K1T
,ooo Moss
,,oo
(MeV)
Fig. 2. The isospin-half P-wave phase shifts.
J. Sd Borges / Kaon-pion scattering
369
8° "2
6d
30
960
lObO
Krl"
-3d
ll6o
M a f-s ( M e V )
,~ o o
fo%
I"ig. 3. The 1 = ~ and I = 3 S-wave phase shifts.
The S- and P-wave phase shifts can be calculated directly from tile partial-wave amplitudes by the definition (2.4)*,
A(s)PK"(s) 167r tg 611(s ) = I + Re-~II~)
½, '
I =
~,
1 = 0.1 .
(3.9)
Our m e t h o d consists in fixing the two real parameters of the I = ½ P-way(. defined in (3.8), by fitting the experimental [8] phase shifts near 90 ° , with the definition (3.9). This done, the other parameter is fixed in order to fit approximately the I = S-wave [8]. We obtain, for ~J01/2 above the K* meson mass, a " d o w n " type ~olution and it is impossible, in our model, to reproduce " u p " solution. The seagull parameters, defined in the appendix, corresponding to the figures, are ~Jl =
0.0531,
~J2 = 0 ' 0 8 3 7 '
~3 = 0 . 0 1 8 8 .
The corrections e l, to Weinberg's S-wave scattering length a w, calculated from
* As elastic unitarity is not exactly .satisfied, the definition of phase shift is quite arbitrary.
J. Sd Borges / Kaon-pion scattering
370
(2.6), a I = a)V( l + el), are el/2 = 0.035,
e3/2 = 0.350.
Therefore, the correction to the current algebra a~/2 -- a3/2 W is about 14%, which is bigger than other theoretical predictions [12I.
4. Conclusion Estimating vertex and two-point functions, we have exhibited the connection between the soft-meson Kn scattering amplitude and its exact representation derived by Ford. This way, we have defined a first-order correction to tire current algebra result. Performing this unitarization program of current algebra, we have constructed a model for the low-energy kaon-pion scattering. The current-algebra amplitudes are real and the computation of the first-order correction to current algebra leads to amplitudes that are unitary up to this order. Furthermore, as the isospin-~ P-wave current-algebra amplitude vanishes, its first-order correction is real, in our model. The full amplitude is exactly crossing symmetric and we do not have any arbitrary function to simulate tile left-hand discontinuities of the partial-wave amplitudes. Despite the lack of exact unitarity, this method, already in lowest-order approximation, takes to a sharp determination of the S-wave phase shifts once the I -- ~ Pwave is given in the low-energy region. Among our principal results are the corrections to Weinberg's scattering lengths and the S- and P-wave phase shifts. The isospin-~ Swave phase shifts are found to be of the " d o w n " type and the isospin-~ S- and P-wave results are in agreement with other predictions [9]. Going to higher-order correction would presumably improve the unitari, ty properties of the amplitudes, while preserving exact crossing symmetry. Finally we feel that these ideas, already used for obtaining n n phase shifts, should be applicable to other process for which the Ward identities of chiral current algebra play a role.
Appendix We can express the seagull amplitudes t +, and t - , respectively symmetric and antisymmetric in the exchange o f s by u, in terms of three real parameters/~1, ~2 and ~3 as
t+(1)(s, t, u) = ~ l ( t - - 2 , n 2 ) ( t - 2m2) + ~2[(s ~ M2) 2 + (u - M2)21, t-(l)(s,t,u)=~3[(s-
M2) 2 ( u
M2)2].
371
J. Sd Borges ] Kaon-pion scattering The total amplitudes of the model, with defined isospin in the s-channel are
7.(l,& l/2W.t. t , ) t= +{l) + _"~t (l) + - - - 3-
32t :4
,__( 32F 4
1 . 5tt.
96F4
+ -~,l~(S
)
, 2M 2
-
(t
) hK.(S ) tt + _ns44_ "' ,
(
,
u ] 1/2(u)
+
6
(M 2
u)h3/2(u)
u)hp(t) ,
T(1). 3/2(S, t, u) = l +( 1)
t (1) + _81F4(M2 .- s)h3/2(s)
m4) hl/2(u)
4864
u ] ~
+
2
14 (M 2 -. u) h3/2(u) -- 41,~1 ~ (s
tt)hp(t),
M2 hK,(S ) = ~
h i/2(s) =
- 23I 2+
1 (5s
G(s)+
~ m4
25• 2 ,_ 3 7 4 ) (;(S) + ~-~2 ( 7 M4.
s)G(s)
ttp(t ) = (2m K- t)g K(t)
M2=,,4+ m2
----
647r 2 '
1) +
tTl4
•
h3/2(s )= 2(M 2
~
s
3 M2 647r 2 '
5/4
16rr 2 m 4 t
967r2'
m2 _-4
'"2. •
From tile expression (2.5), we have defined S2
a(~)= ~
f (mK+mrr)2
PKrr(X)
dx-----
X2(X S)
resulting in
16n 2 G(s) = - p KTr(S) In
S -- M 2 + SPKTr(S) 2 mKl?lrt
m 2~ ] m K \m 2
-s
}
n nz. + 1 + iTrPK~(S)
372
J. Sd Borges [ Kaon-pion scattering
t t e r e , g K is the equal mass limit of G (in this case the kaon mass). The r and IP c o m p o n e n t s of the partial waves in (3.7) can be obtained introducing the above formulas in (2.2). This done, the second-degree polynomial P I 1/2, contains only two independent c o m b i n a t i o n s of the three parameters of the model.
References [ 1 ] M. GelI-Mann and M. Levy, Nuovo Cimento 16 (1960) 705; • Y. Nambu, Phys. Rev. 4 (1960) 380. [2] S. Weinberg, Phys. Rev. Letters 17 (1966) 616. 13] I.S. Gerstein and H.J. Schnitzer, Phys. Rev. 170 (1968) 1638; 1.S. Gerstein, J.H. Schnitzer and S. Weinberg, Phys. Rev. 175 (1968) 1873; I.S. Gerstein and l-l.J. Schnitzer, Phys. Rev. 175 (1968) 1876. [4] tt.J. Schnitzer, Phys. Rev. D2 (1970) 1621. [5] P. Ford, Nucl. Phys. B57 (1975) 25. [6] J. S~ Borges, Nucl. Phys. B51 (1973) 189. [7] R.W. Griffith, Phys. Rev. 176 (1968) 1705. [81 R. Mercer et al., Nucl. Phys. B32 (1971) 381. [9] C. Lovelace, Proc. Conf. on rrn and Krr interactions (Argonne National Laboratory, 1969) p. 562. [10] J.L. Petersen, Phys. Reports 2 (1971) no. 3. [1 l ] M. Gell-Mann, Physics 1 (1964) 63. [12] D.H. Boal, R.H. Graham and B. Weisman, Phys. Rev. DI2 (1975) 1472. [13] P. Ford and H.J. Schnitzer, Phys. Rev. D4 (1971) 2402.