K→π, K→2π and K→3π amplitudes with unitarity

Volume 207, number

PHYSICS

4

K+ IT,K-t 2n AND K+3x AMPLITUDES

LETTERS

B

30 June 1988

WITH UNITARITY

Tran N TRUONG

Recelvcd

25 March

1988

Due to the unltarlty which IS neglected III all recent works, the Al= ;Krc amplitude 1s shown to be 40% smaller than that given data by the tree lagranglan calculation The K- 3rr rates and spectra are m excellent agreement with experlmental

The reason why the umtarlty conditions must be imposed on the soft plon theorem m processes mvolvmg two or more plon emlsslon m the final state 1s not only a matter of prmclple but 1s also a matter of necessity to reconcile the current algebra calculation without umtarlty correction and the expenmental data [4-81 For example, the current algebra or choral lagranglan calculation by Weinberg and others [ 93 for the S wave I= 0 scattenng length, the K-tnxev and the 17’3~ decays are too low by 40% or more (in amplitudes) compared with experimental results It was previously shown that the disagreement between the theoretical predictions and experimental results m these apparently unrelated processes 1s due to the neglect of the umtarlty m the final state multlplon system [4-S] The essential point is that the I= 0, S wave low energy in interaction 1s large and therefore introduces large corrections to the current algebra theorems The umtarlty correction for these processes was carried out using the exact solution of the integral equation of the Muskhehshvllh-Omnis type which is of course non-perturbatlve The final results can be expressed m terms of the I= 0, S wave nn phase shift which 1s experimentally known and the final results agree with experimental data [ 4-7 ] In this approach, the final state theorem, which is one of the few rigorous results m strong interaction physics (valid to first order in the electromagnetic or weak mteractlons but to all orders of the strong mteractlon) 1s fully used to check various approxlmatlons This 1s in contrast with the fashionable choral perturbation theory which IS not only cumbersome but 1s

A precise knowledge of the KJ~:amplitude and its relation with K+2n and K+3rr: amplitudes are of a fundamental Importance m the study of the origin of the AI= f rule, the CPvlolatlon effect in the standard model, the K,_-KS mass difference and the wave decays of KL and KS A detailed study of these amphtudes also serves as a useful purpose of providing a possible solution to some problems encountered in the lattice gauge calculation of the AZ= 4, K-t2n decay due to the euchdean nature of the lattice gauge calculation It 1s usually assumed in the literature that the Kn amplitude can be obtained straightforwardly from the K+27t amplitude using the usual current algebra technique #’ (which could give a wrong result d one was not careful) or more directly from the non linear d model choral lagranglan at the tree level approxlmatlon [ 31 It 1s shown here that the Kz amplitude obtained by these methods, which are widely used m the literature, 1s 3540% too high This 1s so because the umtarlty correction to the Z=O, S wave XX scattering 1s usually neglected m calculatmg the K+2z amplitude As a support for this claim, a complete K+3n calculation with the umtarlty correction 1s given Good agreement with experimental data 1s obtamed to a level of a few percent which 1s an lmprovement over earlier tree lagrangian calculations yielding a K+ 37t amplitude to be 20% too small see e g ref [ 1] m this book IS too large by a of choral symmetry [ 21

‘I For a review of the current algebra calculations The K-X couphng determmed factor of two due to a vlolatlon

0370-2693/88/s (North-Holland

03 50 0 Elsevler Science Publishers Physics Pubhshmg Dlvlslon )

BV

495

Volume 207, number

PHYSICS

4

Just a perturbation approach where the final state theorem cannot be satlsfactorlly implemented #* The simplest way to Implement chn-al symmetry on K-+27t and K-+3x 1s to use the non lmear choral lagranglan [ 31 In the followmg the AZ= 1 amphtudes for the K-n, K+2x and K+3n are calculated and the results are to be compared with the corresponding AZ= 4 amplitudes extracted from the experimental data Usmg the effective choral lagranglan and denotmg respectively, M( &( k) --tit+ (p)rc- (q) ) and M(KL+n+ (41 )n- (q2)x”(q3) ZZ,(s, t, u), we have

)> by F,(k** q*, P’),

F,(k’,q*,p’)=tlCf,(2k*-p2-q*),

(la)

H,(.&& ~)=(C/Jz)(~-P2L

(lb)

M&+X’)=

- (Cl&)f;q(N

q(K)>

(lc)

wheres= (ql +q2)*, t= (q2+q3)2, u= (q, +qs)2 with s+ t + u = 3so = m2 + 3~’ where m and p are respectively K and 7cmasses and the subscript t refers to the tree approxlmatlon Eq ( 1b) 1s valid when the three plans are on their mass shell It satisfies the current algebra theorem for taking the “odd” plon soft (q3+O) Eq ( lb) 1s the sum of the three Feynman amplitudes, the contact, the K pole and n pole amphtudes [ 31 The plon pole graph 1s the product of the KX amplitude and the analytic contmuatlon of the R~C+X~C amplitude Using the experimental determination of the AZ= f ampl~tudea,,2=(0469~0006)~10-3MeV [ll], It is found that C= 1 26x lo-” MeV-*, which is the standard value widely used in recent pubhcatlons The K+31r amplitude determmed by eq (la) with this value of C 1s

LETTERS

B

30 June 1988

amplitude to be 20% too low, and the odd plan slope to be 12% too low This determination of the value of C is obviously not reliable as it depends on the vahdlty of eq ( 1a) which does not satisfy the umtarlty constraint This 1s so because the umtarlty and the time reversal mvariance requires the K+2n, AZ= f amplitude having the phase So, Z=O, S wave 7cn:scattering which amounts to approximately 35 O-45’ at the K mass and 1s by no means neghglble #3 In the presence of the strong mteractlon, the effective choral lagranglan such as given on the RHS of eq ( 1a) makes sense only m the region where there is no singularity With the two plans on their mass shell the function F( k2, p2, ,L?) IS an analytic function m the cut s= k” complex plane, with a cut on the real axis starting at the threshold s=4~* Eq ( la) should be interpreted as the first two terms of a power series expansion which IS only vahd for s < 4,u* It states that F has a zero at s=p*, Its derlvatlve at this pomt is C With these two boundary condltlons we can write a subtracted dispersion relation for E(s) = F(s, ,u2, ,u2) F(s) =C(s-p2) Im F(s’ ) ds’ (s’ -/L2)2(s’ -s-1c)

(3)

Using the elastic umtarlty approxlmatlon Im F(s) =F(s)e-““sm So which IS a good approxlmatlon m eq (3), we obtam an integral equation of the Muskhehshvllh-Omnls type whose solution is (4a)

F(~)=C(J-P~)GO(~,P*), where co

8, (s’ ) ds’ (s’ -_cL2)(s’ --S-16) =7 43x lo-‘[ as compared

l+O 233(s-so)y-‘1,

with the experimental

empirical

[Ill

=9 10x lo-‘[

From the defimtlon

IS0

264(s--~~)fi-~]

It 1s seen that the tree lagranglan *’ Standard

choral perturbation onclle with dlsperslon theory

496

(4b)

value

theory

[ lo]

(2)

yields the KL+ 3rc

can be Improved

to rec-

of eq (4b) it IS clear that F(s)

43 For a review of the rm phase shift see ref [ 121 The recent I= 0, S wave KX phase shift analysis by Blswas et al [ 13 ] gives slgmticantly lower values of 6, below 800 MeV These data are contradicted by the recent measurement of Clark et al [ 141 The data of Blswas et al on the I=O, S wave scattermg length also contradicts with that given by the better experiment of Rosselet et al [ 151 In this article we ignore the data of Blswas et al

Volume 207, number

PHYSICS

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has the phase S, as required by the umtanty and satisfies the boundary condltlons as imposed by eq ( 1) Because Go(s, p’) IS related to &, we could evaluate Go(s, ,LL*)using the expenmental data on the S wave I=O, xlr phase shift It 1s however more convenient to parametrlse G,(s, ,u*) m terms of a constructed S wave nc7[:partial wave amphtudefO(s) which 1s consistent with the Weinberg low energy expansion so that Its choral property 1s explicit [ 6,7] We adopt the followmg simple parametrlsatlon for the S wave amplitude h(s) =e”” sin &/p(s) where p(s) = (s4~2)11zls1r2 fo(J)=tL(s-4y2)lD(s),

(5)

where O(J) = 1 +b, (s-f/?) x

+ fL(s-

f/L*)

wlthf,=

Wn)J’6%%In[(,b+J’~)P~l

133 MeV and h(s)=

A

good fit to experimental data on S,, 1s obtamed with bO= -0 04 ,L-’ The scattering length a, 1s 0 2 1 ,L-* and the phase shift at s = mz is 40’ We have Go (s, $)

=D(fi’)lD(s)

(6)

From eq (6) we can deduce G,,(s=m2,~2)=1

40ei4””

(7)

and using eq (4a), we have C=O 90x lo-”

MeV-*,

(8)

which 1s 40% lower than the currently used value The Kn amplitude eq (lc) requires, on the other hand, no correction except for the self energy correction m the propagators of K and n which are assumed to be negligible here The real test for the new value of eq (8) lies m our successful calculation of the K+ 3n: amplitude which is presented below Before doing so, it IS worthwhile to show the difference between the present approach and the more popular choral perturbation theory In the last method, instead of solving the integral equation (3), one iterates it once by approxlmatmg fo(s) z iL(s4~‘) This perturbatlve approach [ 81 is not sufficiently accurate for this problem It can be shown however that the usual choral perturbation theory can be modzjied to give an identical result as given by the non-perturbatlve method presented here

1101

30 June 1988

B

The K+ 371problem From the choral lagranglan of the current algebra, assuming H, transforms as an octet, it 1s straightforward to deduce the following boundary condltlons llm M(K,~~+(q,)n-(q,)K’(q,))=O y,-0 (orY* +O)

(9a)

and hm M(K,-tKfX-zo(q3)) 43m.O

=-

(1

l&&W&+x+0

(9b)

By inspection, we could try to construct the followmg approximate KL+n+n-~O amphtude HO(s, t, u) H”(s,t,u)=(CIJZ)(s-~u’)Go(s,~L2)

th(s)-h(lP*)-v(s)l,

where L= (47&)-

LETTERS

(10)

Eq (10) satisfies the boundary condltlon Imposed by eqs (9) Using C determined by eq (8), the K,+3n amplitude 1s found to be 25% lower than the Al= f experimental value given above, the odd plan spectrum 1s however m perfect agreement with the experimental data (eq (2) ) The fact that eq ( 10) does not give the correct rate is not surpnsmg smce it satisfies only the two-body umtarlty relation, which 1s a poor approxlmatlon To take mto account the three-body interaction, we can make use of the KhunTrelman and Sawyer-Wall integral equation [ 16,17 ] written a long time ago and first used by Neveu and Scherk [ 181 m combmatlon with the current algebra to study the odd plon spectrum For slmphclty, we neglect the I= 2, S wave and the I= 1, P wave nrc mteractlons The function H( s, t, u) now depends on a single variable s and the umtanty relation reads ImH(s)=[H(s)+jH(J)]e-“‘“sm6,,

(lla)

and H(s)=;

Jg

[H(t)+H(u)],

(1 lb)

which 1s the S wave projection The resulting Integral equation for H(s) 1s much more comphcated than eq (3 ) No exact solution is known but an approxlmate solution can be found [ 5,16 ] Defining a correction function g(s) H(s) =Ho (s) +g(s) mg(s) 1s to be constructed out of Go(s, p’), eqs (4) and vanishes at s=pz and s=m2 497

Volume 207, number

4

x[G,(s,~‘)-l](m’-s)/(m”-so)

PHYSICS

(12)

Putting eq ( 12) into eqs ( 11) and making the lmear approxnnatlon I?(s) = I?( 4 ( 3s0 -s) ), we have

(13) with this value of 1, the K,-t37c rate 1s m perfect agreement with the experimental data, but the calculated odd plon spectrum 1s 20% lower This low value of the odd plon spectrum 1s expected because the P wave ~?r: interaction m the form of the p resonance was not taken into account m the umtarlty relation The P wave interaction changes only the odd plan spectrum but not the K-+3x rate Tbls situation IS similar to the previous study of 1-t 3n: [ 5 ] where, using the standard current algebra technique, the socalled bremsstrahlung t?rm was pomted out m the form of the p resonance giving a correct odd plon spectrum (The main difference IS the transformation property of the tadpole term m the IJ+ 3x amphtude which 1s not the same as H, ) One can incorporate the idea of the p dominance in the K-*~II problem m a choral invariant way by studying the plon pole contrlbutlon There IS no problem of introducing the vector meson in the 7cz+nz amphtude The choral invariant prescription was given a long time ago by Weinberg [ 19 ] and more recently by others [ 20,2 1 ] The additional contnbutlon to the nn invariant amplitude 1s (l/~~)[(~--)(~--_y2)l(~;-~) +(S--U)(t-fi*)I(m;-t)l

(14)

Taking into account this term, and a slmllar one for the K pole and the p contact term in the K-371 amplitude, we have finally H(J, t, u)=(CI$){(S-~~)G&,~~) +~[(~2-~)/(~2-~o)l[Go(~,~2)-ll} +d[(s-t)(u-~2)/(,~-,)

+(S--U)(t-~2)I(m~-t)l,

(15)

where C, Go and ;1 are given, respectively, by eqs (8 ), (6 ) and ( 13 ) Expanding eq ( 15 ) around s=so the matrix element for M(KL+n+nC-~‘) IS

498

LETTERS

B

~8 86x lo-‘[

30 June 1988

l+O 25O(s-so)fi--]I,

(16)

and IS dominantly real The odd plon spectrum IS lmear Comparing eq ( 16) with eq (2), it 1s seen that the agreement between the theory and the expenment is excellent The numerical results of eq ( 16) remam unchanged even when G,(s, ,u’) (eq (4b) ) IS parametrlsed in a different manner, but with the constraint that the phase shift So agrees with the experimental data m the low energy region below 800 MeV [4] References [ 1 ] R E Marshak, Rlazuddm and C P Ryan, Theory of weak mteractions m partxle physics ( Wiley-Interscience, New York, 1969) [2] TN Pham and I DuPont, Phys Rev D 29 (1984) 1369 [3] JA Cromn,Phys Rev 161 (1967) 1483 [4] TN Truong, Phys Lett B 99 ( 1981) 154 [ 51 C Roiesnel and T N Truong, Nucl Phys B 187 ( 198 1) 293, Ecole Polytechmque preprmt A5 15-0982 ( 1982), unpublished, C Rolesnel, These d’Etat, Umverslte de Pans Sud (Orsay) (1982) [6] T N Truong, Act Phys Pol B 15 (1984) 633 [ 71 T N Truong, Modern application of dispersion relation choral perturbation versus chsperslon technique, Contrib Festschrlft for Professor K Nishyima, m Wandermg in the fields, eds K Kawarabayashl and A Ukawa (World &entlfic, Smgapore, 1987) [S] J Gasser and H Leutwyler, Ann Phys (NY) 158 ( 1984) 142,Nucl Phys B250 (1985) 517 [9]S Wemberg,Phys Rev Lett 17(1966)336,616,18 (1987) 1178(E), Phys Rev Dl1 (1975) 3583, Y Tomozawa, Nuovo Clmento 46A ( 1966) 707, N Cablbbo and L Malam, in Evolution m particle physics (Academic Press, New York, 1970), G Paris1 and M Testa, Nuovo Cimento 67A ( 1970) 13 IO] T N Truong, to be published 111 T J Devhn and J 0 Dickey, Rev Mod Phys 51 ( 1979) 237 121 B R Martin D Morgan and G Shaw, Pion pion mteractlon m particle physics (Academic Press, New York, 1986) [13]NN Biswasetal,Phys Rev Lett 47(1981) 1378 [14]RK Clarketal,Phys Rev D32(1985) 1061 [15]L Rosseletetal,Phys Rev D15(1977)574 [16]NN KhunandSB Treiman,Phys Rev 119(1960) 1115 [17]RF SawyerandKC Wah,Phys Rev 119 (1960) 1429 [ 181A Neveu and J Scherk, Ann Phys 57 (1970) 39 [ 191 S Wemberg, Phys Rev 166 (1968) 1568 [20] M Mashaal, T N Pham and T N Truong, Phys Rev D 34 (1986) 3484 [21]M Bandoetal,Phys Rev Lett 54(1985) 1215