Kaon decays and predictions of chiral symmetry

University of Massachusetts Amherst From the SelectedWorks of John Donoghue 1989 Kaon decays and predictions of chiral symmetry John Donoghue, University of Massachusetts - Amherst Barry Holstein, University of Massachusetts - Amherst Available at: https://works.bepress.com/john_donoghue/ 92/ PHYSICAL REVIE%' D VOLUME 40, NUMBER 11 1 DECEMBER 1989 Kaon transitions and predictions of chiral symmetry John Department of Physics and F. Donoghue Astronomy, R. Holstein of Massachusetts, Amherst, Massachusetts 01003 and Barry University (Received 7 August 1989) We describe a set of kaonic electromagnetic and semileptonic weak decay processes which are completely predicted within the framework of chiral symmetry (and, therefore, of low-energy QCD), emphasizing where present problems exist and suggesting future experiments. I. INTRODUCTION The study of quantum chromodynamics (QCD) has occupied a central role in the program of modern particle physics. ' Traditional experiments have involved the machines, very-highest-energy using the asymptoticfreedom property of QCD (Ref. 2) in order to confront QCD perturbation theory, which is an expansion in I /lnE. It is also possible to pose significant tests of QCD at low [s &(500 MeV) ] energies using the feature of i.e., invariance under separate global chiral symmetry, rotations among left- and right-handed u, d, and s quarks. This SU(3)L II SU(3)t, symmetry is broken, of course, by the existence of a nonzero quark mass, but such effects can be included perturbatively and it is possible to construct an effective Lagrangian which describes the lowenergy interact. ions of 0 Goldstone bosons of the theory in a rigorous fashion provided that the underlying quark-gluon interactions are indeed chirally symmetric. ' Here predictions are given in an expansion in powers of the energy and/or quark masses. As in the case of many high-energy tests there exist drawbacks within such an specifically that absolute predictions are in approach general not possible, only relations between empirical observables. Also the restriction to low-energy processes poses a serious limitation. Nevertheless this technique, called chiral perturbation theory, offers an attractive avenue by which to probe the underlying structure of strong interactions in a regime wherein relatively precise measurements are presently possible. In an earlier paper we studied one aspect of such a weak and electromagnetic interactions of the program and identified various relations between charged pion pionic interaction parameters required by chiral symmetry. Confrontation with experimental results revealed good agreement except in the case of the pion polarizability, which strongly suggests remeasurement of this parameter. In this paper we extend our discussion of lowenergy tests of chiral symmetry to include the kaonic sector, which, because of the greater kaon mass and plethora of semileptonic decay modes, allows for a much richer program of experimental probes. Indeed, as we shall show, there exist kaonic analogs of each of the pionic tests discussed in Ref. 6 as well as many additional experimental possibilities allowed by the existence of KI3p +I3yy EI4, etc. , modes. We shall consider only semileptonic processes here, as nonleptonic and nonleptonic-radiative — — have been treated elsewhere. kaon transitions The greater kaon mass also presents a diSculty for chiral perturbation theory in that the corrections to the lowestorder predictions will be larger than they are in the case of pionic transitions. In some situations, chiral perturbation theory will use SU(3) breaking in order to predict these modifications. In known examples, the size of the corrections is about 25 —30%. It is important to note that these modifications do not invalidate chiral perturbation theory, but fit naturally into the framework using the energy expansion. In the next section then we present for completeness a brief outline of the effective Lagrangian methods used in our analysis. In Sec. III we examine the kaonic analogs of the pionic processes of Ref. 6 while in Sec. IV, we study the richer vein ofFered by %13, I( &3& studies. Finally, we summarize our findings in a concluding Sec. V. II. FORMALISM Since the chiral Lagrangian technique has been carefulin other works, there is no need to belabor this formalism here. Nevertheless for completeness we procedure which will be outline the Gasser-Leutwyler employed in subsequent sections. In the limit that the u-, d-, and s-quark masses vanish, QCD possesses an exact global SU(3)L @SU(3)~t chiral symmetry: ly explained 8 ql ~exp i g Aja j=l qit ~exp g j=i — q~=Lq~, 8 i A, JP& qit=Lqtt, where (2) denotes a three-component column vector and k; are the Gell-Mann matrices. This chiral SU(3)I SU(3)~ invariance is dynamically broken to SU(3)v and Goldstone's =0 theorem implies the existence of eight "massless" Goldstone bosons, which are identified with m, E, and g (Ref. g). The axial U(l ) ~ transformation s J 1989 The American Physical Society KAON TRANSITIONS AND PREDICTIONS OF CHIRAL SYMMETRY q —+e ioy& q (3) invariance and is is, however, not a quantum-mechanical In'addiassociated with the well-known QCD anomaly. tion the quarks are not massless and the inclusion of appropriate mass terms introduces a small explicit breaking of the chiral symmetry, which can be accounted for via a perturbative expansion in the energy. ' The manifestation of chiral symmetry within the interactions of the (pseudo-)Goldstone bosons is most succinctly realized in terms of the nonlinear order parameter U 8 l =exp (4) J J where P. are the pseudoscalar fields and F =94 MeV is the pion decay constant. Under SU(3)L SU(3)R the matrix U is defined to transform as +LUR~ . U— Including the effects of quark mass, the simplest Lagrangian consistent with chiral and Lorentz invariance is then F2 4 Here F„, are external FL, R PV F' and Q F2 TrD P UD"Ut+ R FL, V field-strength Qg PL, R P 4 & ) . I m =2Bo 0 0 m& 0 0 0 m, R FL, R [FL, ] V P 9 A„, The mass matrix m characterizes chiral-symmetry break- (9) E Obviously the first piece of X' ' contains the meson kinetic energy and is chiral invariant, while the second term includes the pseudo scalar masses and transforms as (3I, 3R )C3(3L, 3R ) under chiral rotations. Comparison with experimental values of the 0 masses yields the nor- malization" +m„ m, 6m 2m 2mK 2Bo m&+ „ m„m„+ m&+4m, Such effective Lagrangians have been known for over two decades and the tree-level evaluation of L' ' yields m )] the familiar Weinberg mm scattering am[to plitude' 0(p, —1 F2 [gab/cd($ m 2 ) + Qac$bd( r U +F„+"U D U)+L, OTrF„UF" L„. Here the coe%cients . . , L, 2 are arbitrary and unphysical (bare) since they can be used in order to absorb divergent loop contributions from X' '. The physical (renormalized) couplings are found to be L;"(p) =L; + I; —+ln4~+ — y where y is Euler's dimensionality, and r&= 6 14' & 7 constant, ', (l3) represents the p 1 32m' e=d —4 2 ) which is roughly borne out experimenta11y, as well as other successful predictions. Of course, loop diagrams associated with X.' ' are required for unitarity and produce effects of higher order [O(p, p m, m")] along with divergences. The infinities can be eliminated by renormalization of phenomenological chiral couplings of order 4. Gasser and Leutwyler have given the most general such Lagrangian X' '= L, (TrD„UD" U )2+L2(TrD„UD U ) +L3Tr(D„UD"Ut) +L4Tr(D„UD"Ut)Trm(U+Ut)' +L~TrD„U D" U (m U+ U m)+Lb[Trm ( U+ U )] +L7[Trm ( U+ U )] +Ls Tr(m Um U+ m U iL9Tr(F„+"U—D m +5'"6 '(u —m )] tensors defined via covariant derivative, defined by ]+i D„=d„i[V„, — m„0 (6) =V +A D„ is the ing and is given by (S i 0 ) Trm(U+U 3701 U m Ut) +L»TrF„,F" +L,2Trm (12) are constants chosen in order to cancel the divergences. Finally, we must include also the effects associated with the anomaly, which is also of order 4. Including the gauge dependence only for the photon field A„, we find' S,„, = f d5x e'~"' TrL, L LkLIL +fdxe""8P A V + f d x A„j ~ A a TP with '„r3=0, —, ~ g r4=-,', r = ', L„=B„UU', Z„= U'a„U, , (14) 4s s ~ ri2 24 j "=e" ~Tr(QL„L L&+QR, R R&), (16) T&=Tr(Q L&+Q R&+ ,'QUQUtL&+ ,'QU"Q—URp), — 3702 JOHN F. DONOGHUE AND BARRY R. HOLSTEIN terms in a brief appendix. and g= 0 —' o 0 —, III. 0 (17) As mentioned above, the kaon, because of its greater mass, offers a much expanded laboratory for theoretical and experimental analysis than does its pionic counterpart. Before exploring some of the additional processes which the kaonic system offers, however, it is useful to emphasize that each of the pionic reactions discussed in Ref. 6 has a direct kaonic counterpart. Thus we de6ne the following: (i) Electromagnetic form factor: 3 being the quark charge matrix. The full non-Abelian anomaly has also been given but will not be needed here. ' The use of such an effective chiral Lagrangian in order to make contact with experimental quantities has been described elsewhere. ' Thus, we quote henceforth only &K+(p2)~ the results obtained from such evaluations. For simplicity of exposition we shall omit X' ' loop contributions in the main body of the paper. Indeed such terms are 0 (1/N) with respect to the leading tree effects and explicit calculation confirms the dominance of the tree-level X' ' coefficients over their X' ' loop counterparts. ' For completeness, however, we include the effect of loop M„(q,p)= J d x e'~' ( T[ 0~ = v 2F»(p —q) (2p V™(x)J, "( 0)]~ K+( with f v'2F»g p. +I A— [(p . —, + + &q Radiative ~8 decay (19) —K + ~e + v, y, 8 V~8 — '(r + 2+ q), q —,'(r— )+ 2F»(p &q q— +h..ep, pp ) q]— gp—(p & & "(p —q)' —m, q)pq. V™ ~K+(p, ) =f»+(q')(p, +p, )„ ):—1+ '(r +(q (ii) q)„(1+ — p) KAONIC ANAI. OGS OF PIONIC PROCESSES —qpqy)— rA(g pyq' q & Here h z, rz represent axial structure constants while h v is a form factor which arises from the polar vector current. course, the axial structure constant rA is relevant only for the e+v, e+e mode. (iii) Kaonic Compton scattering: —Jd x e ' (K+(P2)iT[V'„(x)V„' (0)]iK+(p, )& A„(pl, q„q2): (P q2 Pl )Tp(P2 P2+q I ) T/l(pl ql Pl )~ 1 (p, —q, )' m» —ql„ql + —(r» &(qlg +2g — +qzg~ where Tlt (p 2 p1) (p 1 +p 2 )p ( 1 + ' —, & r» & ' — ' q ) q „-, & r» & (p 1 —p 2 ) q2„q2 ( r2K & ~theor = y» . — (23) That the electric (az ) and magnetic (PM ) polarizabilities must be negatives of one another is associated with the chiral-symmetry requirement, as described in Ref. 6. In terms of the effective chiral Lagrangian outlined above we can represent Pve of these experimental quantities in terms of just two of the chiral parameters L9 and 2 io — ~ —ql q2„)+. . . (21) 12L9 theor is the (off-shell) kaon electromagnetic vertex As dis-. cussed in Ref. 6, yK represents the kaonic polarizability with ) Qf ) )+2y'»(ql. — q2g& (22) a~(K+ ) = PM(K+ (P2 P2+'q2 (p, —q, )' —m» (20) =32~ (L9+L,o), theor =32m. hv L, 9, (24) a~(K)+pM(K) ~'""'=0 (K) theor— i 4O ~K+K2 Staying strictly within the kaonic system we observe that chiral symmetry requires three relations to be valid: KAON TRANSITIONS AND PREDICTIONS OF CHIRAL SYMMETRY 40 8' 2 Fp( "Kp) aE(K) = —p~(K) = ( r2 (25) m m~F~ ~y 8m hv+h~ i'"P'=(0. 043+0. 003)m ~az(K)~'"P'(2X10 fm vs i h i'"'"=0.' 014m ih V — vs A = 3 + 2 + 1 =033 fm +)= m2 m2 m2 vs az(K)i'"'"=5. 8X10 IV. K(3, E(3y DECAY PROCESSES We have observed in the previous section a series of experiments involving the charged kaon which can be expressed in terms of only two chiral parameter, L9 and L &p which have already been determined in the pion sector. A second class of kaonic reactions . +ivy, of L9, L, p, and K a third paramecan be described in terms ter 5. Since the latter is well determined by the measured value of kaonic-to-pionic decay constants' I = 1+ z (mz (28a) P and with the chiral-symmetry (r2 + ) =(r + ) prediction = 0. 44+0. 03 fm (28b) there exists very little else which is precisely known charged-kaon electroweak The properties. K+ ~e+v, e+e has not been observed, so that rz known. The radiative process K+~e+v, y has studied, but only the sum of h ~ and h ~ is known precision about decay is unbeen with —m „)L5=1.22+0. 02 (31) chiral symmetry also makes unambiguous predictions for these KI3, K~3~ processes. —l l+v, , Ki In the case of K» decays, K+ ~~ n only a crude upper limit is available from the kaonic atom measurements In view of this situation it is clearly out of the question to meaningful test of chiral symmetry at the present time. On the other hand, the possibility of producing such a chiral comparison should serve as a challenge to The gathering of such data ought cerexperimentalists. tainly to be pursued as part of the experimental program of the new kaon facilities which may be coming on line in the future. (One difficulty in this regard is that the expected polarizability is smaller than that of the pion by the factor F&mziF m„-4. 8. Hopefully this can be overcome. ) K» prediction hv+h„~'"' "=0.038m eject a mlv, (27) (29) & O. 1m ) —0. 8m ' Finally, in the case of the polarizability K(3.. K 34+Q. Q5 fm2 2 &rJc+ — i'"P'= Q and agrees with both the vector-dominance (26) and verifying these predictions constitutes an important i.e. , QCD. While the ab test of the underlying theory solute values of r~ /hv or of aE(K) requires the use of a specific model, the relations between these parameters must obtain in any chiral model. However, we can go even further. Since L9, L&p were already determined in the pionic system, we can make here an absolute prediction for each of these experimental quantities in terms of the corresponding pion values. Unfortunately the present kaonic data base is quite limited. While the K charge radius has been measured' h ih v — i ) iexPt 3703 ~~ fm (30) it is traditional to parametrize ments in terms of form factors &~(p, )l V„ IK(p, =' the hadronic matrix ele(q ): f + (q ), f )& +f li2lf+(q')(pi+p2) ('q )(pi p2— )„1. (32) Alternatively, one often finds results expressed in terms of s- and p-wave projections in the cross channel: s wave fo(q )=f+(q )+ 2 m m& — f (q ), (33) p wave f+(q ) . In either case it is common to employ a linear extrapolation in order to characterize the momentum-transfer dependence: f(q')=f(0)(1+-,'& '&q'+ . =f(0)(1+A.q + ) . ) (34) The chiral-symmetry prediction can be obtained from the effective Lagrangian of Eqs. (6) and (12) yielding f+(pi f P2 q')=1+ (pi~p2~q )= F2 —m —)L~ — 2 (m~ Tl 2L, 9 (35) (pi — p~) 37D4 JOHN This form is consistent with Callan-Treiman condition ' (and '(p )Il'„ I&+(p, ) F. DONOGHUE AND BARRY R. HOLSTEIN by) the required & — — ' &ol[g;, v„]I@+(p,)& p &= — &ol~„ P le+(p, ) 1 2 F p,1P„ (36) first derived by current-algebra —PCAC (partial conservationary axial-vector-current) techniques. We note that f+(p'1 o pi)+f —(pi o pi)= except for the case of A, o(K„+z) for which a significant work is discrepancy exists. Additional experimental strongly suggested in order to resolve this problem. Two other points should be noted here. First, one might at first be surprised that the predicted E,3 charge radius ~ (37) p2)=2- (38) 2& (43) Q2 is identical to that for the electromagnetic form factor, since from vector dominance one would expect difFering values Similarly the soft-kaon condition f+(O, p2, p2)+f (O, p2, f 2 we use mass-shell condi- m 2 p)theor (mK, m, p)'thepr"'= FK —l — 9 (mK —m F„ (39) ) The difFerence, however, between these two charge radii would arise only at order X' ' and hence is outside the scope of the present investigation. A second point to be noted is that when the u, d mass di8'erence is included, a slight di6'erence in the expected rates for charged and neutral E13 rates is introduced due to m. -g mixing. Thus~ = —0. 13 . f l~'& =coselp'&+»nelg'& m, We observe that o) vanishes in the SU(3)(mK, symmetry limit as expected and that the I.9 contribution is required in order to yield the negative sign indicated by present K„3 experiments: f (mK, m, o) f+ (mK,2 m, o) —0. 11+0.09 from E„3, —0. 35+0. 15 from E„+3 . with (40) fK 2 f K~rr (p) are quite consistent mentally: —1 =0.040 fm 4 m, k+ 0. 060+0. 003 fm 0. 056+0. 008 fm 0. 068+0. 010 fm 0. 067+0. 017 fm 0.050+0. 012 fm 0. 008+0. 014 fm from EP3, from E,3 from E„3, from E„+3, from from EP3, E+, , ) experi- md m„ —'(m„+md) = 1.021 which is in good agreement violation fK T m„ —, —, = 1+ 3 with the values determined md —'(m„+md (45) =cose+i 3sine (O) =0.067 fm —m 2 4 m, theor tr (41) mK e= Q3 leading to Likewise the predicted slopes 2L9 mp K is obtained. In comparing with experiment tions, yielding (m (44) mKe f e K~rr (46) —, with the measured isospin expt (p) (p) =1.029+0. 010 . (47) %'e have overall then an excellent picture of the %13 system within the chiral framework except for A, o(IC&& ). A second process which can be understood simply in terms of 1-9, I-~p Lg is that of radiative E13 decay for the which we find (we quote only the E+~m result +m result is similar) It. L — — KAON TRANSITIONS AND PREDICTIONS OF CHIRAL SYMMETRY (p&, q&, qz)= A fd x e ' V™(x) V""(0)j~K+(p, )) (m (P2)~T[ q— ) p) , ~„(p) +&I/2 — +4 g )&~'(p& L9+Lio L9 P& ~2 tq„(»'q& )I q»= q & '(p»lV". "I&+(p pt'q&) qz„p2 (49) (q&, p&) . (50) v, y; Er au) 1(Z+ ) 10 MeV) &5.3X1O q2g„)+4 +qi pi (q]„q] —q„q] ) +pr p2„1 p&„pz (48) — — ed. Careful readers will note that we have not inc1uded the process in our discussion, for which E&4 semileptonic some considerable and precise experimental data are available. This omission was purposeful, since analysis of four additional chiral parameters, %&4 decay involves . . , L4, and it is not simply predicted in terms of available data. Rather it is the other way around. The very careful K+~m+~ e+v, data can be used in order to place restrictions on the size of the chira1 parameters which are simply unavailable by other means. This is a very important program and will be described in a separate communication. We observe that in addition to the pole term associated there exist substantial and with inner bremsstrahlung, distinctive structure-dependent contributions required by chiral symmetry. The size of this structure dependence is completely determined in terms of known properties of the charged pion and verification of such structure would constitute an additional and a dependence significant test of low-energy chiral symmetry. As with all such radiative processes, experimentally isolating the structure dependence from the large and generally dominant inner bremsstrahlung component is none too simple but could be within the reach of a dedicated experiment. In fact the best present limit ~~ e & — and soft-pion requirements I sD(K —p&) measure two of these parameters L9 and L, o. In this work we have demonstrated how these values of L9 L]o plus one additional quantity L~, which is given in terms of Fz/F can be used in order to determine an entire range of electromagnetic and semileptonic weak kaon transitions. These are explicit predictions of chiral symmetry and large violation of any such prediction would be dificult to understand within the framework of QCD. Certainly in this regard experimental work at present and/or future kaon facilities is strongly suggest- —q»& —( '(p, )IV"„"l~+(p, )) 1 A„(p„q„qz) —0 2F„ M q, V".'I&+(p& (q] qp„—q] It is straightforward to verify that this form satisfies both the gauge invariance q"~„.(p 3705 L„. ' ACKNOWLEDGMENTS This research was supported Science Foundation. is tantalizingly close to the level, at which such structure dependence should begin to be observed. in part by the National APPENDIX V. CONCLUSIONS As stated in Sec. I, at low energies the loop corrections arising from the lowest-order Lagrangian X' ' are in general small and make only a small modification to the tree-level order X' '+X' ' predictions. For completeness we list here the X' ' loop contributions to various quanti- We have used the fact that higher-order e6'ects in models which possess chiral symmetry (such as QCD) can be expressed in terms of a small number of phenomenological parameters . . , L &0. In a previous paper (Ref. 6) we discussed how experiments involving the chiral pion could be used in order to test chiral symmetry and to L„. f +(q)=1+q . F + 32~ 2 2 F 2 x —4 j q ~H 2 mx ties. form factor: (i) Charged-kaon + 2H q m ——ln 3 2 m~ p 6 ln m p2 (A 1) where H(x) = —~4+ ', x —, 'x) ', — +( — —, 1/2 inQ (x) (A2) 3706 4Q —4+ &+ —4 —&x &x &x x For F. DONOGHUE AND BARRY R. HOLSTEIN JOHN (A3) &(m we then find q 16m I' 2 where t =-(q, 1 — ln X . 4(L9+L",0) —q2 ) = —2q, .q2 2 p p -0.01. Note that 12L9-0.084)) 3/32~ (ii) Kaonic Compton scattering: a~(K)= 2 2 1 3 — + ln mK + — ln 1 ——+ — 2 (A5) Q Since for small arguments transfer. is the momentum 2 1 m~ ln +— 2 t ln Q we have X + Q(x)= —1 —12 (A6) we see that a~(IC)= 0.' 4(L9+Lio)+ 2 mKFK so that the tree-order amplitude (iii) t 1 384m 2 m 2 + (A7) obtains at low momentum transfer. K» (Ref. 20): . - (q')+ ,'F-~„(q'), f+(q') =1+ 'F, , fo(q )= 1+-8y' +, 4(mz 5q —2(mz+m —m , ) (5I. ) 2m~ 2 —3 3t 2 2 Jz q' (q + 24F' 3q —2(mz+m ) 2 — — 2 2 Jzz(q q' ) +— „)— (A8) Note that where F, (q )= = m; 2 )— L;,lj (q L9+M;".(q — (A12) ) (A9) p, ) 32m ln I' m; so that f+ (0) = 1 in the SU(3)-symmetry limit. However, using physical values for the masses we find f + (0) = 0. 978 . p and the functions Jj Mj Lj have been defined in Ref. 5(b). These forms are somewhat complex. However, a simple result is obtained for + (0): f f + (0) = 1+ —3F~ (0) + —F~„(0), of the vector Similarly we can address the q dependence form factor: i( 2) gKvr 1 m 2 2 m~ 128 2r2 2 mK 5 +— ln 6 (Alo) ', (A13) 2 mK m 2 + —ln m„ —2 mK where (A14) where F~(0)= — 2 1 128m F2 (m, +m, ) 1+ m; m m4 ln w, (x) m, m'j =- —3x —3x + 1 1 lnx + 2 (x —1) x x +1 —1 3 (A15) m4 j (A 1 1) We then predict a small reduction prediction: from the tree-order KAON TRANSITIONS AND PREDICTIONS OF CHIRAL SYMMETRY A, ""=0.067 fm -+A, "'P=0.060 fm (A16) 2 For the scalar form factor we find F~ 1 0 2 m~ m~ m $92~ 2 1 2 X 3x (1 — x) x) (1 — tree l82 0 Q4Q fm2~/looP 0 034 (A18) 11Ilx Numerically we again find a slight reduction tree-order prediction 2 Q2 3 1+x tu, (x) =— 3707 from the fm2 (A19) pyz 19m&+3m 6(m~+m „ „) Pl 2~ Pl 2~ —3 (A17) where 'See, e.g. , K. Gottfried and V. F. Weisskopf, Concepts of Particle Physics (Oxford University Press, New York, 1986},Vol. II. D. Gross and F. Wilczek, Phys. Rev. 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In Ref. 5(a) Gasser and Leutwyler use a Zweig-rule argument in order to deduce in L], . . . , L4 from mm scattering measurements. J. F. Donoghue and B. R. Holstein (in preparation). 23V. N. Bolotov et al. , Yad. Fiz. 44, 108 (1986) [Sov.