UAB-FT-335/94 UG-FT-41/94 November 11, 1994 EFFECTIVE CHIRAL LAGRANGIANS WITH AN SU(3)-BROKEN VECTOR-MESON SECTOR A. Bramon 1 Grup de Fsica Teorica, U

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1 UAB-FT-335/94 UG-FT-4/94 November, 994 EFFECTIVE CHIRAL LAGRANGIANS WITH AN SU(3)-BROKEN VECTOR-MESON SECTOR A. Bramon Grup de Fsica Teorica, Universitat Autonoma de Barcelona, 893 Bellaterra (Barcelona), Spain A. Grau Departamento de Fsica Teorica y del Cosmos, Universidad de Granada, 87 Granada, Spain and G. Pancheri 3 INFN, Laboratori Nazionali di Frascati, P.O.Box 3, 44 Frascati, Italy Abstract HEP-PH-9469 The accuracy of eective chiral lagrangians including vector-mesons as (hidden symmetry) gauge elds is shown to be improved by taking into account SU(3)-breaking in the vector-meson sector. The masses, M V, and couplings to the photon eld and pseudoscalar pairs, f V and g VPP, are all consistently described in terms of two parameters, which also t the values for the pseudoscalar decay-constants, f P, and charge radii, <rp >. The description of the latter is further improved when working in the context of chiral perturbation theory. 644::BRAMON 6487::IGRAU 3 IRMLNF

2 The question of including spin- mesons in eective chiral lagrangians has been frequently discussed in the past and has enjoyed considerable renewed interest during the last few years. This is partly due to the various proposals for new low-energy, highluminosity e + e colliders whichhave been put forward and approved, as in the case of the DANE -factory, presently under construction in Frascati. DANE is expected to provide very accurate and exhaustive experimental information on -decays, as well as on a number of low-energy parameters, both in the kaon system and in other pseudoscalar and vector meson decays. This recent revival, spurred by the expectation for new experimental data, has had its counterpart on the theoretical front. Indeed, traditional ideas concerned with Vector Meson Dominance, somehow associating spin- mesons with gauge bosons of local symmetries, have been further investigated and theoretically developed. The so-called \massive Yang-Mills approach" and \hidden symmetry scheme" recently reviewed by Meissner [] and Bando [] are two excellent examples. The parallel developement of Chiral Perturbation Theory (ChPT) [3] and its successful description of the low-energy interactions of the pseudoscalar-meson octet { a success considerably enlarged when spin- mesons are appropriately taken into account [4] { also points in the same direction. Our purpose in the present note is to go one step further by rening the predictions of the above schemes through the introduction of a well-known eect in hadron physics, namely, SU(3)-breaking in the vector-meson sector, a renement that forthcoming data will certainly require. Conventional ChPT accounts for the electro-weak and strong interactions of the pseudoscalar meson octet, P, in a perturbative series expansion in terms of their masses or four-momenta [3]. At lowest order in this expansion, the chiral lagrangian starts with the term L () = f 8 Tr(D D y ) () where = = exp(ip=f) and f is the pion decay constant f = 3 MeV = f = f K, at this lowest-order level. Electromagnetic interactions are contained in the covariant derivative D =@ +iea [Q; ] () where A is the photon eld and Q the quark-charge matrix (the extension to weak interactions is trivial). The mass degeneracy is broken via the additional mass term L () m = f 8 Tr(y + y ) (3) with containing the quark-mass matrix M = diag(m u ;m d ;m s ) and transforming asa(3;3 )+(3 ;3) representation of SU(3) L SU(3) R.At this lowest order, ChPT essentially coincides with Current Algebra. The next order piece of the chiral expansion (fourth order) contains one-loop corrections with vertices from () to (3) and a series of ten counterterms required to cancel loop divergencies [3]. Some of them, e.g., L (4) 9 = iel 9 F Tr(Q D D y +Q D y D ) (4)

3 are chirally SU(3)-symmetric, whereas others, e.g., L (4) 5 = L 5 Tr D D y y + y (5) break the symmetry as L () m in eq.(3). At this one-loop level, one obtains [3] f K f =+ 5 4 K f (m K m )L 5 () (6) where loop eects appear through the so-called chiral-logs, P = m P 6 f Similarly, the pseudoscalar electromagnetic charge-radii are found to be [3]! <r +> = 3+ln m 6 f + lnm K + 4 f L 9()! <r K +> = 3+ ln m 6 f + lnm K + 4 f L 9() <r K > = <r +> <r K +>= 6 f lnm K m lnm P. (7) where, again, the divergencies accompanying the chiral-logs are absorbed in one of the counterterms (or low-energy constants), L 9, appearing in L (4). The large number of low-energy constants, which need to be determined along the lines just discussed, considerably reduces the predictive power of ChPT, unless one adopts the so called resonance saturation hypothesis[4]. According to this suggestion, the values of these constants are related to the exchange of the known meson resonances, so far ignored in the original context of ChPT. This is a very attractive possibility which has been successfully veried in many cases. In most of them, particularly in processes involving the Wess-Zumino anomalous action [5], vector-mesons turn out to play the dominant role. Their couplings to pseudoscalar plus photon states, as extracted from the experiments, saturate an important part of the above counterterms. Accurate eective lagrangians incorporating additional properties of vector mesons and further details on their dynamics could therefore be extremely useful not only as a self-contained eective theory but also as an auxiliary lagrangian xing the counterterms in the ChPT context. We follow the \hidden symmetry approach" of Bando et al.[] to write the most general lagrangian containing pseudoscalar, vector and (external) electroweak gauge elds with the smallest number of derivatives. At this lowest-order, it is given by the linear combination L A + al V, a being an arbitrary parameter, of the two lagrangians L A = f 8 Tr D L y L D R y R L V = f 8 Tr D L y L+D R y R (8) 3

4 both possessing a global (chiral) SU(3) L SU(3) R and a local (hidden) SU(3) V symmetry. The SU(3)-valued matrices L and R contain the pseudoscalar elds, P, and the unphysical (or compensator) scalar elds,, that will be absorbed to give a mass to the vector mesons The full covariant derivative is L;R = exp (i=f) exp (ip=f) (9) D L(R) =(@ igv ) L(R) + ie L(R) A Q () where, as before, only the photon eld, A, has been explicitly shown, and P and V stand for the SU(3) octet and nonet matrices P = B@ p + p 6 + K + p + p 6 K K K p 6 CA V = B@ p +! p + K + K p + p! K () Under SU(3) L SU(3) R SU(3) V, the transformation properties of L;R (x) and the SU(3) V gauge-elds V (x) are L;R (x)! h(x) L;R (x)g y L;R and V (x)! ih(x)@ h y (x)+ h(x)v (x)h y (x), where g L;R SU(3) L;R and h(x) parametrizes local, hidden SU(3) V - transformations. The lagrangian L A +al V can be reduced to the chiral lagrangian () for any value of the parameter a []. In fact, working in the unitary gauge, i:e: xing the local SU(3) V gauge by y = L R = = exp (ip=f) to eliminate the unphysical scalar elds, and substituting the solution of the equation of motion for V, the L V part vanishes and L A becomes identical to the non linear chiral lagrangian (). The \hidden symmetry" lagrangian L A + al V (8) can be easily seen to contain, among other things, a vector meson mass term, the pseudoscalar weak decay constants, the vector-photon conversion factor and the couplings of both vectors and photons to pseudoscalar pairs. The latter can be eliminated xing a =, thus incorporating conventional vector-dominance in the electromagnetic form-factors of pseudoscalars. With this lagrangian one can therefore attempt to describe the following sets of data (that we take from [6] neglecting error bars when negligible): the vector meson mass spectrum the weak decay constants M ;! ; M K ; M =:6; :8; :4 GeV ; () f = 3 MeV; f K = 6 MeV ; (3) the ;! and conversion factors, f V. This xes the gauge coupling g, once the respective quark charges in V! e + e have been taken into account, to g =3:6:; 4::; 4::; (4) 4 K CA

5 and the g = p g, g K K = p 3 g= p and g KK = p g decay constants, which lead similarly to g =4:3; 4:5; 4:6 (5) with negligible errors. From [7] we take the following experimental values for the electromagnetic (e.m.) charge radii <r +> = :44 :3 fm <r K + > = :3 :5 fm <r K > = :54 :6 fm (6) and the combined result, free from most systematic errors (see Dally et al. Amendolia et al. in ref.[7]) and <r +> <r K +>=:3 :4 fm (7) The most immediate possibility to account for the above sets of data consists in using the \hidden symmetry lagrangian" (8) as a self contained eective theory in the exact SU(3) limit. In this case, the SU(3)-breaking eects shown by some of the above data remain unexplained, but two successful relations can be obtained for the nonstrange sector. The lagrangian (8) predicts M;! =g f in the :5 :6 GeV range when the values (3, 4, 5) for f and g are used. This agrees with the value obtained from the direct measurement of the ;! mass (). Moreover, it also agrees with that coming from the pion charge radius, leading in our approach to M =6=<r>=:5 GeV. Alternatively, one can regard the lagrangian (8) in a ChPT context, include the chiral-logs from the loop corrections and saturate the nite part of the required counterterms with (8). This maintains the successful relation M;! =g f and gives L 9 = f =4M, with a vanishing value for the SU(3)-breaking low-energy counterterm L 5. With this value for L 9, and evaluating the chiral-log correction at the conventional value = M, (7) predicts <r >=:46 fm in complete agreement with experiment (6). The next step is to include SU(3)-breaking eects. Since the pseudoscalar sector is known to break the symmetry as in eq.(3), i:e:, proportionally to Tr( R M y L + LM y ), R we incorporate SU(3) symmetry breaking, as already attempted in ref.[], via a similar hermitian combination ( L y R + R L) y in both L A and L V terms, i:e:, and L A +L A = L V +L V = + L A y R+ R A y L i D f 8 Tr L y L D R Rh y D f 8 Tr L y L +D R Rh y + L V R+ y R V i y L The matrix A(V ) is taken to be A(V ) = diag(; ;c A(V) ), where c A;V are the SU(3) -breaking real parameters to be determined. Notice that the SU(3)-breaking terms, 5 (8) (9)

6 L A;V, are hermitian, unlike those in ref.[]. Working in the unitary gauge y L = R = = exp (ip=f), ( = ), and expanding in terms of the pseudoscalar elds, one observes that the kinetic terms in L A have to be renormalized. This is simply achieved by rescaling the pseudoscalar elds [] q p +ca K! K +c A =3! () where an - mixing angle of -9:5 has been used for the case. The physical content of this new, SU(3)-broken \hidden symmetry" lagrangian (8) and (9) can now be easily worked out. From a (L V +L V ), we obtain the conventional SU(3)-splitting for the vector meson masses (a = ). M = M! =g f ; M K =M ( + c V ); M = M (+c V ) () For the V- couplings, the corresponding part of the lagrangian is explicitly given by: L V = egf A Tr[fQ; V g (+ V )] p em;! = p A " +! ( + c V ) g 3 3 () The new terms in the lagrangian (8,9) also induce an SU(3) symmetry breaking in the g VPP coupling constants. One obtains g = p g g KK = g!kk = g p +c A g KK = p g +c V +c p A 3 g K K = p g ( + c V ) p (3) +ca where the c A -dependence comes from the symmetry breaking in the L A +L A lagrangian due to the renormalization of the pseudoscalar elds (see eq.()). This redenition of the pseudoscalar elds also implies symmetry breaking in the pseudoscalar meson decay constants, namely, f K = p +c A f f q = +c A =3 f (4) One can now attempt a description of the whole set of data (-6) solely in terms of the SU(3)-broken lagrangian (8) and (9). This xes the values of the two new 6

7 free parameters to c V =:3 and c A =:45. The t is completely satisfactory for the four sets of data quoted in eqs.() to (5). For the pseudoscalar charge radii, one gets from eqs.(, 3, 4) <r +> = 6 M <r K +> = <r +> ( + c A ) <r K > = <r + > ( + c A ) 3 +(+c V) M ;! M 3 (+c V) M ;! M!! (5) and the above values of c V and c A imply <r +>=:39 fm ; <r K +>=:6 fm ; <r K >=: fm (6) somewhat below, by one or two 's, the experimental data (6). As previously discussed, an alternative, more sophisticated possibility is to use our SU(3)-broken lagrangian (8,9) in conjunction with ChPT. This can only modify the predictions for f P (4) and <rp >(5) related to processes without vector-mesons in the external legs. The chiral-logs in eq.(6), evaluated at the conventional value = M, account now for some 35% of the observed dierence between f K and f,thus requiring a smaller contribution from the L 5 counterterm and, hence, a smaller value for c A. Accordingly, the best global t is now achieved by the slightly modied values c V =:8 c A =:36 (7) which preserve the goodness of the preceding t for M V, f V, g VPP and f P, while improving the agreement in the <rp >sector. Indeed, adding the chiral loop contributions to the charge radii (5), one obtains <r +> = 6 f <r K > = 6 f <r +> <r K +> = 6 f 3+ ln m lnm K m lnm K m + +c A " + 6 M! + lnm K + 6 M ( + c V ) M M " + ( + c V ) +c A M M (8) The chiral-logs enhance the previous predictions for <r P >and we nd <r +>=:46fm ; <r K +>=:3fm ; <r K >=:43fm (9) in better agreement with the data (6). 7

8 Notice that the last two expressions in eq.(8), which explicitly contain SU(3) breaking eects, considerably improve the one-loop ChPT results [3] quoted in the last line of eq.(7), namely <r K >=<r + > <r K + >=:36 fm, which was signicantly below the data (7). This improvement stems from the fact that in saturating the ChPT counterterms with our SU(3)-broken lagrangian one goes beyond the fourth order counterterms in ChPT, such as the SU(3)-symmetric counterterm L 9, and introduces corrections from its SU(3)-breaking analogues belonging to sixth order counterterm(s) in L (6). This is explicitly seen in eqs.(8), which reduce to the conventional ChPT result (7) only in the exact SU(3) limit c A = c V = and M ;! = M.We summarize the above intable I, where we indicate the results obtained for the charge radii in the three models so far discussed, i:e:, Chiral Perturbation Theory (ChPT) and SU(3) broken \Hidden Symmetry" scheme (HSS) with and without chiral loops. Table I (all data are in fm ) exp:[7] ChPT[3] SU(3)broken SU(3)broken HSS HSS + Ch loops <r > + :44 :3 :44 :39 :46 <r K > + :3 :5 :4 :6 :3 <r K> :54 :6 :36 : :43 <r +> <r K +> :3 :4 :36 :3 :4 One can easily extend the above results for the e.m. charge radii of pseudoscalars to include their weak analogue in K l3 decays. Sirlin's theorem [8], valid up to rst order in SU(3)-breaking, and requiring <r K > <r K >= <r +>+ <r K +> (3) provides the clue. The data (6) and the experimental value [6] <r K >=:36 : fm are fully compatible with Sirlin's theorem (3). On the other hand, one immediately sees that the predictions of our rst-order SU(3)-breaking lagrangian verify the theorem, thus providing a check of our calculations and their automatic extension to the K case. From the expression <r K >= +c V 6 p + chiral loop contributions (3) +ca MK where the contribution from chiral loops is the one to be found in ref. [3], we obtain the very acceptable value <r K >=:33 fm. The above equation can be compared with the chiral perturbation theory result [3] <r K >= chiral loop contributions + 4 f L 9() (3) 8

9 which gives <r K >=:38 fm. An extension of the present treatment to processes related to the anomaly or Wess-Zumino lagrangian is also possible and will be discussed elsewhere. Here we simply note that introducing our value c A =:36, eq.(7), in eq.(4) for f, leads to (! ) = :53 KeV quite in line with the measured decay width [6], i.e. (! )=:46 :5 KeV. In summary, we have presented a model for low energy hadronic interactions in which SU(3)-breaking eects in the vector-meson sector are introduced in chiral lagrangians including vector-mesons as (hidden symmetry) gauge elds. By comparing a set of low energy data with the model predictions, we have shown that it is possible to extract values for the SU(3) breaking parameters which are very reasonable and which can lead to predictions for the electromagnetic charge radii of the pseudoscalar mesons in good agreement with experimental data. Further applications, primarily to the anomalous sector of the lagrangian, are being studied and will be presented in a forthcoming paper. Acknowledgements Thanks are due to M. Lavelle for a critical reading of the manuscript. This work has been supported in part by the Human Capital and Mobility Programme, EEC Contract CHRX-CT96, and by CICYT, AEN 93-5(A.B.) and AEN 93-65(A.G.). References [] U.-G. Meissner, Phys. Rep. 6 (988) 3. [] M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B59 (985) 493, and Phys. Rep. 64 (988) 7. [3] J. Gasser and H. Leutwyler, Nucl. Phys. B5 (985) 465, 57, 539. [4] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B3 (989) 3. J.F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D39 (989) 947. [5] J. Bijnens, Int. J. Mod. Phys. A8 (993) 345 and references therein. [6] Particle Data Group, Phys. Rev. D45 (99). [7] A. Quenzer et al., Phys. Lett. B76 (977) 5. E.B. Dally et al., Phys. Rev. Lett. 39 (977) 76. 9

10 E.B. Dally et al., Phys. Rev. Lett. 48 (98) 375. S.R. Amendolia et al., Phys. Lett. B78 (986) 435. W.R. Molzon et al., Phys. Rev. Lett. 4 (978) 3. C. Erkal and M.G. Olsson, J. Phys. G: Nucl. Phys. 3 (987) 355. [8] A. Sirlin, Phys. Rev. Lett. 43 (979) 94; Ann. Phys.(N.Y.) 6 (97) 94.