The ultraviolet behaviour of the generating functional for hadronic weak
interactions with $|\Delta S| =1, 2$ is investigated to one loop for a generic
number of flavours and in the quenched approximation. New quenched chiral
logarithms generated by the weak interactions can be accounted for via a
redefinition of the weak mass term in the $\Delta S=\pm 1$ weak effective
Lagrangian at leading order. Finally, we illustrate how chiral logarithms are
modified by the quenched approximation in $K\to\pi\pi$ matrix elements with
$\Delta I=1/2$ and 3/2.Comment: LATTICE98(matrixelement), 3 page

Bernard

Bijnens

Colangelo

Dawson

E. Pallante

Ecker

Gasser

Kambor

Testa

English

arXiv

The ultraviolet behaviour of the generating functional for hadronic weak interactions with |Î”S| = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the Î”S = Â±1 weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation in K â†’ Ï€Ï€ matrix elements with Î”I = 1/2 and 3/2

Pallante, E.

NARCIS 

The quenched generating functional for hadronic weak interactions

Crossref

University of Groningen

   University of GroningenThe quenched generating functional for hadronic weak interactionsPallante, E.Published in:Nuclear Physics B - Proceedings SupplementsDOI:10.1016/S0920-5632(99)85060-9IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Pallante, E. (1999). The quenched generating functional for hadronic weak interactions. Nuclear Physics B- Proceedings Supplements, 73(1). https://doi.org/10.1016/S0920-5632(99)85060-9CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-04-2023ELSEVIER IUCLEAR PHYSIC~ Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 P R O C E E D I N G S  S U P P L E M E N T S  The quenched generating functional for hadronic weak interactions E. Pallante ~* aInsti tut  fiir Theoretische Physik, Universit~it Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland The ultraviolet behaviour of the generating functional for hadronic weak interactions with IASI = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the AS = =kl weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation in K --+ 7rcr matrix elements with A I  = 1/2 and 3/2. 1. I n t r o d u c t i o n  The computat ion of weak matr ix  elements at long distances is one of the main and still open problems in lattice QCD. Much progress has been done, al though their computat ion still suffers from the presence of major  sources of systematic errors: finite volume effects, unphysical quark masses, the need of computing unphysical matrix elements [1] (see also [2] for recent alternative pro- posals) and finally the fact tha t  still most of the lattice evaluations are done in the quenched ap- proximation. Here we report  on some results con- cerning formal aspects of the quenched approxi- mat ion in the ultraviolet regime. The framework we use is known as Chiral Per turbat ion Theory (ChPT)  [3] and quenched C h P T  [4,5], extended to full (unquenched) weak interactions in [6] and to the quenched case in [7]. The advantages of deriving the generating functional of the effective low energy theory in the full and quenched case are at least two: first, it allows for a systematic control on the quenched modifications and sec- ond, it gives in one step the coefficients of all chiral logarithms for any Green's function or S- matr ix  element. with IASI = 1, 2 and at leading order in the chiral expansion can be written as follows /~AS=I ---- Vs((I)o)str(As32u,,u~) -FV5 (¢o)str(As32X+s) +Vo(¢o)str(A832u,8)str(u~) + h.c + V27( ¢o )tiJ'atstr( Asijuus)str( As~lu~ ) (1) and the same last te rm for AS  = 2 with the appropriate  t ij,kt tensor. The projection ma- trix onto the octet and 27-plet components is • , ~.. z---~,t  with the graded matr ix  Aaij = ~8-,~ 2 ~8, ()~ij)a b : 5iaf~jb. The graded fields u~8 = : u t V  U t -I- usx?sUs iuIDuU~u i = u~ and X+. . . ~ . .2 contain the dynamical meson field Us -~ u 8 : exp(v~ i  ¢/F) in the usual way. The field Xs = 2BoAds + . . .  contains the graded quark mass ma- trix. The potentials ~ ( ~ 0 )  are real and even functions of the super-~f field ~0 = str(q~) and we keep them up to order ~2. The ultraviolet diver- gences of the weak generating functional to one loop can be derived as it was done in the strong sector [5], provided an expansion in powers of GF is performed. For degenerate quark masses we get for the AS = =kl octet contributions at order GF: 2. T h e  w e a k  g e n e r a t i n g  f u n c t i o n a l  t o  o n e  l o o p  The quenched generalization of the weak effec- tive Lagrangian for nonleptonic weak interactions *supported by Schweizerisches Nationalfonds 1 Z ~ 8 ) -  (47r)2(d_4) fdx {lm~(gls-2gs)k(Ax+) +¼gsW4 + 8w0 +  (gs - 4)  - (w,2 + w36) + [-  i gs 0 -  vl) + 1 ' gs (1 - 4vl)] w7 0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All fights reserved. PII S0920-5632(98)00585-4 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 319 1 (o 1) +~-~(gs - 8~)s)Ws + ~-~(2gs - g~) - ~98 W10 1 t 1 +[(~+~6)(g~8-gs) - -~v2(gs-2gs) -[v5  +~gs W~ + ~98 (W~l The operators Wi are listed in Appendix A of [7]. The 27-plet contribution is given by Z(27) - -  (41r)2(d_4) dx927 D1 + -~D2 ~4 1 1 1 3 -1D3 - D4 - ~D~ + ~D6 + ~D7 + ~D8 3D 1 ( l - ~ a )  D12 1 (l_2"v27"~D1o _ _ - - lOl l  -- ~---~D13 7u ~ D 1 4 -  ~--~D15- ~--~D16 - ~D19 - ~D2o + ~D21 +~D22 1 1 } + ~D23 + ~D24 q- O(G2), where the operators Di are listed in Appendix A of [7]. m~ and a are the usual singlet param- eters. The couplings gs, g~, gs and g2~ are the first terms in the expansion of the weak poten- tials 1)/, i = 8,5,0,27, while the parameters vi are the coefficients of the Oo 2 term. The quenched generating functional for AS = 2 interactions has exactly the same structure of Z~27), with the ap- propriate t ij'kl tensor. The quenched approximation largely reduces the ultraviolet divergent contribution to the octet sector at one loop. Of the initially divergent 25 octet operators only 10 remain in the quenched approximation. The octet operators Ws, ... W~2 a n d  W38 contribute to K --+ 27r decays. In the 27-plet sector all the unquenched chiral invariants Di survive to the quenched approximation. Only the operators D8, ...  D12 contribute to K --4 27r decays. The systematic cancellation of the flavour num- ber dependence follows the rules outlined in [5,8]. The cancellation of the 1/N, 1/N 2 terms is pro- vided by the sum of the non-singlet contributions and the singlet contributions within the bosonic sector. This type of cancellation is entirely due to the presence of a dynamical singlet field. The linear flavour number dependence is cancelled by the fermionic ghost determinant as expected. 2.1. Q u e n c h e d  chiral  logar i thms The most relevant behaviour of the quenched generating functional to one loop is the appear- ance of quenched chiral logarithms, i.e. of the type m~logm 2, that are pure artefacts of the quenched approximation. As it was discussed at length in [5], quenched chiral logarithms ap- pearing in the strong sector can be formally re- absorbed in a redefinition of the Bo-parameter. In the weak sector an analogous mechanism oc- curs. The quenched chiral logarithms which ap- pear through the first term in the equation for Z[s ) can be formally reabsorbed into a redefini- tion of the weak mass term coupling g~ of the leading order Lagrangian (1). To remove the mg divergence one has to add to the lowest order pa- rameter g~ a d-dependent part proportional to m02 that has a pole at d=4 and the rescaled coupling can be defined as follows: gsn = g~(1 m2 (1 2g8~ M2 'gts(#) ) 48  F2 -  /log-j + The rescaling of the coupling g~ together with the rescaling of the parameter Bo -4/~o defined in [5] in the tree level contribution to any weak observ- able can be used as a short-cut procedure to unre- veal the presence of quenched chiral logarithms, generated when the quenched approximation is implemented to one loop. 3. K --4 zrzr ma t r ix  e l e m e n t s  Using the one loop expression of K --4 7rzr ma- trix elements as derived in [9] and the quenched counterpart of the ultraviolet divergences derived here, we can produce a few quantitative estimates of quenching effects on the coefficients of chiral logarithms that contribute to K -+ zrzr ampli- tudes. We work in the infinite volume limit for illustrative purpose. In [10] we shall report on the analysis of unphysical choices of the kinemat- ics used for the computation of lattice K --4 zr~r matrix elements. 320 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 Table 1 The coefficients of the one loop chiral logarithms in K -~ ~r~r amplitudes, full and quenched. Full (rnK -- mr) Full (mr = 0) Quenched Ao - 4  (1 - A2 11. __3 13 2 4 4- We decompose the K --~ ~r~r matrix ele- ments into definite isospin invariant amplitudes: ~mAo = .~m(A~ + A  27) with AI  ---- 1/2, and ~mA2 with AI = 3/2. In Table (1) we com- pare the coefficients of chiral logarithms for the three amplitudes in the full theory and in the quenched approximation. For the full amplitudes we consider two extreme mass configurations in the one loop corrections: a) degenerate masses, i.e. m g  ---- mr and b) mr = 0. The numeri- cal analysis of the physical non degenerate mass case together with the comparison with unphysi- cal choices of the matrix elements on the lattice will be given in [10]. The coefficients of chiral logarithms in the full 27-plet amplitudes .~mA~ 7 and ..~mA2 are quite large in the degenerate mass limit. In addition, by the comparison of sec- ond and third column in Table (1) one can con- clude that all the coefficients in the full ampli- tudes are extremely sensitive to the variation of masses. Using a ~ 0.6 and disregarding for now the unknown contributions to the octet amplitude in the quenched case we find the following pat- tern going from the full degenerate mass case to the quenched one. Quenching reduces by a tiny amount (from 0.6 to 0.54) the coefficient of the chiral logarithm in the octet amplitude Ao s, while reduces the one in A27 by about 34% in absolute value and 41% in the case of A2. Note that the pattern is opposite for A2 when we compare the quenched amplitude to the mr = 0 limit of the full amplitude. This analysis (see [7,10] for more details) shows that quenching largely affects the coefficient of the chiral logarithm in the 27-plet AI = 3/2 and AI = 1/2 amplitudes. In addition, the comparison with the mr = 0 limit of the full amplitudes (expected to be the most approximate to the physical value) shows that the modification induced by quenching follows a pattern that tends to suppress the AI = 1/2 dominance. R E F E R E N C E S  1. C. Bernard, T. Draper, G. Hockney and A. Soni, Nucl. Phys. (Proc. Suppl.) 4 (1988) 483. 2. C. Dawson et al., Nucl. Phys. B514 (1998) 313. M. Testa, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 38. 3. J. Gasser, and H. Leutwyler, Ann. of Phys. (N.Y.) 158 (1984) 142. 4. C.W. Bernard, and M.F.L. Golterman, Phys. Rev. D 46 (1992) 853. 5. G. Colangelo and E. Pallante, Nucl. Phys. B520 (1998) 433; G. Colangelo and E. Pal- lante, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 299. 6. J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B 346 (1990) 17; G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101. 7. E. Pallante, "The generating functional for hadronic weak interactions and its quenched approximation", preprint BUTP-98/18, hep- lat/9808018. 8. G. Colangelo and E. Pallante, Phys. Lett. B 409 (1997) 455. 9. J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B521 (1998) 305. 10. E. Pallante, "K ~ ~rlr matrix elements: ChPT and the lattice", preprint BUTP- 98/19. 

   University of GroningenThe quenched generating functional for hadronic weak interactionsPallante, E.Published in:Nuclear Physics B - Proceedings SupplementsDOI:10.1016/S0920-5632(99)85060-9IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Pallante, E. (1999). The quenched generating functional for hadronic weak interactions. Nuclear Physics B- Proceedings Supplements, 73(1). https://doi.org/10.1016/S0920-5632(99)85060-9CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019ELSEVIER IUCLEAR PHYSIC~ Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 PROCEEDINGS SUPPLEMENTS The quenched generating functional for hadronic weak interactions E. Pallante ~* aInstitut fiir Theoretische Physik, Universit~it Bern, Sidlerstrasse 5,CH-3012 Bern, Switzerland The ultraviolet behaviour of the generating functional for hadronic weak interactions with IASI = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the AS = =kl weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation i  K --+ 7rcr matrix elements with A I  = 1/2 and 3/2. 1. In t roduct ion  The computation of weak matrix elements at long distances is one of the main and still open problems in lattice QCD. Much progress has been done, although their computation still suffers from the presence of major sources of systematic errors: finite volume effects, unphysical quark masses, the need of computing unphysical matrix elements [1] (see also [2] for recent alternative pro- posals) and finally the fact that still most of the lattice evaluations are done in the quenched ap- proximation. Here we report on some results con- cerning formal aspects of the quenched approxi- mation in the ultraviolet regime. The framework we use is known as Chiral Perturbation Theory (ChPT) [3] and quenched ChPT [4,5], extended to full (unquenched) weak interactions in [6] and to the quenched case in [7]. The advantages of deriving the generating functional of the effective low energy theory in the full and quenched case are at least two: first, it allows for a systematic control on the quenched modifications and sec- ond, it gives in one step the coefficients of all chiral logarithms for any Green's function or S- matrix element. with IASI = 1, 2 and at leading order in the chiral expansion can be written as follows /~AS=I ---- Vs((I)o)str(As32u,,u~) -FV5 (¢o)str(As32X+s) +Vo(¢o)str(A832u,8)str(u~) + h.c + V27( ¢o )tiJ'atstr( Asijuus)str( As~lu~ ) (1) and the same last term for AS = 2 with the appropriate t ij,kt tensor. The projection ma- trix onto the octet and 27-plet components is • , ~.. z---~,t with the graded matrix Aaij = ~8-,~ 2 8, ()~ij)a b : 5iaf~jb. The graded fields u~8 = : u tV  U t -I- usx?sUs iuIDuU~u i = u~ and X+. . .~ . .2 contain the dynamical meson field Us -~ u 8 : exp(v~i ¢/F) in the usual way. The field Xs = 2BoAds +. . .  contains the graded quark mass ma- trix. The potentials ~(~0)  are real and even functions of the super-~f field ~0 = str(q~) and we keep them up to order ~2. The ultraviolet diver- gences of the weak generating functional to one loop can be derived as it was done in the strong sector [5], provided an expansion in powers of GF is performed. For degenerate quark masses we get for the AS = =kl octet contributions at order GF: 2. The  weak  generat ing  funct iona l  to  one  l oop The quenched generalization of the weak effec- tive Lagrangian for nonleptonic weak interactions *supported by Schweizerisches Nationalfonds 1 Z~8) -  (47r)2(d_4) fdx {lm~(gls-2gs)k(Ax+) +¼gsW4 + 8w0 +  (gs - 4) - (w,2 + w36) + [- i gs 0 -  vl) + 1 ' gs (1 - 4vl)] w7 0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All fights reserved. PII S0920-5632(98)00585-4 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 319 1 (o 1) +~-~(gs - 8~)s)Ws + ~-~(2gs - g~) - ~98 W10 1 t 1 +[(~+~6)(g~8-gs)--~v2(gs-2gs)-[v5 +~gs W~ + ~98 (W~l The operators Wi are listed in Appendix A of [7]. The 27-plet contribution is given by Z(27) - -  (41r)2(d_4) dx927 D1 + -~D2 ~4 1 1 1 3 -1D3 - D4 - ~D~ + ~D6 + ~D7 + ~D8 3D 1 ( l -~a)  D12 1 (l_2"v27"~D1o _ _ -- lOl l  -- ~---~D13 7u~D14-  ~--~D15- ~--~D16 - ~D19 - ~D2o + ~D21 +~D22 1 1 } + ~D23 + ~D24 q- O(G2), where the operators Di are listed in Appendix A of [7]. m~ and a are the usual singlet param- eters. The couplings gs, g~, gs and g2~ are the first terms in the expansion of the weak poten- tials 1)/, i = 8,5,0,27, while the parameters viare the coefficients of the Oo 2term. The quenched generating functional for AS = 2 interactions has exactly the same structure of Z~27), with the ap- propriate t ij'kl tensor. The quenched approximation largely reduces the ultraviolet divergent contribution to the octet sector at one loop. Of the initially divergent 25 octet operators only 10 remain in the quenched approximation. The octet operators Ws, ... W~2 and W38 contribute to K --+ 27r decays. In the 27-plet sector all the unquenched chiral invariants Di survive to the quenched approximation. Only the operators D8, ... D12 contribute to K --4 27r decays. The systematic cancellation of the flavour num- ber dependence follows the rules outlined in [5,8]. The cancellation of the 1/N, 1/N 2 terms is pro- vided by the sum of the non-singlet contributions and the singlet contributions within the bosonic sector. This type of cancellation is entirely due to the presence of a dynamical singlet field. The linear flavour number dependence is cancelled by the fermionic ghost determinant as expected. 2.1. Quenched chiral logarithms The most relevant behaviour of the quenched generating functional to one loop is the appear- ance of quenched chiral logarithms, i.e. of the type m~logm 2, that are pure artefacts of the quenched approximation. As it was discussed at length in [5], quenched chiral logarithms ap- pearing in the strong sector can be formally re- absorbed in a redefinition of the Bo-parameter. In the weak sector an analogous mechanism oc- curs. The quenched chiral logarithms which ap- pear through the first term in the equation for Z[s ) can be formally reabsorbed into a redefini- tion of the weak mass term coupling g~ of the leading order Lagrangian (1). To remove the mg divergence one has to add to the lowest order pa- rameter g~ a d-dependent part proportional to m02 that has a pole at d=4 and the rescaled coupling can be defined as follows: gsn = g~(1 m2 (1 2g8~ M2 'gts(#) ) 48  F2 -  /log-j + The rescaling of the coupling ~ together with the rescaling of the parameter Bo -4/~o defined in [5] in the tree level contribution to any weak observ- able can be used as a short-cut procedure to unre- veal the presence of quenched chiral logarithms, generated when the quenched approximation isimplemented to one loop. 3. K --4 zrzr matr ix e lements  Using the one loop expression of K --4 7rzr ma- trix elements as derived in [9] and the quenched counterpart of the ultraviolet divergences derived here, we can produce a few quantitative estimates of quenching effects on the coefficients of chiral logarithms that contribute to K -+ zrzr ampli- tudes. We work in the infinite volume limit for illustrative purpose. In [10] we shall report on the analysis of unphysical choices of the kinemat- ics used for the computation of lattice K --4 zr~r matrix elements. 320 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 Table 1 The coefficients of the one loop chiral logarithms in K -~ ~r~r amplitudes, full and quenched. Full (rnK -- mr) Full (mr = 0) Quenched Ao -4  (1 - A2 11. __3 13 2 4 4- We decompose the K --~ ~r~r matrix ele- ments into definite isospin invariant amplitudes: ~mAo = .~m(A~ +A 27) with AI ---- 1/2, and ~mA2 with AI = 3/2. In Table (1) we com- pare the coefficients of chiral logarithms for the three amplitudes in the full theory and in the quenched approximation. For the full amplitudes we consider two extreme mass configurations in the one loop corrections: a) degenerate masses, i.e. mg ---- mr and b) mr = 0. The numeri- cal analysis of the physical non degenerate mass case together with the comparison with unphysi- cal choices of the matrix elements on the lattice will be given in [10]. The coefficients of chiral logarithms in the full 27-plet amplitudes .~mA~ 7 and ..~mA2 are quite large in the degenerate mass limit. In addition, by the comparison of sec- ond and third column in Table (1) one can con- clude that all the coefficients in the full ampli- tudes are extremely sensitive to the variation of masses. Using a ~ 0.6 and disregarding for now the unknown contributions tothe octet amplitude in the quenched case we find the following pat- tern going from the full degenerate mass case to the quenched one. Quenching reduces by a tiny amount (from 0.6 to 0.54) the coefficient of the chiral logarithm in the octet amplitude Ao s, while reduces the one in A27 by about 34% in absolute value and 41% in the case of A2. Note that the pattern is opposite for A2 when we compare the quenched amplitude to the mr = 0 limit of the full amplitude. This analysis (see [7,10] for more details) shows that quenching largely affects the coefficient of the chiral logarithm in the 27-plet AI = 3/2 and AI = 1/2 amplitudes. In addition, the comparison with the mr = 0 limit of the full amplitudes (expected to be the most approximate to the physical value) shows that the modification induced by quenching follows a pattern that tends to suppress the AI = 1/2 dominance. REFERENCES 1. C. Bernard, T. Draper, G. Hockney and A. Soni, Nucl. Phys. (Proc. Suppl.) 4 (1988) 483. 2. C. Dawson et al., Nucl. Phys. B514 (1998) 313. M. Testa, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 38. 3. J. Gasser, and H. Leutwyler, Ann. of Phys. (N.Y.) 158 (1984) 142. 4. C.W. Bernard, and M.F.L. Golterman, Phys. Rev. D 46 (1992) 853. 5. G. Colangelo and E. Pallante, Nucl. Phys. B520 (1998) 433; G. Colangelo and E. Pal- lante, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 299. 6. J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B 346 (1990) 17; G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101. 7. E. Pallante, "The generating functional for hadronic weak interactions and its quenched approximation", preprint BUTP-98/18, hep- lat/9808018. 8. G. Colangelo and E. Pallante, Phys. Lett. B 409 (1997) 455. 9. J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B521 (1998) 305. 10. E. Pallante, "K ~ ~rlr matrix elements: ChPT and the lattice", preprint BUTP- 98/19. 

The ultraviolet behaviour of the generating functional for hadronic weak interactions with |Î”S| = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the Î”S = Â±1 weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation in K â†’ Ï€Ï€ matrix elements with Î”I = 1/2 and 3/2.

Pallante, E.,

University of Groningen Digital Archive

ARTS repository - University of Groningen

   University of GroningenThe quenched generating functional for hadronic weak interactionsPallante, E.Published in:Nuclear Physics B - Proceedings SupplementsDOI:10.1016/S0920-5632(99)85060-9IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Pallante, E. (1999). The quenched generating functional for hadronic weak interactions. Nuclear Physics B- Proceedings Supplements, 73(1). https://doi.org/10.1016/S0920-5632(99)85060-9CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 06-06-2022ELSEVIER IUCLEAR PHYSIC~ Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 P R O C E E D I N G S  S U P P L E M E N T S  The quenched generating functional for hadronic weak interactions E. Pallante ~* aInsti tut  fiir Theoretische Physik, Universit~it Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland The ultraviolet behaviour of the generating functional for hadronic weak interactions with IASI = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the AS = =kl weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation in K --+ 7rcr matrix elements with A I  = 1/2 and 3/2. 1. I n t r o d u c t i o n  The computat ion of weak matr ix  elements at long distances is one of the main and still open problems in lattice QCD. Much progress has been done, al though their computat ion still suffers from the presence of major  sources of systematic errors: finite volume effects, unphysical quark masses, the need of computing unphysical matrix elements [1] (see also [2] for recent alternative pro- posals) and finally the fact tha t  still most of the lattice evaluations are done in the quenched ap- proximation. Here we report  on some results con- cerning formal aspects of the quenched approxi- mat ion in the ultraviolet regime. The framework we use is known as Chiral Per turbat ion Theory (ChPT)  [3] and quenched C h P T  [4,5], extended to full (unquenched) weak interactions in [6] and to the quenched case in [7]. The advantages of deriving the generating functional of the effective low energy theory in the full and quenched case are at least two: first, it allows for a systematic control on the quenched modifications and sec- ond, it gives in one step the coefficients of all chiral logarithms for any Green's function or S- matr ix  element. with IASI = 1, 2 and at leading order in the chiral expansion can be written as follows /~AS=I ---- Vs((I)o)str(As32u,,u~) -FV5 (¢o)str(As32X+s) +Vo(¢o)str(A832u,8)str(u~) + h.c + V27( ¢o )tiJ'atstr( Asijuus)str( As~lu~ ) (1) and the same last te rm for AS  = 2 with the appropriate  t ij,kt tensor. The projection ma- trix onto the octet and 27-plet components is • , ~.. z---~,t  with the graded matr ix  Aaij = ~8-,~ 2 ~8, ()~ij)a b : 5iaf~jb. The graded fields u~8 = : u t V  U t -I- usx?sUs iuIDuU~u i = u~ and X+. . . ~ . .2 contain the dynamical meson field Us -~ u 8 : exp(v~ i  ¢/F) in the usual way. The field Xs = 2BoAds + . . .  contains the graded quark mass ma- trix. The potentials ~ ( ~ 0 )  are real and even functions of the super-~f field ~0 = str(q~) and we keep them up to order ~2. The ultraviolet diver- gences of the weak generating functional to one loop can be derived as it was done in the strong sector [5], provided an expansion in powers of GF is performed. For degenerate quark masses we get for the AS = =kl octet contributions at order GF: 2. T h e  w e a k  g e n e r a t i n g  f u n c t i o n a l  t o  o n e  l o o p  The quenched generalization of the weak effec- tive Lagrangian for nonleptonic weak interactions *supported by Schweizerisches Nationalfonds 1 Z ~ 8 ) -  (47r)2(d_4) fdx {lm~(gls-2gs)k(Ax+) +¼gsW4 + 8w0 +  (gs - 4)  - (w,2 + w36) + [-  i gs 0 -  vl) + 1 ' gs (1 - 4vl)] w7 0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All fights reserved. PII S0920-5632(98)00585-4 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 319 1 (o 1) +~-~(gs - 8~)s)Ws + ~-~(2gs - g~) - ~98 W10 1 t 1 +[(~+~6)(g~8-gs) - -~v2(gs-2gs) -[v5  +~gs W~ + ~98 (W~l The operators Wi are listed in Appendix A of [7]. The 27-plet contribution is given by Z(27) - -  (41r)2(d_4) dx927 D1 + -~D2 ~4 1 1 1 3 -1D3 - D4 - ~D~ + ~D6 + ~D7 + ~D8 3D 1 ( l - ~ a )  D12 1 (l_2"v27"~D1o _ _ - - lOl l  -- ~---~D13 7u ~ D 1 4 -  ~--~D15- ~--~D16 - ~D19 - ~D2o + ~D21 +~D22 1 1 } + ~D23 + ~D24 q- O(G2), where the operators Di are listed in Appendix A of [7]. m~ and a are the usual singlet param- eters. The couplings gs, g~, gs and g2~ are the first terms in the expansion of the weak poten- tials 1)/, i = 8,5,0,27, while the parameters vi are the coefficients of the Oo 2 term. The quenched generating functional for AS = 2 interactions has exactly the same structure of Z~27), with the ap- propriate t ij'kl tensor. The quenched approximation largely reduces the ultraviolet divergent contribution to the octet sector at one loop. Of the initially divergent 25 octet operators only 10 remain in the quenched approximation. The octet operators Ws, ... W~2 a n d  W38 contribute to K --+ 27r decays. In the 27-plet sector all the unquenched chiral invariants Di survive to the quenched approximation. Only the operators D8, ...  D12 contribute to K --4 27r decays. The systematic cancellation of the flavour num- ber dependence follows the rules outlined in [5,8]. The cancellation of the 1/N, 1/N 2 terms is pro- vided by the sum of the non-singlet contributions and the singlet contributions within the bosonic sector. This type of cancellation is entirely due to the presence of a dynamical singlet field. The linear flavour number dependence is cancelled by the fermionic ghost determinant as expected. 2.1. Q u e n c h e d  chiral  logar i thms The most relevant behaviour of the quenched generating functional to one loop is the appear- ance of quenched chiral logarithms, i.e. of the type m~logm 2, that are pure artefacts of the quenched approximation. As it was discussed at length in [5], quenched chiral logarithms ap- pearing in the strong sector can be formally re- absorbed in a redefinition of the Bo-parameter. In the weak sector an analogous mechanism oc- curs. The quenched chiral logarithms which ap- pear through the first term in the equation for Z[s ) can be formally reabsorbed into a redefini- tion of the weak mass term coupling g~ of the leading order Lagrangian (1). To remove the mg divergence one has to add to the lowest order pa- rameter g~ a d-dependent part proportional to m02 that has a pole at d=4 and the rescaled coupling can be defined as follows: gsn = g~(1 m2 (1 2g8~ M2 'gts(#) ) 48  F2 -  /log-j + The rescaling of the coupling g~ together with the rescaling of the parameter Bo -4/~o defined in [5] in the tree level contribution to any weak observ- able can be used as a short-cut procedure to unre- veal the presence of quenched chiral logarithms, generated when the quenched approximation is implemented to one loop. 3. K --4 zrzr ma t r ix  e l e m e n t s  Using the one loop expression of K --4 7rzr ma- trix elements as derived in [9] and the quenched counterpart of the ultraviolet divergences derived here, we can produce a few quantitative estimates of quenching effects on the coefficients of chiral logarithms that contribute to K -+ zrzr ampli- tudes. We work in the infinite volume limit for illustrative purpose. In [10] we shall report on the analysis of unphysical choices of the kinemat- ics used for the computation of lattice K --4 zr~r matrix elements. 320 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 Table 1 The coefficients of the one loop chiral logarithms in K -~ ~r~r amplitudes, full and quenched. Full (rnK -- mr) Full (mr = 0) Quenched Ao - 4  (1 - A2 11. __3 13 2 4 4- We decompose the K --~ ~r~r matrix ele- ments into definite isospin invariant amplitudes: ~mAo = .~m(A~ + A  27) with AI  ---- 1/2, and ~mA2 with AI = 3/2. In Table (1) we com- pare the coefficients of chiral logarithms for the three amplitudes in the full theory and in the quenched approximation. For the full amplitudes we consider two extreme mass configurations in the one loop corrections: a) degenerate masses, i.e. m g  ---- mr and b) mr = 0. The numeri- cal analysis of the physical non degenerate mass case together with the comparison with unphysi- cal choices of the matrix elements on the lattice will be given in [10]. The coefficients of chiral logarithms in the full 27-plet amplitudes .~mA~ 7 and ..~mA2 are quite large in the degenerate mass limit. In addition, by the comparison of sec- ond and third column in Table (1) one can con- clude that all the coefficients in the full ampli- tudes are extremely sensitive to the variation of masses. Using a ~ 0.6 and disregarding for now the unknown contributions to the octet amplitude in the quenched case we find the following pat- tern going from the full degenerate mass case to the quenched one. Quenching reduces by a tiny amount (from 0.6 to 0.54) the coefficient of the chiral logarithm in the octet amplitude Ao s, while reduces the one in A27 by about 34% in absolute value and 41% in the case of A2. Note that the pattern is opposite for A2 when we compare the quenched amplitude to the mr = 0 limit of the full amplitude. This analysis (see [7,10] for more details) shows that quenching largely affects the coefficient of the chiral logarithm in the 27-plet AI = 3/2 and AI = 1/2 amplitudes. In addition, the comparison with the mr = 0 limit of the full amplitudes (expected to be the most approximate to the physical value) shows that the modification induced by quenching follows a pattern that tends to suppress the AI = 1/2 dominance. R E F E R E N C E S  1. C. Bernard, T. Draper, G. Hockney and A. Soni, Nucl. Phys. (Proc. Suppl.) 4 (1988) 483. 2. C. Dawson et al., Nucl. Phys. B514 (1998) 313. M. Testa, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 38. 3. J. Gasser, and H. Leutwyler, Ann. of Phys. (N.Y.) 158 (1984) 142. 4. C.W. Bernard, and M.F.L. Golterman, Phys. Rev. D 46 (1992) 853. 5. G. Colangelo and E. Pallante, Nucl. Phys. B520 (1998) 433; G. Colangelo and E. Pal- lante, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 299. 6. J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B 346 (1990) 17; G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101. 7. E. Pallante, "The generating functional for hadronic weak interactions and its quenched approximation", preprint BUTP-98/18, hep- lat/9808018. 8. G. Colangelo and E. Pallante, Phys. Lett. B 409 (1997) 455. 9. J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B521 (1998) 305. 10. E. Pallante, "K ~ ~rlr matrix elements: ChPT and the lattice", preprint BUTP- 98/19. 

   University of GroningenThe quenched generating functional for hadronic weak interactionsPallante, E.Published in:Nuclear Physics B - Proceedings SupplementsDOI:10.1016/S0920-5632(99)85060-9IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Pallante, E. (1999). The quenched generating functional for hadronic weak interactions. Nuclear Physics B- Proceedings Supplements, 73(1). https://doi.org/10.1016/S0920-5632(99)85060-9CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 16-07-2023ELSEVIER IUCLEAR PHYSIC~ Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 P R O C E E D I N G S  S U P P L E M E N T S  The quenched generating functional for hadronic weak interactions E. Pallante ~* aInsti tut  fiir Theoretische Physik, Universit~it Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland The ultraviolet behaviour of the generating functional for hadronic weak interactions with IASI = 1, 2 is investigated to one loop for a generic number of flavours and in the quenched approximation. New quenched chiral logarithms generated by the weak interactions can be accounted for via a redefinition of the weak mass term in the AS = =kl weak effective Lagrangian at leading order. Finally, we illustrate how chiral logarithms are modified by the quenched approximation in K --+ 7rcr matrix elements with A I  = 1/2 and 3/2. 1. I n t r o d u c t i o n  The computat ion of weak matr ix  elements at long distances is one of the main and still open problems in lattice QCD. Much progress has been done, al though their computat ion still suffers from the presence of major  sources of systematic errors: finite volume effects, unphysical quark masses, the need of computing unphysical matrix elements [1] (see also [2] for recent alternative pro- posals) and finally the fact tha t  still most of the lattice evaluations are done in the quenched ap- proximation. Here we report  on some results con- cerning formal aspects of the quenched approxi- mat ion in the ultraviolet regime. The framework we use is known as Chiral Per turbat ion Theory (ChPT)  [3] and quenched C h P T  [4,5], extended to full (unquenched) weak interactions in [6] and to the quenched case in [7]. The advantages of deriving the generating functional of the effective low energy theory in the full and quenched case are at least two: first, it allows for a systematic control on the quenched modifications and sec- ond, it gives in one step the coefficients of all chiral logarithms for any Green's function or S- matr ix  element. with IASI = 1, 2 and at leading order in the chiral expansion can be written as follows /~AS=I ---- Vs((I)o)str(As32u,,u~) -FV5 (¢o)str(As32X+s) +Vo(¢o)str(A832u,8)str(u~) + h.c + V27( ¢o )tiJ'atstr( Asijuus)str( As~lu~ ) (1) and the same last te rm for AS  = 2 with the appropriate  t ij,kt tensor. The projection ma- trix onto the octet and 27-plet components is • , ~.. z---~,t  with the graded matr ix  Aaij = ~8-,~ 2 ~8, ()~ij)a b : 5iaf~jb. The graded fields u~8 = : u t V  U t -I- usx?sUs iuIDuU~u i = u~ and X+. . . ~ . .2 contain the dynamical meson field Us -~ u 8 : exp(v~ i  ¢/F) in the usual way. The field Xs = 2BoAds + . . .  contains the graded quark mass ma- trix. The potentials ~ ( ~ 0 )  are real and even functions of the super-~f field ~0 = str(q~) and we keep them up to order ~2. The ultraviolet diver- gences of the weak generating functional to one loop can be derived as it was done in the strong sector [5], provided an expansion in powers of GF is performed. For degenerate quark masses we get for the AS = =kl octet contributions at order GF: 2. T h e  w e a k  g e n e r a t i n g  f u n c t i o n a l  t o  o n e  l o o p  The quenched generalization of the weak effec- tive Lagrangian for nonleptonic weak interactions *supported by Schweizerisches Nationalfonds 1 Z ~ 8 ) -  (47r)2(d_4) fdx {lm~(gls-2gs)k(Ax+) +¼gsW4 + 8w0 +  (gs - 4)  - (w,2 + w36) + [-  i gs 0 -  vl) + 1 ' gs (1 - 4vl)] w7 0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All fights reserved. PII S0920-5632(98)00585-4 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 319 1 (o 1) +~-~(gs - 8~)s)Ws + ~-~(2gs - g~) - ~98 W10 1 t 1 +[(~+~6)(g~8-gs) - -~v2(gs-2gs) -[v5  +~gs W~ + ~98 (W~l The operators Wi are listed in Appendix A of [7]. The 27-plet contribution is given by Z(27) - -  (41r)2(d_4) dx927 D1 + -~D2 ~4 1 1 1 3 -1D3 - D4 - ~D~ + ~D6 + ~D7 + ~D8 3D 1 ( l - ~ a )  D12 1 (l_2"v27"~D1o _ _ - - lOl l  -- ~---~D13 7u ~ D 1 4 -  ~--~D15- ~--~D16 - ~D19 - ~D2o + ~D21 +~D22 1 1 } + ~D23 + ~D24 q- O(G2), where the operators Di are listed in Appendix A of [7]. m~ and a are the usual singlet param- eters. The couplings gs, g~, gs and g2~ are the first terms in the expansion of the weak poten- tials 1)/, i = 8,5,0,27, while the parameters vi are the coefficients of the Oo 2 term. The quenched generating functional for AS = 2 interactions has exactly the same structure of Z~27), with the ap- propriate t ij'kl tensor. The quenched approximation largely reduces the ultraviolet divergent contribution to the octet sector at one loop. Of the initially divergent 25 octet operators only 10 remain in the quenched approximation. The octet operators Ws, ... W~2 a n d  W38 contribute to K --+ 27r decays. In the 27-plet sector all the unquenched chiral invariants Di survive to the quenched approximation. Only the operators D8, ...  D12 contribute to K --4 27r decays. The systematic cancellation of the flavour num- ber dependence follows the rules outlined in [5,8]. The cancellation of the 1/N, 1/N 2 terms is pro- vided by the sum of the non-singlet contributions and the singlet contributions within the bosonic sector. This type of cancellation is entirely due to the presence of a dynamical singlet field. The linear flavour number dependence is cancelled by the fermionic ghost determinant as expected. 2.1. Q u e n c h e d  chiral  logar i thms The most relevant behaviour of the quenched generating functional to one loop is the appear- ance of quenched chiral logarithms, i.e. of the type m~logm 2, that are pure artefacts of the quenched approximation. As it was discussed at length in [5], quenched chiral logarithms ap- pearing in the strong sector can be formally re- absorbed in a redefinition of the Bo-parameter. In the weak sector an analogous mechanism oc- curs. The quenched chiral logarithms which ap- pear through the first term in the equation for Z[s ) can be formally reabsorbed into a redefini- tion of the weak mass term coupling g~ of the leading order Lagrangian (1). To remove the mg divergence one has to add to the lowest order pa- rameter g~ a d-dependent part proportional to m02 that has a pole at d=4 and the rescaled coupling can be defined as follows: gsn = g~(1 m2 (1 2g8~ M2 'gts(#) ) 48  F2 -  /log-j + The rescaling of the coupling g~ together with the rescaling of the parameter Bo -4/~o defined in [5] in the tree level contribution to any weak observ- able can be used as a short-cut procedure to unre- veal the presence of quenched chiral logarithms, generated when the quenched approximation is implemented to one loop. 3. K --4 zrzr ma t r ix  e l e m e n t s  Using the one loop expression of K --4 7rzr ma- trix elements as derived in [9] and the quenched counterpart of the ultraviolet divergences derived here, we can produce a few quantitative estimates of quenching effects on the coefficients of chiral logarithms that contribute to K -+ zrzr ampli- tudes. We work in the infinite volume limit for illustrative purpose. In [10] we shall report on the analysis of unphysical choices of the kinemat- ics used for the computation of lattice K --4 zr~r matrix elements. 320 E. Pallante/Nuclear Physics B (Proc. Suppl.) 73 (1999) 318-320 Table 1 The coefficients of the one loop chiral logarithms in K -~ ~r~r amplitudes, full and quenched. Full (rnK -- mr) Full (mr = 0) Quenched Ao - 4  (1 - A2 11. __3 13 2 4 4- We decompose the K --~ ~r~r matrix ele- ments into definite isospin invariant amplitudes: ~mAo = .~m(A~ + A  27) with AI  ---- 1/2, and ~mA2 with AI = 3/2. In Table (1) we com- pare the coefficients of chiral logarithms for the three amplitudes in the full theory and in the quenched approximation. For the full amplitudes we consider two extreme mass configurations in the one loop corrections: a) degenerate masses, i.e. m g  ---- mr and b) mr = 0. The numeri- cal analysis of the physical non degenerate mass case together with the comparison with unphysi- cal choices of the matrix elements on the lattice will be given in [10]. The coefficients of chiral logarithms in the full 27-plet amplitudes .~mA~ 7 and ..~mA2 are quite large in the degenerate mass limit. In addition, by the comparison of sec- ond and third column in Table (1) one can con- clude that all the coefficients in the full ampli- tudes are extremely sensitive to the variation of masses. Using a ~ 0.6 and disregarding for now the unknown contributions to the octet amplitude in the quenched case we find the following pat- tern going from the full degenerate mass case to the quenched one. Quenching reduces by a tiny amount (from 0.6 to 0.54) the coefficient of the chiral logarithm in the octet amplitude Ao s, while reduces the one in A27 by about 34% in absolute value and 41% in the case of A2. Note that the pattern is opposite for A2 when we compare the quenched amplitude to the mr = 0 limit of the full amplitude. This analysis (see [7,10] for more details) shows that quenching largely affects the coefficient of the chiral logarithm in the 27-plet AI = 3/2 and AI = 1/2 amplitudes. In addition, the comparison with the mr = 0 limit of the full amplitudes (expected to be the most approximate to the physical value) shows that the modification induced by quenching follows a pattern that tends to suppress the AI = 1/2 dominance. R E F E R E N C E S  1. C. Bernard, T. Draper, G. Hockney and A. Soni, Nucl. Phys. (Proc. Suppl.) 4 (1988) 483. 2. C. Dawson et al., Nucl. Phys. B514 (1998) 313. M. Testa, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 38. 3. J. Gasser, and H. Leutwyler, Ann. of Phys. (N.Y.) 158 (1984) 142. 4. C.W. Bernard, and M.F.L. Golterman, Phys. Rev. D 46 (1992) 853. 5. G. Colangelo and E. Pallante, Nucl. Phys. B520 (1998) 433; G. Colangelo and E. Pal- lante, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 299. 6. J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B 346 (1990) 17; G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101. 7. E. Pallante, "The generating functional for hadronic weak interactions and its quenched approximation", preprint BUTP-98/18, hep- lat/9808018. 8. G. Colangelo and E. Pallante, Phys. Lett. B 409 (1997) 455. 9. J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B521 (1998) 305. 10. E. Pallante, "K ~ ~rlr matrix elements: ChPT and the lattice", preprint BUTP- 98/19. 

The quenched generating functional for hadronic weak interactions

Abstract

Similar works

Full text

Available Versions

NARCIS

Crossref

University of Groningen

University of Groningen

University of Groningen Digital Archive

ARTS repository - University of Groningen

ARTS repository - University of Groningen