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Chiral perturbation theory for the quenched approximation
Nuclear Physics B (Pros. Suppl.) 26 (1992) 360-362 North-Holland
C I
L PERTU
.~TION THEORY FOR THE QUENCHED APPROXIMATION
Claude Bernard and Maarten Golterman Department of Physics, Washington University, St . Louis, MO 63130, USA We describe a technique for constructing the effective chiral theory for quenched QCD. The effective theory which results is a Lagrangian one, with a graded symmetry group which mixes Goldstone bosons and fermions, and with a definite set of Feynman rules. The techniques have been used to calculate chiral logarithms in fK/f*, m,., and InK.
'l'he quenched apprex:mation (QA) is a "necessary evil" that will be with us for the foreseeable future. Even with a Teraflop Machine[1], the QA will be needed for the crucial corners of the parameter space: large volumes, physical quark masses, and "deep scaling ." We therefore need to learn as much as possible analytically about the QA in order to have good control over the systematics of such calculations . In the full theory, chiral perturbation theory (ChPT) is a key analytic tool. It gives: e The detailed form of the the approach to the chiral limit. The universal terms ("chiral logs) can be calculated order by order in the loop expansion. Comparison with this expected chiral behavior provides, e.g., a crucial check of lattice weak matrix element calculations . ® The leading finite-volume corrections at large volume[2] . As the lightest particles, the pseudoscalar mesons clearly control these corrections; ChPT is simply the effective theory of their interactions.
It is therefore clear why one would like to have a ChPT corresponding to the QA. In fact, there have been several previous attempts to calculate quenched chiral logs. Morel[3] and Sharpe[4] use the strong coupling and 1/d expansions ; Kilcup et al.[5] and Sharpe[6] trace quârb ûow through ordinary ChPT diagrams, eliminating those diagrams which have virtual quark loops. The latter Presented by C. Bernard 0920-5632132/$05 .00
1992- Elsevier Science
paper emphasizes the importance of quenched ChPT and describes many of the key issues. In the quark-flow approach one needs a unique identification of meson diagrams in ordinary ChPT with quark-flow diagrams in QCD. To do this, one must keep the q' explicitly in the chiral theory. Only with all 9 q-q combinations allowed as mesons can one make a 1-1 correspondence . The rl' must be kept for a second reason, too. In the full theory the q' gets the singlet part of its mass (_ it) from iteration of quark loops joined by gluons. Assuming p » moctet, the q' may be neglected . In the QA, however, only the diagram linear in p2 ("two hairpins joined by gluons") survives. Since the diagram is not iterated, 112 appears in the numerator of the q' propagator . Thus the q' cannot be neglected . In many cases, it is then clear which full ChPT diagrams should be dropped in the QA. Consider, e.g., the first correction to a 7r+ propagator: a meson tadpole. When the tadpole is a K+, then the diagram is absent in the QA, since the s quark must come from a virtual loop. When the tadpole is a 7r+, however, things are less clear. If the previous s quark has just been replaced by a d, then again the diagram is absent in the QA. Instead, however, the valence quarks could make the tadpole. The vertex in such a diagram is a
Publishers B.V All rights reserved.
C. Bernara M. Goher»ran l Chiral perturfiation theoryfor the quenched apprax*rurtion
meson-meson scattering vertex with no quark exchange . This vertex in fact vanishes at O(p2 ), although we have not been able to prove this in full detail within this approach. Thus 7r+ tadpoles are also absent in the QA. Indeed, the only correction to the 7r+ propagator at this order comes from an V tadpole with one two-hairpin vertex. The end result of this approach can be described as a "Lagrangian + rules." The Lagrangian is the ordinary chiral Lagrangian corresponding to full QCD; the rules tell which diagrams to drop in the QA. We find this approach unsatisfactory, however. First of all, it is difficult to make the application of the rules routine. More importantly, the presence of the "rules" implies that this is not a true Lagrangian theory. For example, the invariance of the physics under field redefinitions is not guaranteed . Such redefinitions are needed to reduce the number of q' terms in the Lagrangian - see [7]. Similarly, it is not clear that the rules will respect the usual cancellation of the 6(4)(0) terms. A true Lagrangian framework is however possible. To make a QCD Lagrangian which describes the QA[3], take the ordinary QCD Lagrangian and add, for each quark qa (a = u, d, s), a scalar (ghost) quark qo with the same mass. The ghost determinant then cancels the quark determinant. The low-energy effective theory for this quenched QCD Lagrangian may now be constructed. It will have all possible pseudoscalar bound states of quarks or scalar quarks with their antiparticles: ordinary qq mesons (denoted generically by 0); ghost qq mesons (~); and fermionic mesons qq and qq (X and Xt). As in ordinary ChPT, the symmetries at the quark level determine the form of the interactions among the mesons. The symmetry is U(313)L x U(313)R, where U(313) is "almost" a U(6) among u, d, s, û, d, i, but has a graded structure since it mixes fermions and bosons. If
we write a matrix U E U(3I3) in block form as U=1 A C, ,then A and B are matrices of D LI / commuting numbers; C and D, of anticommuting. Complex conjugation is defined to itch the order of anticommuting variables; in terms of this complex conjugation, Hermitian conju tion and unitarity have their usual definitions. There is also a cyclic "supertrace» defined by str(U) = tr(A) - tr(B), and a "superdeterminant," sdet(U) = exp str In U, with the property sdet(Ul U2) = sdet(Ul ) sdet(U2) . Now define the Hermitian field and the mass matrix M by
4 - ( d xt 1 x á where M is the usual quark mass matrix. The unitary field E = exp(2il~/f) then transforms as E -+ ULEUR. The ChPT Lagrangian invariant under the full U(313)L x U(313)R, is then given by Ginv
=
2
str(a EOOEt) + v str(ME + MEt) .
The anomaly breaks the symmetry group down to SU(3,3)L x SU(313)R x U(1). The anomalous field is A~o = (V' - 4')/V2. Under the reduced group, 4~o oc str In E = In sdetE is invariant, so arbitrary functions of A~o can be included in the full Lagrangian, G. However, one can now redefine E to simplify G, much as in [7]. The result is L = Gin, + ,C-0., with C.*a = Vo (~o) +Vi( '~bo)str(a,~Ear`Et) +V2(OPO)str(ME + MEt) +
Vs(P0)(c?N%)2,
where the functions T; can be chosen to be real and even . We need below only the quadratic terms in 4. : G = &_ + a(a o)2 -p 2 2o + . . .~ where a = %(0) and p2 = -(1/2)Vö (0). One can calculate straightforwardly with C.
C.
M. Gohernwn /Chiralperturfiodon theoryfor the quenched appraaimation
One unusual feature: in the V', ii' sector, the quadratic terms from Li o aid Coo cannot be sim'litaneousl-y diagonalized in a mornenturnindependent way. This leads :iii' to treat the quaGratic terms from C.&o as :rertices. Iterations of these vertices on the same line then vanish due to cancellation between r!' and the negative metric 4'. (The negative metric comes from the definition of str.) This is a manifestation of the fact that the iteration of the two-hairpin vertex is forbidden in quenched QCD. The Goo vertices can in principle appear more than once in a diagram if they occur on different lines. However, it is our philosophy to treat the parameters a and p2 as small . This is certainly true in the 1/N, expansion. Moreover, it appears that the real expansion parameters are a/3 and U2 /3 (see below). To estimate their size, take full QCD with a =- 0, neglect rl- q' mixing, and use the physical ri' mass. One gets P 2 /3 ^= (5()() McV)2 MK, which indicates that quenched ChPT may be roughly as good as full ChPT for the kaon. So far, we have calculated ra.,r, MK, fA, fK, (emu), and (ss) at 1-loop, for infinite volume . Here is a representative sample of these results[8]: ( ra 1A-loop)2
= m 22 (l+
1
81r2j 2
(
a A2 _ p2 3 3
2a , + 3 mÁ + ( ~ - 3 m~ ln(AZ/m~) » 2
fa-loop = f
(f fi
/f'r)1-'oop =
1
m2 _ 3 K
l'+ 161r2 ƒ2
~mx - 3 mir (2MK 2 - m~) 2 2 (mK
21
In
2:TtK _ 2 niir
2
3 + 1
where A is the cutoff, and MK, m,r, and f are the bare parameters . Using a = 0 and p as estimated above, (fK/fx) i-"op 1.07, indicating that quenched ChPT is working well. Note how-
ever that (MIloop/mr)2 1.5 for A a5 1GeV, though this ratio is not directly physical and the large correction is perhaps not worrisome . Some comments on these results and directions for future work: 121
® The absence of chiral logs in m. seen in [4] is presumably a feature of the leading term in the 1/d expansion . Indeed, the q' diagrams which give such logs are mentioned in [6] . The calculation of (fK/f*)1-1°°p, repeated at finite volume, will give the leading finite size dependence in the QA . For fixed volume, however, it is probably not a numerical prediction since terms quartic in momenta presumably affect the ratio . e -loop blows up as m ti , and -" (fKlf*)' 0 with m, fixed . This is an IR effect coming from the double pole of the two-hairpin diagram, and is absent in the full theory where the vertex can be iterated . ® A Gasser- Leutwyler[7] program for quenched ChPT at 1-loop is possible: there may be interesting numerical relations involving only computable (on the lattice) quan-
tities. Such relations could give quantitative insight into the effects of quenching . e Extension of these ideas to weak matrix elements seems to be straightforward.
We thank Steve Sharpe for many useful discussions. This work was supported in part by the US Department of Energy. References [1]
Physics Goals of the QCD Teraflop Project, S . Aoká et at., Intl . J. Mod . Phys . C , to be published . [2] See, e .g ., J . Gasser & H . Leutwyler, Phys . Lett. 184B (1987) 83; H . Neuberger, Nucl . Phys. B300 (1988)
180 ; P . Hasenfratz & H . Leutwyler, Nucl . Phys . B343 (1990) 241 .
[3] A . Morel, J . Physique 48 (1987) 111 . [4] S . Sharpe, Phys . Rev . D41 (1990) 3233 .
[5] G . Kilcup et at ., Phys . Rev. Lett. 64 (1990) 25. [6] S . Sharpe, Nucl . Phys . B17 (Pros . Sup .) ('90) 146. [7] J .Gasser & H .Leutwyler, Nucl . Phys. B250 ('85) 465 . [8]
Except for the A2 term, a terms are actually higher order in a combirt:û expansion in 1/N, and M .
C I
L PERTU
.~TION THEORY FOR THE QUENCHED APPROXIMATION
Claude Bernard and Maarten Golterman Department of Physics, Washington University, St . Louis, MO 63130, USA We describe a technique for constructing the effective chiral theory for quenched QCD. The effective theory which results is a Lagrangian one, with a graded symmetry group which mixes Goldstone bosons and fermions, and with a definite set of Feynman rules. The techniques have been used to calculate chiral logarithms in fK/f*, m,., and InK.
'l'he quenched apprex:mation (QA) is a "necessary evil" that will be with us for the foreseeable future. Even with a Teraflop Machine[1], the QA will be needed for the crucial corners of the parameter space: large volumes, physical quark masses, and "deep scaling ." We therefore need to learn as much as possible analytically about the QA in order to have good control over the systematics of such calculations . In the full theory, chiral perturbation theory (ChPT) is a key analytic tool. It gives: e The detailed form of the the approach to the chiral limit. The universal terms ("chiral logs) can be calculated order by order in the loop expansion. Comparison with this expected chiral behavior provides, e.g., a crucial check of lattice weak matrix element calculations . ® The leading finite-volume corrections at large volume[2] . As the lightest particles, the pseudoscalar mesons clearly control these corrections; ChPT is simply the effective theory of their interactions.
It is therefore clear why one would like to have a ChPT corresponding to the QA. In fact, there have been several previous attempts to calculate quenched chiral logs. Morel[3] and Sharpe[4] use the strong coupling and 1/d expansions ; Kilcup et al.[5] and Sharpe[6] trace quârb ûow through ordinary ChPT diagrams, eliminating those diagrams which have virtual quark loops. The latter Presented by C. Bernard 0920-5632132/$05 .00
1992- Elsevier Science
paper emphasizes the importance of quenched ChPT and describes many of the key issues. In the quark-flow approach one needs a unique identification of meson diagrams in ordinary ChPT with quark-flow diagrams in QCD. To do this, one must keep the q' explicitly in the chiral theory. Only with all 9 q-q combinations allowed as mesons can one make a 1-1 correspondence . The rl' must be kept for a second reason, too. In the full theory the q' gets the singlet part of its mass (_ it) from iteration of quark loops joined by gluons. Assuming p » moctet, the q' may be neglected . In the QA, however, only the diagram linear in p2 ("two hairpins joined by gluons") survives. Since the diagram is not iterated, 112 appears in the numerator of the q' propagator . Thus the q' cannot be neglected . In many cases, it is then clear which full ChPT diagrams should be dropped in the QA. Consider, e.g., the first correction to a 7r+ propagator: a meson tadpole. When the tadpole is a K+, then the diagram is absent in the QA, since the s quark must come from a virtual loop. When the tadpole is a 7r+, however, things are less clear. If the previous s quark has just been replaced by a d, then again the diagram is absent in the QA. Instead, however, the valence quarks could make the tadpole. The vertex in such a diagram is a
Publishers B.V All rights reserved.
C. Bernara M. Goher»ran l Chiral perturfiation theoryfor the quenched apprax*rurtion
meson-meson scattering vertex with no quark exchange . This vertex in fact vanishes at O(p2 ), although we have not been able to prove this in full detail within this approach. Thus 7r+ tadpoles are also absent in the QA. Indeed, the only correction to the 7r+ propagator at this order comes from an V tadpole with one two-hairpin vertex. The end result of this approach can be described as a "Lagrangian + rules." The Lagrangian is the ordinary chiral Lagrangian corresponding to full QCD; the rules tell which diagrams to drop in the QA. We find this approach unsatisfactory, however. First of all, it is difficult to make the application of the rules routine. More importantly, the presence of the "rules" implies that this is not a true Lagrangian theory. For example, the invariance of the physics under field redefinitions is not guaranteed . Such redefinitions are needed to reduce the number of q' terms in the Lagrangian - see [7]. Similarly, it is not clear that the rules will respect the usual cancellation of the 6(4)(0) terms. A true Lagrangian framework is however possible. To make a QCD Lagrangian which describes the QA[3], take the ordinary QCD Lagrangian and add, for each quark qa (a = u, d, s), a scalar (ghost) quark qo with the same mass. The ghost determinant then cancels the quark determinant. The low-energy effective theory for this quenched QCD Lagrangian may now be constructed. It will have all possible pseudoscalar bound states of quarks or scalar quarks with their antiparticles: ordinary qq mesons (denoted generically by 0); ghost qq mesons (~); and fermionic mesons qq and qq (X and Xt). As in ordinary ChPT, the symmetries at the quark level determine the form of the interactions among the mesons. The symmetry is U(313)L x U(313)R, where U(313) is "almost" a U(6) among u, d, s, û, d, i, but has a graded structure since it mixes fermions and bosons. If
we write a matrix U E U(3I3) in block form as U=1 A C, ,then A and B are matrices of D LI / commuting numbers; C and D, of anticommuting. Complex conjugation is defined to itch the order of anticommuting variables; in terms of this complex conjugation, Hermitian conju tion and unitarity have their usual definitions. There is also a cyclic "supertrace» defined by str(U) = tr(A) - tr(B), and a "superdeterminant," sdet(U) = exp str In U, with the property sdet(Ul U2) = sdet(Ul ) sdet(U2) . Now define the Hermitian field and the mass matrix M by
4 - ( d xt 1 x á where M is the usual quark mass matrix. The unitary field E = exp(2il~/f) then transforms as E -+ ULEUR. The ChPT Lagrangian invariant under the full U(313)L x U(313)R, is then given by Ginv
=
2
str(a EOOEt) + v str(ME + MEt) .
The anomaly breaks the symmetry group down to SU(3,3)L x SU(313)R x U(1). The anomalous field is A~o = (V' - 4')/V2. Under the reduced group, 4~o oc str In E = In sdetE is invariant, so arbitrary functions of A~o can be included in the full Lagrangian, G. However, one can now redefine E to simplify G, much as in [7]. The result is L = Gin, + ,C-0., with C.*a = Vo (~o) +Vi( '~bo)str(a,~Ear`Et) +V2(OPO)str(ME + MEt) +
Vs(P0)(c?N%)2,
where the functions T; can be chosen to be real and even . We need below only the quadratic terms in 4. : G = &_ + a(a o)2 -p 2 2o + . . .~ where a = %(0) and p2 = -(1/2)Vö (0). One can calculate straightforwardly with C.
C.
M. Gohernwn /Chiralperturfiodon theoryfor the quenched appraaimation
One unusual feature: in the V', ii' sector, the quadratic terms from Li o aid Coo cannot be sim'litaneousl-y diagonalized in a mornenturnindependent way. This leads :iii' to treat the quaGratic terms from C.&o as :rertices. Iterations of these vertices on the same line then vanish due to cancellation between r!' and the negative metric 4'. (The negative metric comes from the definition of str.) This is a manifestation of the fact that the iteration of the two-hairpin vertex is forbidden in quenched QCD. The Goo vertices can in principle appear more than once in a diagram if they occur on different lines. However, it is our philosophy to treat the parameters a and p2 as small . This is certainly true in the 1/N, expansion. Moreover, it appears that the real expansion parameters are a/3 and U2 /3 (see below). To estimate their size, take full QCD with a =- 0, neglect rl- q' mixing, and use the physical ri' mass. One gets P 2 /3 ^= (5()() McV)2 MK, which indicates that quenched ChPT may be roughly as good as full ChPT for the kaon. So far, we have calculated ra.,r, MK, fA, fK, (emu), and (ss) at 1-loop, for infinite volume . Here is a representative sample of these results[8]: ( ra 1A-loop)2
= m 22 (l+
1
81r2j 2
(
a A2 _ p2 3 3
2a , + 3 mÁ + ( ~ - 3 m~ ln(AZ/m~) » 2
fa-loop = f
(f fi
/f'r)1-'oop =
1
m2 _ 3 K
l'+ 161r2 ƒ2
~mx - 3 mir (2MK 2 - m~) 2 2 (mK
21
In
2:TtK _ 2 niir
2
3 + 1
where A is the cutoff, and MK, m,r, and f are the bare parameters . Using a = 0 and p as estimated above, (fK/fx) i-"op 1.07, indicating that quenched ChPT is working well. Note how-
ever that (MIloop/mr)2 1.5 for A a5 1GeV, though this ratio is not directly physical and the large correction is perhaps not worrisome . Some comments on these results and directions for future work: 121
® The absence of chiral logs in m. seen in [4] is presumably a feature of the leading term in the 1/d expansion . Indeed, the q' diagrams which give such logs are mentioned in [6] . The calculation of (fK/f*)1-1°°p, repeated at finite volume, will give the leading finite size dependence in the QA . For fixed volume, however, it is probably not a numerical prediction since terms quartic in momenta presumably affect the ratio . e -loop blows up as m ti , and -" (fKlf*)' 0 with m, fixed . This is an IR effect coming from the double pole of the two-hairpin diagram, and is absent in the full theory where the vertex can be iterated . ® A Gasser- Leutwyler[7] program for quenched ChPT at 1-loop is possible: there may be interesting numerical relations involving only computable (on the lattice) quan-
tities. Such relations could give quantitative insight into the effects of quenching . e Extension of these ideas to weak matrix elements seems to be straightforward.
We thank Steve Sharpe for many useful discussions. This work was supported in part by the US Department of Energy. References [1]
Physics Goals of the QCD Teraflop Project, S . Aoká et at., Intl . J. Mod . Phys . C , to be published . [2] See, e .g ., J . Gasser & H . Leutwyler, Phys . Lett. 184B (1987) 83; H . Neuberger, Nucl . Phys. B300 (1988)
180 ; P . Hasenfratz & H . Leutwyler, Nucl . Phys . B343 (1990) 241 .
[3] A . Morel, J . Physique 48 (1987) 111 . [4] S . Sharpe, Phys . Rev . D41 (1990) 3233 .
[5] G . Kilcup et at ., Phys . Rev. Lett. 64 (1990) 25. [6] S . Sharpe, Nucl . Phys . B17 (Pros . Sup .) ('90) 146. [7] J .Gasser & H .Leutwyler, Nucl . Phys. B250 ('85) 465 . [8]
Except for the A2 term, a terms are actually higher order in a combirt:û expansion in 1/N, and M .