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Second order lattice fermions
Volume 126B, number 3,4
PHYSICS LETTERS
30 June 1983
SECOND ORDER LATTICE FERMIONS Anthony C. LONGH1TANO and Benjamin SVETITSKY
Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA Received 13 April 1983
In an attempt to circumvent the species doubling problem of lattice fermions, we consider a second-order lagrangian formalism. We conclude that the lattice regulator breaks the underlying chiral symmetry of the massless theory. Adjusting the coefficient of the induced Pauli coupling does not restore the symmetry in the continuum limit.
Second-order formulations of the Dirac theory [1 ] have been advanced as cures for some of the diseases of lattice fermion theories [2]. As is well known, the common methods used for circumventing the species doubling problem [3] either explicitly break continuous chiral symmetries [4] or involve non-local lagrangians [5,6]. In this note we report that secondorder theories as well fall short of their promise. One begins [2] with Nf flavors of ordinary continuum Dirac fermions coupled to a gauge field. The euclidean vacuum functional is defined as
z=f
z =f
Expressing (2) as a fermion integral yields
z= f q)Au*×~ exp(-Sg[A]
(1)
where Sg [A ] is the usual gauge-field action, including any gauge-fixing and F a d d e e v - P o p o v terms. (Our Dirac matrices are hermitian, and the covariant derivative D u - O, + iA u, where A u is the Lie-algebra-valued gauge field.) Integrating over the fermionic variables yields Z = f @ A ~z exp {- Sg [A ] } { Det (I~ + m)] Nf . Since det(l~ + m ) = det T5(l~ + m ) 75 = det ( - I ~ + m ) = [det(-I~I~ + m2)] 1/2 , we can write 0 031-9163/83/0000
.
(2)
c~ Au"7) ff c/) ~ exp ( - Sg [A ]
- fd4x~(l~+m) t~) ,
A. exp {-Sg [A ] } [Det (-I~I~ + m2)] Nf/2
0000/$ 03.00 © 1983 North-Holland
-
f
\
d4x ~-(-I~l~ + rn2)x)
,
(3)
where now there are Nf/2 flavors of the X fermions. The stratagem of a second-order lattice formalism is to impose the lattice cutoff not in (1) as usual, but rather in (2) or (3). Since the quantity -l~I~ + m 2 = - D u D u + m 2 + t FU~, Our
(4)
is just the gauged Klein--Gordon operator augmented by a Pauli coupling (recall F u r = i -1 [Du, Dr]), the lattice theory, while local, has no species doubling. This property, along with the positivity of (4) in the continuum theory, gives this formulation the potential for being uniquely suited to the study of dynamical chiral symmetry breakdown and of chiral gauge theories via Monte Carlo techniques [7]. However, the realization of chiral symmetry on the fields presents a puzzle. It is not at all obvious how the chiral symmetry of the m = 0 Dirac theory acts in the second-order ff)rmalism. In fact, the mass term in (3) does not break any symmetry which is locally realized on X and ~-. Of course, we know that in the continuum, the sec259
PttYSICS LETTERS
Volume 126B, number 3,4
+
r
(~
-
+
:
30 June 1983
~'q~
O"
-
+
O"
Fig. I. One-loop contributions to the fermlon self-energy in the second-order theory are illustrated. Vertices labelled tr come from the Pauli coupling. The last two diagrams give zero in the continuum theory and in the lattice models we consider. ond order theory "remembers" the chiral symmetry of the Dirac theory so that, although not manifest, the symmetry remains intact. On the other hand, the lattice version of the second order action cannot be derived from a first order lattice action, so that we must determine by hand whether the lattice regulator respects the chiral symmetry of the continuum theory. Our criterion for this determination rests upon the perturbative calculation of the fermion self-energy ~ ( p ) in the theory (3) with m -- 0. In the continuum theory (e.g., dimensionally regularized), the mass shift 6m 2 = Y.(0) is zero because of a cancellation in the diagrams of fig. 1 (and similarly in higher orders) which is ultimately due to the underlying chiral symmetry. Note: (1) 6m 2 is relevant even if one uses (3) to calculate only Green functions without fermion legs because it locates the beginnings of cuts in those Green functions. Chiral symmetry implies that the m = 0 theory will have cuts starting at the origin. (2) l~(p) is (naively) quadratically divergent in the ultraviolet, and its leading divergence, which gives 6rn 2, is infrared finite when p = m = 0. The next to leading part is logarithmically divergent in both the ultraviolet and the infrared and, when regulated with a lattice
spacing a and p ~ 0, is proportional to p2 log p 2 a 2 . It does not contribute to 5m 2, but only to wave-function renormalization (when subtracted off-shell). For definiteness we consider the non-compact abelian gauge theory [6]. The gauge action (in a covariant gauge) and the gauged K l e i n - G o r d o n piece of the fermion action are put on a lattice in the customary manner, giving in perturbation theory the Feynman rules displayed in fig. 2. [We have defined S u ( k ) = (2/a) sin ½k~,a, C u ( k ) = 2 cos ~ k u a , and D ( k ) = Y-,, S u ( k ) 2 . ] In putting the Pauli term on a lattice there is, of course, great latitude. Two prescriptions are: Prescription (1):
where the sum is around the contour of fig. 3a. This gives a single-photon vertex ate 1. Sv(2k) q,(k) o, with k the incoming photon momentum. Prescription (2):
I.tV
X ( U + p - I t v ~ - U v U +"~ - ~i-1 , n-Ilvn-p n n+1o-ps ,
•
~
v
p
:>'
pl
p
pl
~p.v" (I- ~) Sp.(k)Sv(k) / D(k) I:) (k) (a)
I/O{k)
(b)
-e Sp(2 p+k)
(c)
-eeCu(2P + k,+ k~,)
(d)
Fig. 2. Feynman rules arising from the gauged Klein-Gordon part of the second-order lattice theory. 260
Volume 126B, number 3,4
PttYSICS LETTERS ,J'
i
A
A
n-F+v
i
i !
(a)
n
(b)
Fig. 3. (a) Contour for defining F/av in the Pauli term. Both ~, and ~ axe at n. (b) Alternative prescription: ~ is at n, ~ at n - li+ ~,. One takes the difference of the two paths from to ~.
which is illustrated in fig. 3b. This arises from the re1 + -iplacement in (4) of Iklk by ~{1~ , I~- }, where Du is + the forward difference Dumbn = Unu ~bn+u - ~n and Du is the analogous backward difference. This leads to a single-photon vertex - i e c o s [ ( p v + ~t kv - Pu - 2ku)a]Sv(k) t °u=' ' with momenta as shown in fig. 2c. For greater generality, we also multiply the Pauli term by a coefficient K, as allowed by gauge invariance [2]. K -- 1 is the value which naturally arises in the continuum theory as a consequence of minimal coupling. The result of evaluating the three non-zero graphs of fig. I a t p = 0 is that 6m 2 ~ 0 i f K = 1, when either of the above prescriptions is employed for the Pauli coupling. Specifically, the first of these leads to the following result for arbitrary K:
30 June 1983
just K accordingly, order by order in perturbation theory. This has been suggested [2] as a means of restoring chiral symmetry in the continuum limit. However, this scheme is inconsistent with the usual renorrealization of the theory, and does not lead to a finite limit. The usual renormalization procedure is to demand that the dressed vertices of fig. 2 and the dressed Pauli vertex all stay finite as a -+ 0. This defines the one-loop /3 functions for the running bare couplings a de2/da = - (e4/127r 2)(3K 2 - 1) N f ,
(6)
a dK2/da = (e2/47r 2) K 2 (1 - K 2 ) .
(7)
Here e and K are the couplings at the cutoff scale a. The ultraviolet (a ~ 0) flows * l are shown in fig. 4. Imagine now starting with initial values of e and K which satisfy 6m 2 = 0, namely K 2 = 3.6327 + O(e2), according to (5). By differentiating (5), we see that K and e flow off the 5rn 2 = 0 surface, and the "chiral symmetry" condition is not maintained. A bare mass term is then necessary for a finite theory. Alternatively, one may demand 6m 2 = 0 as e runs according to (6). Then (5), rather than (7), determines the running K. The difference between the flows of (5) and those of (7) implies that, in this scheme, finiteness of the dressed Pauli vertex is lost. In the continuum theory, on the other hand, 5m 2 ~xe2(1 _ K 2) A 2, with A an ultraviolet cutoff, and it is clear from (7) that K = I and 5m 2 = 0 are maintained together.
6m 2 =~3 e2a -2 [CI(K2 _ 1 ) + C 2 K 2 ] ,
C1 = j
d x [ e x p ( _ x ) l o ( x ) ] 4 o=0 . 3 0 9 8 7 ,
I The asymptotic freedom of the theory near K = 0 should not be surprising, since the K = 0 theory is scalar electrodynamics augmented with minus signs for fermion loops.
0
C2 = f o
d x [ e x p ( _ 2 X ) i o ( x ) i i ( x ) ] 2 _ ~ o_-0.22457, (5)
where In(X ) is the modified Bessel function of order n. The C 2 term, which survives at K = 1, demonstrates that the lattice regulator breaks the hidden chiral symmetry of the second-order theory. (As a check of these ideas, it is straightforward to show that 5m 2 = 0 when the action is the square of a chirally symmetric firstorder lattice action, such as the naive doubled action or the SLAC action [5]). Of course, one may demand that 5m 2 = 0 and ad-
J
JJ
O e
Fig. 4. Schematic ultraviolet flows in the (e, K) plane, derived from the one-loop renormalization group equations. 261
Volume 126B, number 3,4
PHYSICS LETTERS
We thank L. McLerran for introducing us to the second-order idea. This work benefitted from hard questions asked by M. Peskin and K.G. Wilson. We also thank H. Kawai and J. Sapirstein for assistance. This work was supported by the National Science Foundation under Grant No. NSF PHY77-22336.
References [ 1] R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; L.M. Brown, Phys. Rev. 111 (1958) 957. [2] T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103;
262
[3]
[4]
[5] [6] [7]
30 June 1983
T. Banks, in: Proc. Johns Hopkins Workshop on Current problems in particle theory 4 (Bonn, 1980), Johns Hopkins University Report. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185 (1981) 20; B193 (1981) 173; J.M. Rabin, Nucl. Phys. B201 (1982) 315. K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977); L. Susskind, Phys. Rev. DI6 (1977) 3031. S.D. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev. D14 (1976) 1627. J.M. Rabin, Phys. Rev. D24 (1981) 3218. J. Kuti, Phys. Rev. Left. 49 (1982) 183.
PHYSICS LETTERS
30 June 1983
SECOND ORDER LATTICE FERMIONS Anthony C. LONGH1TANO and Benjamin SVETITSKY
Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA Received 13 April 1983
In an attempt to circumvent the species doubling problem of lattice fermions, we consider a second-order lagrangian formalism. We conclude that the lattice regulator breaks the underlying chiral symmetry of the massless theory. Adjusting the coefficient of the induced Pauli coupling does not restore the symmetry in the continuum limit.
Second-order formulations of the Dirac theory [1 ] have been advanced as cures for some of the diseases of lattice fermion theories [2]. As is well known, the common methods used for circumventing the species doubling problem [3] either explicitly break continuous chiral symmetries [4] or involve non-local lagrangians [5,6]. In this note we report that secondorder theories as well fall short of their promise. One begins [2] with Nf flavors of ordinary continuum Dirac fermions coupled to a gauge field. The euclidean vacuum functional is defined as
z=f
z =f
Expressing (2) as a fermion integral yields
z= f q)Au*×~ exp(-Sg[A]
(1)
where Sg [A ] is the usual gauge-field action, including any gauge-fixing and F a d d e e v - P o p o v terms. (Our Dirac matrices are hermitian, and the covariant derivative D u - O, + iA u, where A u is the Lie-algebra-valued gauge field.) Integrating over the fermionic variables yields Z = f @ A ~z exp {- Sg [A ] } { Det (I~ + m)] Nf . Since det(l~ + m ) = det T5(l~ + m ) 75 = det ( - I ~ + m ) = [det(-I~I~ + m2)] 1/2 , we can write 0 031-9163/83/0000
.
(2)
c~ Au"7) ff c/) ~ exp ( - Sg [A ]
- fd4x~(l~+m) t~) ,
A. exp {-Sg [A ] } [Det (-I~I~ + m2)] Nf/2
0000/$ 03.00 © 1983 North-Holland
-
f
\
d4x ~-(-I~l~ + rn2)x)
,
(3)
where now there are Nf/2 flavors of the X fermions. The stratagem of a second-order lattice formalism is to impose the lattice cutoff not in (1) as usual, but rather in (2) or (3). Since the quantity -l~I~ + m 2 = - D u D u + m 2 + t FU~, Our
(4)
is just the gauged Klein--Gordon operator augmented by a Pauli coupling (recall F u r = i -1 [Du, Dr]), the lattice theory, while local, has no species doubling. This property, along with the positivity of (4) in the continuum theory, gives this formulation the potential for being uniquely suited to the study of dynamical chiral symmetry breakdown and of chiral gauge theories via Monte Carlo techniques [7]. However, the realization of chiral symmetry on the fields presents a puzzle. It is not at all obvious how the chiral symmetry of the m = 0 Dirac theory acts in the second-order ff)rmalism. In fact, the mass term in (3) does not break any symmetry which is locally realized on X and ~-. Of course, we know that in the continuum, the sec259
PttYSICS LETTERS
Volume 126B, number 3,4
+
r
(~
-
+
:
30 June 1983
~'q~
O"
-
+
O"
Fig. I. One-loop contributions to the fermlon self-energy in the second-order theory are illustrated. Vertices labelled tr come from the Pauli coupling. The last two diagrams give zero in the continuum theory and in the lattice models we consider. ond order theory "remembers" the chiral symmetry of the Dirac theory so that, although not manifest, the symmetry remains intact. On the other hand, the lattice version of the second order action cannot be derived from a first order lattice action, so that we must determine by hand whether the lattice regulator respects the chiral symmetry of the continuum theory. Our criterion for this determination rests upon the perturbative calculation of the fermion self-energy ~ ( p ) in the theory (3) with m -- 0. In the continuum theory (e.g., dimensionally regularized), the mass shift 6m 2 = Y.(0) is zero because of a cancellation in the diagrams of fig. 1 (and similarly in higher orders) which is ultimately due to the underlying chiral symmetry. Note: (1) 6m 2 is relevant even if one uses (3) to calculate only Green functions without fermion legs because it locates the beginnings of cuts in those Green functions. Chiral symmetry implies that the m = 0 theory will have cuts starting at the origin. (2) l~(p) is (naively) quadratically divergent in the ultraviolet, and its leading divergence, which gives 6rn 2, is infrared finite when p = m = 0. The next to leading part is logarithmically divergent in both the ultraviolet and the infrared and, when regulated with a lattice
spacing a and p ~ 0, is proportional to p2 log p 2 a 2 . It does not contribute to 5m 2, but only to wave-function renormalization (when subtracted off-shell). For definiteness we consider the non-compact abelian gauge theory [6]. The gauge action (in a covariant gauge) and the gauged K l e i n - G o r d o n piece of the fermion action are put on a lattice in the customary manner, giving in perturbation theory the Feynman rules displayed in fig. 2. [We have defined S u ( k ) = (2/a) sin ½k~,a, C u ( k ) = 2 cos ~ k u a , and D ( k ) = Y-,, S u ( k ) 2 . ] In putting the Pauli term on a lattice there is, of course, great latitude. Two prescriptions are: Prescription (1):
where the sum is around the contour of fig. 3a. This gives a single-photon vertex ate 1. Sv(2k) q,(k) o, with k the incoming photon momentum. Prescription (2):
I.tV
X ( U + p - I t v ~ - U v U +"~ - ~i-1 , n-Ilvn-p n n+1o-ps ,
•
~
v
p
:>'
pl
p
pl
~p.v" (I- ~) Sp.(k)Sv(k) / D(k) I:) (k) (a)
I/O{k)
(b)
-e Sp(2 p+k)
(c)
-eeCu(2P + k,+ k~,)
(d)
Fig. 2. Feynman rules arising from the gauged Klein-Gordon part of the second-order lattice theory. 260
Volume 126B, number 3,4
PttYSICS LETTERS ,J'
i
A
A
n-F+v
i
i !
(a)
n
(b)
Fig. 3. (a) Contour for defining F/av in the Pauli term. Both ~, and ~ axe at n. (b) Alternative prescription: ~ is at n, ~ at n - li+ ~,. One takes the difference of the two paths from to ~.
which is illustrated in fig. 3b. This arises from the re1 + -iplacement in (4) of Iklk by ~{1~ , I~- }, where Du is + the forward difference Dumbn = Unu ~bn+u - ~n and Du is the analogous backward difference. This leads to a single-photon vertex - i e c o s [ ( p v + ~t kv - Pu - 2ku)a]Sv(k) t °u=' ' with momenta as shown in fig. 2c. For greater generality, we also multiply the Pauli term by a coefficient K, as allowed by gauge invariance [2]. K -- 1 is the value which naturally arises in the continuum theory as a consequence of minimal coupling. The result of evaluating the three non-zero graphs of fig. I a t p = 0 is that 6m 2 ~ 0 i f K = 1, when either of the above prescriptions is employed for the Pauli coupling. Specifically, the first of these leads to the following result for arbitrary K:
30 June 1983
just K accordingly, order by order in perturbation theory. This has been suggested [2] as a means of restoring chiral symmetry in the continuum limit. However, this scheme is inconsistent with the usual renorrealization of the theory, and does not lead to a finite limit. The usual renormalization procedure is to demand that the dressed vertices of fig. 2 and the dressed Pauli vertex all stay finite as a -+ 0. This defines the one-loop /3 functions for the running bare couplings a de2/da = - (e4/127r 2)(3K 2 - 1) N f ,
(6)
a dK2/da = (e2/47r 2) K 2 (1 - K 2 ) .
(7)
Here e and K are the couplings at the cutoff scale a. The ultraviolet (a ~ 0) flows * l are shown in fig. 4. Imagine now starting with initial values of e and K which satisfy 6m 2 = 0, namely K 2 = 3.6327 + O(e2), according to (5). By differentiating (5), we see that K and e flow off the 5rn 2 = 0 surface, and the "chiral symmetry" condition is not maintained. A bare mass term is then necessary for a finite theory. Alternatively, one may demand 6m 2 = 0 as e runs according to (6). Then (5), rather than (7), determines the running K. The difference between the flows of (5) and those of (7) implies that, in this scheme, finiteness of the dressed Pauli vertex is lost. In the continuum theory, on the other hand, 5m 2 ~xe2(1 _ K 2) A 2, with A an ultraviolet cutoff, and it is clear from (7) that K = I and 5m 2 = 0 are maintained together.
6m 2 =~3 e2a -2 [CI(K2 _ 1 ) + C 2 K 2 ] ,
C1 = j
d x [ e x p ( _ x ) l o ( x ) ] 4 o=0 . 3 0 9 8 7 ,
I The asymptotic freedom of the theory near K = 0 should not be surprising, since the K = 0 theory is scalar electrodynamics augmented with minus signs for fermion loops.
0
C2 = f o
d x [ e x p ( _ 2 X ) i o ( x ) i i ( x ) ] 2 _ ~ o_-0.22457, (5)
where In(X ) is the modified Bessel function of order n. The C 2 term, which survives at K = 1, demonstrates that the lattice regulator breaks the hidden chiral symmetry of the second-order theory. (As a check of these ideas, it is straightforward to show that 5m 2 = 0 when the action is the square of a chirally symmetric firstorder lattice action, such as the naive doubled action or the SLAC action [5]). Of course, one may demand that 5m 2 = 0 and ad-
J
JJ
O e
Fig. 4. Schematic ultraviolet flows in the (e, K) plane, derived from the one-loop renormalization group equations. 261
Volume 126B, number 3,4
PHYSICS LETTERS
We thank L. McLerran for introducing us to the second-order idea. This work benefitted from hard questions asked by M. Peskin and K.G. Wilson. We also thank H. Kawai and J. Sapirstein for assistance. This work was supported by the National Science Foundation under Grant No. NSF PHY77-22336.
References [ 1] R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; L.M. Brown, Phys. Rev. 111 (1958) 957. [2] T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103;
262
[3]
[4]
[5] [6] [7]
30 June 1983
T. Banks, in: Proc. Johns Hopkins Workshop on Current problems in particle theory 4 (Bonn, 1980), Johns Hopkins University Report. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185 (1981) 20; B193 (1981) 173; J.M. Rabin, Nucl. Phys. B201 (1982) 315. K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977); L. Susskind, Phys. Rev. DI6 (1977) 3031. S.D. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev. D14 (1976) 1627. J.M. Rabin, Phys. Rev. D24 (1981) 3218. J. Kuti, Phys. Rev. Left. 49 (1982) 183.