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Supersymmetric lattice gauge theory
Volume 122B, number 1
PHYSICS LETTERS
24 February 1983
SUPERSYMMETRIC LATTICE GAUGE THEORY Ikuo ICHINOSE l
Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo, Japan 125 Received 8 September 1982
A supersymmetric lattice gauge model is proposed. By using the resummation method, the strong coupling behaviour of this model is investigated, and it is shown that fermion-antifermion and scalar-scalar condensates are formed. However, cancellation between these condensates occurs, and supersymmetry is not broken dynamically.
Recently, supersymmetry [1 ] has been investigated renewedly from the standpoint of unified theories. If supersymmetry is a real symmetry of nature, it must be broken at least below the present experimental energy scale. It is possible to break supersymmetry explicitly in some way without losing the "naturalness" o f the theories (soft breaking of supersymmetry), but there remains another problem to explain the origin of this explicit breaking. On the other hand, supersymmetry would be broken spontaneously, but it is known that supersymmetry will not be broken in any finite order of perturbation expansion unless it is broken in the tree level. Dynamical breaking of supersymmetry by nonperturbative effect is the most attractive one and it might explain the hierarchy problem, and there has been given some consideration on the possibility of it [2,3] and realistic models have been proposed by assuming the dynamical breaking of supersymmetry by the non-abelian interaction [4]. To formulate strongly interacting field theory on a lattice is very useful, and especially recent result of computer experiment strongly indicates that in QCD quarks are confined and dynamical breaking of chiral symmetry actually occurs. So it is natural to desire to formulate supersymmetric theory on a lattice and to investigate whether dynamical symmetry breaking occurs, but unfortunately it is a difficult problem though there have been several attempts on it [5,6]. Generally, it is difficult t~ construct lattice theories which have the exactly same boson-fermion symmetry exists in the corresponding continuum theory, mainly because on a lattice the Leibniz rule does not hold. Instead, we have attempted to find lattice supersymmetry which is characteristic in lattice theory and gives the analogous constraint on the form of the action in the naive continuum limit. In this note, we propose one of these models which has such lattice supersymmetry, and investigate it in the strong coupling limit. The model consists of the SU(N) (or U(N)) gauge field Uu(x), Dirac fields ~ in the fundamental representation and complex scalar fields q~_+ as its supersymmetric partner. There does not appear a Majorana field which usually appears as the suppersymmetric partner of the gauge field. It seems significant to investigate what condensates form and whether supersymmetry is broken by these condensates in this model. We take the following naive lattice action for the Dirac field ~fl (flavour indices i = 1 ..... Nf, color indices a = 1 .... N),
(1) where we take the lattice spacing a = 1. By the transformation, 1 Research fellow of Japan Society for the Promotion of Science. 68
0 031-9163/83/0000-0000/$ 03.00 © ! 983 North-Holland
24 February 1983
PHYSICS LETTERS
Volume 122B, number 1
(2)
~(x) rf(x),
(x) = T(x) ×(x), where
T(x) =
(vl) x,
... ( ~ 4 ) x 4 ,
the action (1) becomes 1
S D = S x = ~- ~ [X(x)~ u (x)Uu(x)x(x + It) -
~(x
+ tl)~u(x)U?v(x) X(x)] + M ~
X(x) X(x)
= ~ X ( X ) 1 9 X(x) + M ~ ' x ( x ) x ( x ) ,
(3)
where ~/u(x) = (-1)xW -1 =1xv , and
bx(x) =-~1 ~
[~u(x)Uu(x)x(x + #) - *2u(x - v)U~(x - 12)X(X -/2)1 •
(4)
The action (3) is equivalent to the n -- 4Nf Susskind lattice fermion action. As the supersymmetfic partner of the Dirac field, two complex scalar fields $i(i = -+) are introduced, and their action is
S , = ~ e)ti DlS(~i + m 2 ~ ¢~ e)i
= !
4 v
t* *,
+
- g
-
- ,.,)j t u , . ( x ) , , ( x
(x + , , ) v X (x) - * *, (x -
+
u.(x
+
-
-
-
-
"* I.t :# O
X [Uv(x ) $i(x + v) - Utv(x - v)$i(x - v)] + m 2 ~
$~$i,
(5)
where
0*D(x)=~1 ~). [ n . ( x ) O t
(x + u)Ut~(x) - n . ( x - ul~* (x - u l V . ( x
-
.11
The first term of the action (5) is the naive lattice action for the scalar field which is obtained by replacing au)'(x ) -+ (1/2a) [f(x + U) - f ( x - U)] (where twice the lattice spacing appearing is related to the species doubling of naive lattice fermion), and it can be shown that the oscillating second term becomes the surface term in the classical continuum limit [6]. The total action of the system is given by S = (1/g 2) ~
tr(UUUtU#) + S x + S o ,
(6)
plq.
and when m = M, it is invariant under the lattice supersymmetry (in euclidean space)
69
Volume 122B, number 1
~¢,+(x) = ~± x ~ ( x ) ,
PHYSICS LETTERS
24 February 1983
8¢,~(~) = f ~ ( x ) ~ + , =
=
-
(7)
,
where e is an anticommuting Majorana spinor, and subscript + on the spinors means 75X± = -X±, etc. (of course, by the species doubling of lattice fermion, X+ also contains left-handed particles). Up to now, we considered the theory in 4 euclidean dimension, but eqs. (6) and (7) can be extended to an arbitrary d dimensions• By the strong coupling expansion [7,8] and more precisely by the recent Monte Carlo calculation [9], it has been shown for QCD on a lattice that fermion-antifermion condensate forms, i.e. (~ff)4: 0, and chiral symmetry is spontaneously broken. If also in the present model fermion-antifermion condensate forms, it might also indicate the breakdown of supersymmetry, for from eq. (7), the following identity holds
Q.o,,
~i? X~ = X X + ~ CM
•
. l
- ~ C
. l
where Qa is the supersymmetry charge, and ~b)× + h.c. might become the Nambu-goldstino. Here, we shall investigate the present model in the strong coupling limit by the resummation method which was used in [7] .We first rewrite the action (5) by introducing "auxiliary fields" ~o_+,
where ~bi' = m 1/ 2 ~i' and equivalence between S~ and S~,: is easily verified. Neglecting the pure gauge term in eq. (6) for g2 ~ 0% and treating the nearest-neighbour term in eqs. (3) and (9) as interaction term, the following boots-trap-type equations for ( ¢ [ t ~ ) and (¢it¢i) (not summed) are obtained (see fig. 1) .
'
'
= -
_
m n=0
.
--
.
+d/2)(~0
=N/[m
2m
.
(10a)
~oi)/N]
and
(:~soi) = N/[m + (d/2) (cblt ~al)/N] ,
(10b)
(the "loop diagrams" are neglected in the leading order in 1/N [7,10] ). The solution to eqs. (10a) and (10b) is easily obtained as
( q~t ~}/N = (~o[~oi)/N = - m i d + ( 2/d + m2 /d2) 1/2 ,
(11) ,
_
and it is also proved that no other scalar-scalar condensates are forward, especially (~0i ~b~)- 0. Similarly, for the 14- I
-- o
+
~ ,,
+
....
+
--k
..;-
",o." +
<~tPi> ~II
,,~....
: o + ~o~+ . . . . + , ~ . . . . ~ ,
,~ Fig. I. Resummation equations for (O;.t
70
Volume 122B, number 1
24 February 1983
PHYSICS LETTERS
<~X>
Cx
Fig. 2. Resummation equation for (xx).
Fig. 3. A typical graph which contributes to the ~t x-propagator.
Dirac field the following equation is obtained (see fig. 2), (XX) = - N C / [ M - (d/2) (XX)INC] ,
(12)
where C = 2[ d/2] , and therefore in the present approximation, the feature of the chiral symmetry breaking is not affected by the presence of the scalar fields. Eq. (12) has two solutions, and physically correct one is
(XX)/NC = M/d - (2/d + M2 /d2) 1/2 .
(13)
From the above calculations, chiral symmetry is broken also in the present supersymmetric extension of QCDlike theory by formation of the fermion-antifermion condensate. Also the scalar-scalar condensate is formed, and surprisingly it is derived from eqs. (8), (11) and (13) that (01 Q<~, . ~Tx?
/ I O ) : ( x x ) + i ClM ~ - ~ % t i ~ i ) = O . .
(14)
!
Therefore, cancellation between the fermion-antifermion and the scalar-scalar condensates occurs and supersymmerry is not broken, though chiral symmetry is broken. For completeness, we shall calculate the propagator of the composite field ~t X and practically verify that it cannot be the Nambu-goldstino. In the resummation approximation, the ~t × propagator is obtained by calculating the graphs in fig. 3, and in momentum space it is given as
V-1 ~a eikx(x~(x)~l~f
×~(0)) =
5i/6a#G(k),
X
((X X)/2NC) (c)~~~J;)/2N
G(k) = 4 N - -
(15)
1 - (2]~u cos ku)2((XX)/2NC)2((¢;t 4b;)/2N) (tp; ¢i)]2N Expanding eqs. (11) and (13) in powers of 1/d, G(k) behaves, for small ku, as
G(k) = [d((~×)/2NC) (~7 ~i)/2N ] - 1N/[k2 - (2d) 1/2(M + m)] ,
(16)
and therefore it does not have the massless pole ven when the action (6) is completely invariant under the supertransformation (7), i.e. m = M. It is significant to compare our result with the arguments by Witten [3] and Nilles [ 11 ]. Witten has argued that under some condition it is sufficient to investigate the vacuum structure of the theory in the weak coupling to know whether dynamical supersymmetry breaking occurs, and in any simple Lie group gauge theory with matter fields in real representations the breaking of supersymmetry does not occur. Nilles has investigated some model, using the 't Hooft anomaly condition and complementarity between the Higgs and the confinement phase, and showed that fermion-antifermion condensate might be formed, but the condensed field can be considered as the scalar (not auxiliary) component of some composite superfield and supersymmetry is not broken. It is amusing that though our approach is quite different from the above, the result is qualitatively in agreement. 71
Volume 122B, number 1
PHYSICS LETTERS
24 February 1983
More systematic investigation on the present model is necessary to know the vacuum structure and to obtain the effective potential. And finally, a computer experiment is needed to investigate the weak coupling behavior of the model.
References [1 ] For a review, see for example P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 249; A. Salam and J. Strathdee, Fortschr. Phys. 26 (1978) 57. [2] E. Witten, Nucl. Phys. B188 (1981) 513. [3] E. Witten, Lectures Trieste Conf. Conf. (1981). [4] M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189 (1981) 575; S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353. [5] H. Nicolai and P. Dondi, Nuovo Cimento 41A (1977) 1; T. Banks and P. Windey, Nucl. Phys. B198 (1982) 226. [6] V. Rittenberg and S. Yankielowicz, preprint, ref. TH.3263-CERN (1982). [7] J.M. Blairon, R. Brout, P. Englert and J. Greensite, Nucl. Phys. B180 [FS2] (1981) 439. [8] J. Smit, Nucl. Phys. B175 (1980) 307; J. Greensite and J. Primack, Nucl. Phys. B180 [FS2] (1981) 170; H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B190 [FS3] (1981) 504; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100. [9] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [10] S. Coleman, Lectures Intern. School of Subnuclear Physics, Erice (1979). [11] H.P. Nilles, Phys. Lett. l12B (1982) 455.
72
PHYSICS LETTERS
24 February 1983
SUPERSYMMETRIC LATTICE GAUGE THEORY Ikuo ICHINOSE l
Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo, Japan 125 Received 8 September 1982
A supersymmetric lattice gauge model is proposed. By using the resummation method, the strong coupling behaviour of this model is investigated, and it is shown that fermion-antifermion and scalar-scalar condensates are formed. However, cancellation between these condensates occurs, and supersymmetry is not broken dynamically.
Recently, supersymmetry [1 ] has been investigated renewedly from the standpoint of unified theories. If supersymmetry is a real symmetry of nature, it must be broken at least below the present experimental energy scale. It is possible to break supersymmetry explicitly in some way without losing the "naturalness" o f the theories (soft breaking of supersymmetry), but there remains another problem to explain the origin of this explicit breaking. On the other hand, supersymmetry would be broken spontaneously, but it is known that supersymmetry will not be broken in any finite order of perturbation expansion unless it is broken in the tree level. Dynamical breaking of supersymmetry by nonperturbative effect is the most attractive one and it might explain the hierarchy problem, and there has been given some consideration on the possibility of it [2,3] and realistic models have been proposed by assuming the dynamical breaking of supersymmetry by the non-abelian interaction [4]. To formulate strongly interacting field theory on a lattice is very useful, and especially recent result of computer experiment strongly indicates that in QCD quarks are confined and dynamical breaking of chiral symmetry actually occurs. So it is natural to desire to formulate supersymmetric theory on a lattice and to investigate whether dynamical symmetry breaking occurs, but unfortunately it is a difficult problem though there have been several attempts on it [5,6]. Generally, it is difficult t~ construct lattice theories which have the exactly same boson-fermion symmetry exists in the corresponding continuum theory, mainly because on a lattice the Leibniz rule does not hold. Instead, we have attempted to find lattice supersymmetry which is characteristic in lattice theory and gives the analogous constraint on the form of the action in the naive continuum limit. In this note, we propose one of these models which has such lattice supersymmetry, and investigate it in the strong coupling limit. The model consists of the SU(N) (or U(N)) gauge field Uu(x), Dirac fields ~ in the fundamental representation and complex scalar fields q~_+ as its supersymmetric partner. There does not appear a Majorana field which usually appears as the suppersymmetric partner of the gauge field. It seems significant to investigate what condensates form and whether supersymmetry is broken by these condensates in this model. We take the following naive lattice action for the Dirac field ~fl (flavour indices i = 1 ..... Nf, color indices a = 1 .... N),
(1) where we take the lattice spacing a = 1. By the transformation, 1 Research fellow of Japan Society for the Promotion of Science. 68
0 031-9163/83/0000-0000/$ 03.00 © ! 983 North-Holland
24 February 1983
PHYSICS LETTERS
Volume 122B, number 1
(2)
~(x) rf(x),
(x) = T(x) ×(x), where
T(x) =
(vl) x,
... ( ~ 4 ) x 4 ,
the action (1) becomes 1
S D = S x = ~- ~ [X(x)~ u (x)Uu(x)x(x + It) -
~(x
+ tl)~u(x)U?v(x) X(x)] + M ~
X(x) X(x)
= ~ X ( X ) 1 9 X(x) + M ~ ' x ( x ) x ( x ) ,
(3)
where ~/u(x) = (-1)xW -1 =1xv , and
bx(x) =-~1 ~
[~u(x)Uu(x)x(x + #) - *2u(x - v)U~(x - 12)X(X -/2)1 •
(4)
The action (3) is equivalent to the n -- 4Nf Susskind lattice fermion action. As the supersymmetfic partner of the Dirac field, two complex scalar fields $i(i = -+) are introduced, and their action is
S , = ~ e)ti DlS(~i + m 2 ~ ¢~ e)i
= !
4 v
t* *,
+
- g
-
- ,.,)j t u , . ( x ) , , ( x
(x + , , ) v X (x) - * *, (x -
+
u.(x
+
-
-
-
-
"* I.t :# O
X [Uv(x ) $i(x + v) - Utv(x - v)$i(x - v)] + m 2 ~
$~$i,
(5)
where
0*D(x)=~1 ~). [ n . ( x ) O t
(x + u)Ut~(x) - n . ( x - ul~* (x - u l V . ( x
-
.11
The first term of the action (5) is the naive lattice action for the scalar field which is obtained by replacing au)'(x ) -+ (1/2a) [f(x + U) - f ( x - U)] (where twice the lattice spacing appearing is related to the species doubling of naive lattice fermion), and it can be shown that the oscillating second term becomes the surface term in the classical continuum limit [6]. The total action of the system is given by S = (1/g 2) ~
tr(UUUtU#) + S x + S o ,
(6)
plq.
and when m = M, it is invariant under the lattice supersymmetry (in euclidean space)
69
Volume 122B, number 1
~¢,+(x) = ~± x ~ ( x ) ,
PHYSICS LETTERS
24 February 1983
8¢,~(~) = f ~ ( x ) ~ + , =
=
-
(7)
,
where e is an anticommuting Majorana spinor, and subscript + on the spinors means 75X± = -X±, etc. (of course, by the species doubling of lattice fermion, X+ also contains left-handed particles). Up to now, we considered the theory in 4 euclidean dimension, but eqs. (6) and (7) can be extended to an arbitrary d dimensions• By the strong coupling expansion [7,8] and more precisely by the recent Monte Carlo calculation [9], it has been shown for QCD on a lattice that fermion-antifermion condensate forms, i.e. (~ff)4: 0, and chiral symmetry is spontaneously broken. If also in the present model fermion-antifermion condensate forms, it might also indicate the breakdown of supersymmetry, for from eq. (7), the following identity holds
Q.o,,
~i? X~ = X X + ~ CM
•
. l
- ~ C
. l
where Qa is the supersymmetry charge, and ~b)× + h.c. might become the Nambu-goldstino. Here, we shall investigate the present model in the strong coupling limit by the resummation method which was used in [7] .We first rewrite the action (5) by introducing "auxiliary fields" ~o_+,
where ~bi' = m 1/ 2 ~i' and equivalence between S~ and S~,: is easily verified. Neglecting the pure gauge term in eq. (6) for g2 ~ 0% and treating the nearest-neighbour term in eqs. (3) and (9) as interaction term, the following boots-trap-type equations for ( ¢ [ t ~ ) and (¢it¢i) (not summed) are obtained (see fig. 1) .
'
'
= -
_
m n=0
.
--
.
+d/2)(~0
=N/[m
2m
.
(10a)
~oi)/N]
and
(:~soi) = N/[m + (d/2) (cblt ~al)/N] ,
(10b)
(the "loop diagrams" are neglected in the leading order in 1/N [7,10] ). The solution to eqs. (10a) and (10b) is easily obtained as
( q~t ~}/N = (~o[~oi)/N = - m i d + ( 2/d + m2 /d2) 1/2 ,
(11) ,
_
and it is also proved that no other scalar-scalar condensates are forward, especially (~0i ~b~)- 0. Similarly, for the 14- I
-- o
+
~ ,,
+
....
+
--k
..;-
",o." +
<~tPi> ~II
,,~....
: o + ~o~+ . . . . + , ~ . . . . ~ ,
,~ Fig. I. Resummation equations for (O;.t
70
Volume 122B, number 1
24 February 1983
PHYSICS LETTERS
<~X>
Cx
Fig. 2. Resummation equation for (xx).
Fig. 3. A typical graph which contributes to the ~t x-propagator.
Dirac field the following equation is obtained (see fig. 2), (XX) = - N C / [ M - (d/2) (XX)INC] ,
(12)
where C = 2[ d/2] , and therefore in the present approximation, the feature of the chiral symmetry breaking is not affected by the presence of the scalar fields. Eq. (12) has two solutions, and physically correct one is
(XX)/NC = M/d - (2/d + M2 /d2) 1/2 .
(13)
From the above calculations, chiral symmetry is broken also in the present supersymmetric extension of QCDlike theory by formation of the fermion-antifermion condensate. Also the scalar-scalar condensate is formed, and surprisingly it is derived from eqs. (8), (11) and (13) that (01 Q<~, . ~Tx?
/ I O ) : ( x x ) + i ClM ~ - ~ % t i ~ i ) = O . .
(14)
!
Therefore, cancellation between the fermion-antifermion and the scalar-scalar condensates occurs and supersymmerry is not broken, though chiral symmetry is broken. For completeness, we shall calculate the propagator of the composite field ~t X and practically verify that it cannot be the Nambu-goldstino. In the resummation approximation, the ~t × propagator is obtained by calculating the graphs in fig. 3, and in momentum space it is given as
V-1 ~a eikx(x~(x)~l~f
×~(0)) =
5i/6a#G(k),
X
((X X)/2NC) (c)~~~J;)/2N
G(k) = 4 N - -
(15)
1 - (2]~u cos ku)2((XX)/2NC)2((¢;t 4b;)/2N) (tp; ¢i)]2N Expanding eqs. (11) and (13) in powers of 1/d, G(k) behaves, for small ku, as
G(k) = [d((~×)/2NC) (~7 ~i)/2N ] - 1N/[k2 - (2d) 1/2(M + m)] ,
(16)
and therefore it does not have the massless pole ven when the action (6) is completely invariant under the supertransformation (7), i.e. m = M. It is significant to compare our result with the arguments by Witten [3] and Nilles [ 11 ]. Witten has argued that under some condition it is sufficient to investigate the vacuum structure of the theory in the weak coupling to know whether dynamical supersymmetry breaking occurs, and in any simple Lie group gauge theory with matter fields in real representations the breaking of supersymmetry does not occur. Nilles has investigated some model, using the 't Hooft anomaly condition and complementarity between the Higgs and the confinement phase, and showed that fermion-antifermion condensate might be formed, but the condensed field can be considered as the scalar (not auxiliary) component of some composite superfield and supersymmetry is not broken. It is amusing that though our approach is quite different from the above, the result is qualitatively in agreement. 71
Volume 122B, number 1
PHYSICS LETTERS
24 February 1983
More systematic investigation on the present model is necessary to know the vacuum structure and to obtain the effective potential. And finally, a computer experiment is needed to investigate the weak coupling behavior of the model.
References [1 ] For a review, see for example P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 249; A. Salam and J. Strathdee, Fortschr. Phys. 26 (1978) 57. [2] E. Witten, Nucl. Phys. B188 (1981) 513. [3] E. Witten, Lectures Trieste Conf. Conf. (1981). [4] M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189 (1981) 575; S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353. [5] H. Nicolai and P. Dondi, Nuovo Cimento 41A (1977) 1; T. Banks and P. Windey, Nucl. Phys. B198 (1982) 226. [6] V. Rittenberg and S. Yankielowicz, preprint, ref. TH.3263-CERN (1982). [7] J.M. Blairon, R. Brout, P. Englert and J. Greensite, Nucl. Phys. B180 [FS2] (1981) 439. [8] J. Smit, Nucl. Phys. B175 (1980) 307; J. Greensite and J. Primack, Nucl. Phys. B180 [FS2] (1981) 170; H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B190 [FS3] (1981) 504; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100. [9] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [10] S. Coleman, Lectures Intern. School of Subnuclear Physics, Erice (1979). [11] H.P. Nilles, Phys. Lett. l12B (1982) 455.
72