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Supersymmetric QCD
Volume 158B, number 2
PHYSICS LETTERS
8 August 1985
SUPERSYMMETRIC QCD S. K A L A R A and S. R A B Y Theory Division T-8, MS B285, Los Alamos National Laboratory, Los Alamo& NM 87545, USA
Received 3 May 1985
We demonstrate the existence of a well-defined vacuum state in supersymmetric QCD with finite vacuum expectation values (VEVs) for scalars. Our results differ from those of several authors who find that a supersymmetric vacuum state exists only at infinite VEVs for the scalar fields. Our analysis is an application of previous results obtained using instantons in conjunction with the anomalous supersymmetry transformation laws found by Konishi.
1. Introduction One of the challenges of particle physics is the resolution of the gauge hierarchy problem [ 1 ]. The incorporation of supersymmetry into grand unification schemes has provided us with a possible direction to confront this problem. It has been argued by many authors that supersymmetry could not only provide the resolution of the gauge hierarchy in a technical sense, but also explain the origin of the gauge hierarchy if, for example, supersymmetry is broken dynamically [2]. This exciting possibility of dynamical supersymmetry breaking (DSB) has been explored in the context of supersymmetric gauge theories [ 3 - 6 ] . Recently it has been shown that in supersymmetric QCD (SQCD) the nonrenormalization theorem is violated [3], however the question of DSB is far from settled. A clue to the possibility of DSB in a particular theory can be obtained by calculating its Witten index A. Witten [5] has shown that if a given theory is to admit DSB then A must be zero. This topologically invariant index has been evaluated in certain classes of theories. For example, for SU(N) gauge theories without matter fields or with massive matter, Witten finds A = N; implying that DSB does not occur in such theories. Unfortunately for the interesting cases of massless SQCD or gauge theories with chiral matter fields, A is uncalculable .1. Thus the utility of the Witten index is quite restricted.
, I Except for certain very special choices of massless matter representations, where/X is shown to be nonzero [ 7 ]. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Another approach for studying DSB has been via the analysis of effective lagrangians. Massless SQCD has been studied in this way. Taylor, Veneziano and Yankielowicz [6], Davis, Dine and Sieberg [8], and others [9] find that for SQCD with N f < N one has (~o) ~ oo and (XX) = 0 (where ~ and ~p are the scalar components of the quark superfields and X is the gaugino). Peskin [I0], Nilles [11], and Sharatchandra [12], on the other hand, argue that SQCD allows for a finite nonzero ( ~ ) and dynamical supersymmetry breaking. These conflicting results obtained by different authors are principally due to the different choices of D terms in the effective lagrangians. The identification of the correct vacuum structure of SQCD still eludes us. Recently, instanton calculations have been performed to gain insight into the vacuum structure of SQCD. Using somewhat different techniques Rossi and Veneziano [13] and Affleck, Dine and Sieberg [3] reach the same conclusion, i.e. for SQCD with N f < N , (~p) -~ ~o and (XX) = 0, i.e. SQCD has no stable vacuum state. However, we disagree with the above conclusions. Using the anomalous transformation law of Konishi [ 14] and the superpotential obtained by Affleck et al. [3] we show, in the case N = 2 and N f = I, that there exist supersymmetric vacua for any value of (XX) and (~p) satisfying (XX~) = a A 5 . In the general case, we find supersymmetric vacua for any value of (XX) and (~0) satisfying ((~t) N-Nf X det ~f~0f,) = a ' A 3 N - N f .
131
PHYSICS LETTERS
Volume 158B, number 2
2. The analysis To demonstrate these assertions we start by briefly recapitulating the salient features of the anomalous transformation law [ 14] and the results obtained by the instanton calculations in massless SQCD. For simplicity we consider an SU(2) gauge theory with one flavor, i.e. N = 2, Nf = 1. The quark superfie!ds are Q and Q which transform as a 2 and 2* respectively. They are given by Q=~o+X/~0~+00F,
Q=~+X/20t~+00F.(2.1)
The corresponding lagrangian is ww +
+ fd4OQ*e2gVQ+ f d 4 0 ~ e - 2 g v ~
*,
where W~ = W~aT a and T a [the generator of SU(2) in the fundamental representation[ is normalized by Tr TaT b = ~8 ab and WW - Tr [W~W~]. The global symmetries of the theory are U(.1)y @ U(1)R. U(1)V is defined by Q ~ el~Q, Q ~ e - m Q and U(1)R is defined by. W~ ~ eia.Wa(x, e-i~0), Q ~ e-iaQ(x, e-ia0), -~ e-'~Q(x, e-'~0). 2.1. The anomalous transformation law [14]. The anomalous transformation law, which is a consequence of supersymmetry and gauge invariance, is given by
{0, ~ ) / v ~
= - ~ aw/a~ + (g2/81r2)xx,
(2.2)
where (2 is the supersymmetry generator and XX Tr(X~X~). We note that F = -(aw/a~0)* (using the equations of motion) is the F term of the chiral superfield Q. However, the anomaly means that -(~o aw/a~0) is not the ~2 term of the real superfield Q*e2gVQ. The above anomaly is equivalent to the equation [15] -~D2Q*e2gVQ = - Q 0W/~Q + (g2/81r2)WW,
(2.3)
which means that Q OW/aQ does not (by itself) transform as a chiral superfield. Only the complete term on the RHS of eq. (2.3) transforms correctly. We shall use this important result later. 2.2. Instanton results [3,16]. It has been shown that instantons violate the nonrenormalization theorems and (in this case, N = 2 , N f = 1) induce an effective superpotential Wll : aA 5
f
1/~0,
(2.4)
8 August 1985
where by explicit calculation we find a = g4c/2567r6 and c is a calculated constant resulting from the collective coordinate integrations [17]. The form of the superpotential is uniquely determined by the symmetries of the theory and the result is valid, at least, for large values of the fields. Let us now combine the anomalous transformation law (2.2) with the one instanton effective superpotential (2.4). We immediately see that supersymmetry is unbroken iff
{0, ff }/v
= aA 51~¢ + (g2/STr2)XX -- 0.
(2.5)
Note that without the term in eq. (2.5), coming from the one-instanton effective superpotential (2.4), we would conclude that the only supersymmetric minimum occurs for (XX) = 0. (This was the conclusion of Rossi and Veneziano [13].) However the one-instanton correction cannot be neglected, with it we find
(kTt~tp) = -(8rr2c~/g2)A5 = constant.
(2.6)
Thus (XX) :# 0 in general. Moreover there exist a continuous set of vacua satisfying (2.6). Rossi and Veneziano [13] and Novikov et al. [18] have calculated the Green function (Xk~), in the one-instanton approximation, and have shown it to be non-vanishing. We find by explicit calculation
(XX~tp) = (-g2c/32rr4)A5 .
(2.7)
It is easy to verify that (2.6) and (2.7) are indeed equivalent. The contention that massless SQCD allows for a finite, nonzero value for @~) can be reconciled with the explicit minimization of the scalar potential VII. Affleck et al. [3] argue that VII is given by
VII= lalVli/a~ol 2 + laWn/a~l 2
(2.8)
whose minimum lies at (~0) ~ oo. The key point to note here is that the F component of the superpotential WII contains an anomaly and hence the scalar potential Vll is not just the canonical terms obtained by Affleck et al., but must be supplemented by an anomalous piece. To identify the anomalous piece let us recall that [eq. (2.4)]
f d 2 0 WII =-~D2(aA5/QQ)Io:I__ 0 D - { [o~a5/(~Q)21 (O2 ~Q)10=~=o
D--~ [cIAS/(QQ) 2 ] Io=l=o (D2QQ)Io:I--O • 132
Volume 158B, number 2
PHYSICS LETTERS
However, as we have noted before [see eq. (2.3)], 1/~0 is not the scalar component of a chiral superfield. The correct scalar component Contains the anomaly (g2/87r2)XX, i.e.
f d20 WII = ( 1 / a A 5) [(aA5/~Q) + (g2/8~r 2)WW] 2 Io=0=o X (-~D2)((~Q)I0=0=0 .
(2.9)
The corresponding scalar potential Vll is given by V I I = (1/aA5)2 I[(otAS/~0 ) + (g2/8rr 2)XX] 212(k012 + I~ 12).
(2.10)
The quantity (g2/8rt2)XX is interpreted as an effective composite scalar field. Now the potential VII has minima for all ~o, ~ and XX satisfying (XX~o) = _ ( 8 r r 2 / g 2 ) a A 5 .2 The extension to other N for Nf = N - 1 is straightforward. WII , in these cases, is given by W l l = a a 2N+ 1
fd20 1/det((~fof,).
(2.11)
Plugging this effective superpotential into the anomalous transformation law requires the following Green function to be nonzero (XX(det tp f~0f,)) = bA 2N+ 1 ,
(2.12)
which is indeed the case, as can be verified by explicit calculation [ 16]. For 1 < ~ N f < N - 1, the form of the superpotential can be extracted from symmetry arguments. These theories also have perturbative flat directions, however along these directions the gauge symmetry remains partially unbroken, i.e. WW remains as a dynamical degree of freedom at low energies. Incorporating WW into the effective superpotential, l¥1i is expected to have the following form WII = T A 3 N - N f f d20 1/[(WW) N - N f - l(det ~fQf,)] .
.2 Note that the scalar potential VII [eq. (2.10)] can be obtained in the canonical way from an effective superpotential Wef t . If we def'me T = ~¢/A 2 and S = (g2/81r2)hh/Aa, then Wef t = A a [-a/T+ 2S In(ST/a) +S 2T/s]. Wef t is reminiscent of effective actions studied previously in the literature [6,8-12].
8 August 1985
The nonzero Green's function would then be given by (()t~) N - N f det ~ftpf,) = ~'A 3 N - N f which is exactly what is required from the anomalous transformation laws. The above discussion is not meant to be a derivation but only a plausibility argument.
3. Conclusion We have shown that SQCD with 1 ~
References [1] E. Gildener, Phys, Rev. D14 (1976) 1667; S. Weinberg, Phys. Lett. 82B (1979) 387. [2] S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353; E. Witten, Nucl. Phys. B 188 (1981) 513; M. Dine, W. Fischler and M. Srednieki, Nucl. Phys. B189 (1981) 575. [3] I. Affleek, M. Dine and N. Sieberg, Phys. Rev. Lett. 51 (1983) 1026; Nucl. Phys. B241 (1984) 493. [4] I. Affleck, M. Dine and N. Sieberg, Phys. Rev. Lett. 52 (1984) 1677; Phys. Lett. 140B (1984) 59. 133
Volume 158B, number 2
PHYSICS LETTERS
[5] E. Witten, Nucl. Phys. B202 (1982) 253. [6] T.R. Taylor, G. Veneziano and S. Yankielowicz, NucL Phys. B218 (1983) 493. [7] E. Cohen and C. Gomez, Nucl. Phys. B223 (1983) 183. [8] A.C. Davis, M. Dine and N. Sieberg, Phys. Lett. 125B (1983) 487. [9] G. Veneziano and S. Yankielowicz, Phys. Lett. I10B (1982) 39. [10] M.E. Peskin, in: Problems in unification and supergravity, eds. G. Farrar and F. Henyey (AIP, New York, 1984). [11] H.P. Nilles, Phys. LetL 129B (1983) 103.
134
8 August 1985
[12] M.S. Sharatchandra, Phys~ Lett. 139B (1984) 301. [13] G.C. Rossi and G. Veneziano, Phys. Lett. 138B (1984) 195. [14] K. Konishi, Phys. Lett. 135B (1984) 439. [15] T. Clark, O. Piguet and K. Sibold, Nucl. Phys. 159B (1979) 1. [ 16] D. Amati, G.C. Rossi and G. Veneziano, CERN preprint 3907/84. [17] G. 't Hooft, Phys. Rev. D14 (1976) 3432. [ 18] V.A. Novikov, M.A. Shifman, A.I. VaJnshtein and V.I. Zakharov, Nucl. Phys. B229 (1983) 381,407. [19] Y. Meurice and G. Veneziano, Phys. Lett. 141B (1984) 69.
PHYSICS LETTERS
8 August 1985
SUPERSYMMETRIC QCD S. K A L A R A and S. R A B Y Theory Division T-8, MS B285, Los Alamos National Laboratory, Los Alamo& NM 87545, USA
Received 3 May 1985
We demonstrate the existence of a well-defined vacuum state in supersymmetric QCD with finite vacuum expectation values (VEVs) for scalars. Our results differ from those of several authors who find that a supersymmetric vacuum state exists only at infinite VEVs for the scalar fields. Our analysis is an application of previous results obtained using instantons in conjunction with the anomalous supersymmetry transformation laws found by Konishi.
1. Introduction One of the challenges of particle physics is the resolution of the gauge hierarchy problem [ 1 ]. The incorporation of supersymmetry into grand unification schemes has provided us with a possible direction to confront this problem. It has been argued by many authors that supersymmetry could not only provide the resolution of the gauge hierarchy in a technical sense, but also explain the origin of the gauge hierarchy if, for example, supersymmetry is broken dynamically [2]. This exciting possibility of dynamical supersymmetry breaking (DSB) has been explored in the context of supersymmetric gauge theories [ 3 - 6 ] . Recently it has been shown that in supersymmetric QCD (SQCD) the nonrenormalization theorem is violated [3], however the question of DSB is far from settled. A clue to the possibility of DSB in a particular theory can be obtained by calculating its Witten index A. Witten [5] has shown that if a given theory is to admit DSB then A must be zero. This topologically invariant index has been evaluated in certain classes of theories. For example, for SU(N) gauge theories without matter fields or with massive matter, Witten finds A = N; implying that DSB does not occur in such theories. Unfortunately for the interesting cases of massless SQCD or gauge theories with chiral matter fields, A is uncalculable .1. Thus the utility of the Witten index is quite restricted.
, I Except for certain very special choices of massless matter representations, where/X is shown to be nonzero [ 7 ]. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Another approach for studying DSB has been via the analysis of effective lagrangians. Massless SQCD has been studied in this way. Taylor, Veneziano and Yankielowicz [6], Davis, Dine and Sieberg [8], and others [9] find that for SQCD with N f < N one has (~o) ~ oo and (XX) = 0 (where ~ and ~p are the scalar components of the quark superfields and X is the gaugino). Peskin [I0], Nilles [11], and Sharatchandra [12], on the other hand, argue that SQCD allows for a finite nonzero ( ~ ) and dynamical supersymmetry breaking. These conflicting results obtained by different authors are principally due to the different choices of D terms in the effective lagrangians. The identification of the correct vacuum structure of SQCD still eludes us. Recently, instanton calculations have been performed to gain insight into the vacuum structure of SQCD. Using somewhat different techniques Rossi and Veneziano [13] and Affleck, Dine and Sieberg [3] reach the same conclusion, i.e. for SQCD with N f < N , (~p) -~ ~o and (XX) = 0, i.e. SQCD has no stable vacuum state. However, we disagree with the above conclusions. Using the anomalous transformation law of Konishi [ 14] and the superpotential obtained by Affleck et al. [3] we show, in the case N = 2 and N f = I, that there exist supersymmetric vacua for any value of (XX) and (~p) satisfying (XX~) = a A 5 . In the general case, we find supersymmetric vacua for any value of (XX) and (~0) satisfying ((~t) N-Nf X det ~f~0f,) = a ' A 3 N - N f .
131
PHYSICS LETTERS
Volume 158B, number 2
2. The analysis To demonstrate these assertions we start by briefly recapitulating the salient features of the anomalous transformation law [ 14] and the results obtained by the instanton calculations in massless SQCD. For simplicity we consider an SU(2) gauge theory with one flavor, i.e. N = 2, Nf = 1. The quark superfie!ds are Q and Q which transform as a 2 and 2* respectively. They are given by Q=~o+X/~0~+00F,
Q=~+X/20t~+00F.(2.1)
The corresponding lagrangian is ww +
+ fd4OQ*e2gVQ+ f d 4 0 ~ e - 2 g v ~
*,
where W~ = W~aT a and T a [the generator of SU(2) in the fundamental representation[ is normalized by Tr TaT b = ~8 ab and WW - Tr [W~W~]. The global symmetries of the theory are U(.1)y @ U(1)R. U(1)V is defined by Q ~ el~Q, Q ~ e - m Q and U(1)R is defined by. W~ ~ eia.Wa(x, e-i~0), Q ~ e-iaQ(x, e-ia0), -~ e-'~Q(x, e-'~0). 2.1. The anomalous transformation law [14]. The anomalous transformation law, which is a consequence of supersymmetry and gauge invariance, is given by
{0, ~ ) / v ~
= - ~ aw/a~ + (g2/81r2)xx,
(2.2)
where (2 is the supersymmetry generator and XX Tr(X~X~). We note that F = -(aw/a~0)* (using the equations of motion) is the F term of the chiral superfield Q. However, the anomaly means that -(~o aw/a~0) is not the ~2 term of the real superfield Q*e2gVQ. The above anomaly is equivalent to the equation [15] -~D2Q*e2gVQ = - Q 0W/~Q + (g2/81r2)WW,
(2.3)
which means that Q OW/aQ does not (by itself) transform as a chiral superfield. Only the complete term on the RHS of eq. (2.3) transforms correctly. We shall use this important result later. 2.2. Instanton results [3,16]. It has been shown that instantons violate the nonrenormalization theorems and (in this case, N = 2 , N f = 1) induce an effective superpotential Wll : aA 5
f
1/~0,
(2.4)
8 August 1985
where by explicit calculation we find a = g4c/2567r6 and c is a calculated constant resulting from the collective coordinate integrations [17]. The form of the superpotential is uniquely determined by the symmetries of the theory and the result is valid, at least, for large values of the fields. Let us now combine the anomalous transformation law (2.2) with the one instanton effective superpotential (2.4). We immediately see that supersymmetry is unbroken iff
{0, ff }/v
= aA 51~¢ + (g2/STr2)XX -- 0.
(2.5)
Note that without the term in eq. (2.5), coming from the one-instanton effective superpotential (2.4), we would conclude that the only supersymmetric minimum occurs for (XX) = 0. (This was the conclusion of Rossi and Veneziano [13].) However the one-instanton correction cannot be neglected, with it we find
(kTt~tp) = -(8rr2c~/g2)A5 = constant.
(2.6)
Thus (XX) :# 0 in general. Moreover there exist a continuous set of vacua satisfying (2.6). Rossi and Veneziano [13] and Novikov et al. [18] have calculated the Green function (Xk~), in the one-instanton approximation, and have shown it to be non-vanishing. We find by explicit calculation
(XX~tp) = (-g2c/32rr4)A5 .
(2.7)
It is easy to verify that (2.6) and (2.7) are indeed equivalent. The contention that massless SQCD allows for a finite, nonzero value for @~) can be reconciled with the explicit minimization of the scalar potential VII. Affleck et al. [3] argue that VII is given by
VII= lalVli/a~ol 2 + laWn/a~l 2
(2.8)
whose minimum lies at (~0) ~ oo. The key point to note here is that the F component of the superpotential WII contains an anomaly and hence the scalar potential Vll is not just the canonical terms obtained by Affleck et al., but must be supplemented by an anomalous piece. To identify the anomalous piece let us recall that [eq. (2.4)]
f d 2 0 WII =-~D2(aA5/QQ)Io:I__ 0 D - { [o~a5/(~Q)21 (O2 ~Q)10=~=o
D--~ [cIAS/(QQ) 2 ] Io=l=o (D2QQ)Io:I--O • 132
Volume 158B, number 2
PHYSICS LETTERS
However, as we have noted before [see eq. (2.3)], 1/~0 is not the scalar component of a chiral superfield. The correct scalar component Contains the anomaly (g2/87r2)XX, i.e.
f d20 WII = ( 1 / a A 5) [(aA5/~Q) + (g2/8~r 2)WW] 2 Io=0=o X (-~D2)((~Q)I0=0=0 .
(2.9)
The corresponding scalar potential Vll is given by V I I = (1/aA5)2 I[(otAS/~0 ) + (g2/8rr 2)XX] 212(k012 + I~ 12).
(2.10)
The quantity (g2/8rt2)XX is interpreted as an effective composite scalar field. Now the potential VII has minima for all ~o, ~ and XX satisfying (XX~o) = _ ( 8 r r 2 / g 2 ) a A 5 .2 The extension to other N for Nf = N - 1 is straightforward. WII , in these cases, is given by W l l = a a 2N+ 1
fd20 1/det((~fof,).
(2.11)
Plugging this effective superpotential into the anomalous transformation law requires the following Green function to be nonzero (XX(det tp f~0f,)) = bA 2N+ 1 ,
(2.12)
which is indeed the case, as can be verified by explicit calculation [ 16]. For 1 < ~ N f < N - 1, the form of the superpotential can be extracted from symmetry arguments. These theories also have perturbative flat directions, however along these directions the gauge symmetry remains partially unbroken, i.e. WW remains as a dynamical degree of freedom at low energies. Incorporating WW into the effective superpotential, l¥1i is expected to have the following form WII = T A 3 N - N f f d20 1/[(WW) N - N f - l(det ~fQf,)] .
.2 Note that the scalar potential VII [eq. (2.10)] can be obtained in the canonical way from an effective superpotential Wef t . If we def'me T = ~¢/A 2 and S = (g2/81r2)hh/Aa, then Wef t = A a [-a/T+ 2S In(ST/a) +S 2T/s]. Wef t is reminiscent of effective actions studied previously in the literature [6,8-12].
8 August 1985
The nonzero Green's function would then be given by (()t~) N - N f det ~ftpf,) = ~'A 3 N - N f which is exactly what is required from the anomalous transformation laws. The above discussion is not meant to be a derivation but only a plausibility argument.
3. Conclusion We have shown that SQCD with 1 ~
References [1] E. Gildener, Phys, Rev. D14 (1976) 1667; S. Weinberg, Phys. Lett. 82B (1979) 387. [2] S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353; E. Witten, Nucl. Phys. B 188 (1981) 513; M. Dine, W. Fischler and M. Srednieki, Nucl. Phys. B189 (1981) 575. [3] I. Affleek, M. Dine and N. Sieberg, Phys. Rev. Lett. 51 (1983) 1026; Nucl. Phys. B241 (1984) 493. [4] I. Affleck, M. Dine and N. Sieberg, Phys. Rev. Lett. 52 (1984) 1677; Phys. Lett. 140B (1984) 59. 133
Volume 158B, number 2
PHYSICS LETTERS
[5] E. Witten, Nucl. Phys. B202 (1982) 253. [6] T.R. Taylor, G. Veneziano and S. Yankielowicz, NucL Phys. B218 (1983) 493. [7] E. Cohen and C. Gomez, Nucl. Phys. B223 (1983) 183. [8] A.C. Davis, M. Dine and N. Sieberg, Phys. Lett. 125B (1983) 487. [9] G. Veneziano and S. Yankielowicz, Phys. Lett. I10B (1982) 39. [10] M.E. Peskin, in: Problems in unification and supergravity, eds. G. Farrar and F. Henyey (AIP, New York, 1984). [11] H.P. Nilles, Phys. LetL 129B (1983) 103.
134
8 August 1985
[12] M.S. Sharatchandra, Phys~ Lett. 139B (1984) 301. [13] G.C. Rossi and G. Veneziano, Phys. Lett. 138B (1984) 195. [14] K. Konishi, Phys. Lett. 135B (1984) 439. [15] T. Clark, O. Piguet and K. Sibold, Nucl. Phys. 159B (1979) 1. [ 16] D. Amati, G.C. Rossi and G. Veneziano, CERN preprint 3907/84. [17] G. 't Hooft, Phys. Rev. D14 (1976) 3432. [ 18] V.A. Novikov, M.A. Shifman, A.I. VaJnshtein and V.I. Zakharov, Nucl. Phys. B229 (1983) 381,407. [19] Y. Meurice and G. Veneziano, Phys. Lett. 141B (1984) 69.