Emergent Supersymmetry in the Marginal Deformations of N=4 SYM

2.1 MB·

by  Qingjun Jin

Prepared for submission to JHEP
Emergent Supersymmetry in the Marginal
Deformations of N = 4 SYM
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Qingjun Jina

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aZhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, 310027, P. R. China
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h E-mail: [email protected]
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e Abstract: We study the one loop renormalization group flow of the marginal deforma-
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[ tions of N = 4 SYM theory using the a-function. We found that in the planar limit

some non-supersymmetric deformations flow to the supersymmetric infrared fixed points
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v described by the Leigh-Strassler theory. This means supersymmetry emerges as a result of
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renormalization group flow.
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Contents
1 Introduction 1
2 The a-function and the conformal fixed point 2
3 The RG flow of gauge and Yukawa couplings 3
4 The Leigh-Strassler Theory 5
5 Marginal Operators and Conformal Deformations 6
6 Emergent Supersymmetry 9
7 Discussions 10
Bibliography 11
1 Introduction
Iftherenormalizationgroup(RG)flowofaquantumfieldtheoryhasastableconformalfixed
point, and the fixed point preserves more symmetries(besides conformal symmetry itself)
than the original theory, then these extra symmetries will emerge as a result of RG flow.
Emergent supersymmetry has been found in different contexts, for example, in topological
superconductors [1], in gauge theories with some Yukawa operators [2], and in a class of
models in 1+1 dimensions [3]. In this paper, we show that supersymmetry emerges in a
four dimensional renormalizable quantum field theory: the marginaldeformations ofN = 4
SYM in the planar limit.
The most general superconformal deformation of N = 4 SYM is the Leigh-Strassler
deformation [4]. As a conformal field theory, the Leigh-Strassler deformation is a fixed
subspace in the space of more general deformations. However, it is technically difficult to
determine whether this fixed subspace is stable, given the huge number of parameters in
the Yukawa and quartic scalar couplings.
The a-function (see e.g. [5, 6]) is a proposed quantity which increase monotonically
with the energy scale, and its gradient flow gives the β-functions. The a-function can be
very helpful in the study of RG flow in theories with large number of parameters, because
thecomplicatedbehaviorofRGflowinahighdimensionalspaceischaracterizedbyasingle
function.
We shall start in Section 2 by briefly reviewing the a-function, and discuss the relation
between a-function and conformal fixed points. With the help of a-functions we study the
flow of gauge and Yukawa couplings in gauge theories in Section 3, and show that when
– 1 –
the number of fermions and scalars satisfy a relation, the Yukawa couplings always flow to
conformal fixed points. In Section 4, we briefly review the Leigh-Strassler theory. And in
Section 5 and Section 6, we study the RG flow around the Leigh-Strassler theory, show that
although the Leigh-Strassler theory seems to be a saddle point of generic deformations, it
becomesstableifonlyasubspaceof(butstillnon-supersymmetric)deformationsareturned
on.
2 The a-function and the conformal fixed point
In two dimensional quantum field theories the Zamolodchikov c-theorem [7, 8] identifies a
C-function which satisfies a RG flow equation of the form
dC 3
= G βIβJ, (2.1)
IJ
dlnµ 2
where gI are the couplings corresponding to the operators O , βI are beta functions of
I
gI. G is proportional to the two point functions (cid:104)O O (cid:105), and it is positive definite,
IJ I J
so (2.1) implies that the C function increase monotonically with the energy scale. The
four dimensional analog of C-theorem is the a-theorem, which conjectures that for four
dimensional quantum field theories we can define the a-function, A˜, which satisfies
dA˜
∂ A˜= T βJ, = G βIβJ, (2.2)
I IJ IJ
dlnµ
where G is the symmetric part of T , and a-theorem holds as long as G is positive
IJ IJ IJ
definite. The first evidence of a-theorem appeared in the 1970s [5], and a lot of progresses
have been made in this direction since then (see [9–15] and references therein).
In this paper we are mainly interested in the behavior of RG flow in the infrared
region, and throughout the paper conformal fixed points always means infrared conformal
fixed points unless otherwise specified. Conformal fixed points correspond to the stationary
points of the a-function. This is clear from (2.2): since G is positive definite, dA˜ = 0
IJ dlnµ
if and only if βI = 0. A conformal fixed point can be isolated, in the sense that it is
locally the only conformal fixed point, or it can be located in a D dimensional space of
c
conformalfixedpoints. Anisolatedconformalfixedpointcanbestableorunstable: astable
conformal fixed point corresponds to a local minimum of the a-function, while a unstable
one corresponds to a local maximum or saddle points. An example of stable and unstable
conformal fixed point is shown in Figure 1: g = 1 is a stable conformal fixed point, while
g = 0 is unstable.
The definition for a stable conformal fixed point gI is, any gI = gI +ηI will flow to gI
0 0 0
when energy goes to zero, as long as ηI is small enough. But this is not true if gI located
0
in a D dimensional space of conformal fixed points. So in this case it is more appropriate
c
to discuss the stableness of the conformal fixed subspace. A conformal fixed subspace, C, is
called stable, if any g flow to a point in C at low energies, as long as g is close enough to C.
Stable fixed points(spaces) can be found by solving the differential equations numeri-
cally. ThesetofgI whichflowtothegivenstablefixedpoints(spaces)hasnon-zeromeasure.
– 2 –
0.4
0.3
(cid:76)g 0.2
(cid:72)A
0.1
0.0
(cid:45)0.5 0.0 0.5 1.0 1.5 2.0
g
Figure 1: a-function with A˜(g) = 1g4− 1g3.
4 3
So start with random gI, when energy goes to zero, the possibility that the flow reaches
the stable fixed point(spaces) is not zero.
A particularly interesting case of stable fixed points is when the a-function is bounded
below: the RG flow must stop either when it reaches the lower bound, or trapped in some
fixed point above the lower bound. This means the corresponding quantum field theory
must flow to a conformal fixed point at low energies. In section 3 we will see the the first
two orders of A˜ for gauge theories can be bounded blow with proper choice of fermion and
scalar numbers.
3 The RG flow of gauge and Yukawa couplings
In [16], gauge theories with a Yukawa interaction 1ψTC(Y ) ψ φ + h.c. and a quartic
2 i a ij j a
scalar interaction 1λ φ φ φ φ was studied, and the a-function is computed to 4 loops,
4! abcd a b c d
up to some g6 terms1,
2n 1 1
ds2 =G dgIdgJ = V (1+σg2)dg2+ tr[dyˆ dy ]+ dλ dλ ,
IJ g2 6 a a 144 abcd abcd
A˜(2) =−n β g2,
V 0
1 1
A˜(3) =− n g4(β +σβ )− g2tr[y yˆ Cˆψ]
V 1 0 a a
2 2
1 1 (cid:16) 1 (cid:17) (3.1)
+ tr[y yˆ y yˆ ]+ tr[y yˆ y yˆ ]+ tr[y yˆ ]tr[y yˆ ] ,
a a b b a b a b a b a b
24 12 4
(cid:18) (cid:19)
1 3 1
A˜(4) = λ λ λ + g4(tφtφ) (tφtφ) − tr[y yˆ y yˆ ] λ
λ 8 abcd abef cdef 2 A B ab A B cd 2 a b c d abcd
1 (cid:16) (cid:17)
+ λ λ tr[y yˆ ]−6g2(Cφ) ,
abcd abce e d ed
12
in which
(cid:18) (cid:19)
1 1 1
σ = (102C −20Rψ −7Rφ), β = 11C −2Rψ − Rφ ,
G 0 G
6 3 2
(3.2)
1 (cid:16) (cid:17) 1 2
β = C 34C −10Rψ −Rφ − tr[(Cψ)2]− tr[(Cφ)2].
1 G G
3 n n
V V
1In[16]thecoefficientoflasttermof (3.2)was−4,wemodifieditto−2basedonourowncalculations,
and the results of other authors, e.g. [17].
– 3 –
A˜(4) are the λ-dependent terms in A˜(4), we did not present the other terms here because
λ
we will not need them in this work. Our conventions for the group invariants and various
other constants completely follows [16], and for compactness we will not list them here.
A˜(2) onlydependong,andifβ < 0,g = 0isatheminimumofA˜(2),soatlowenergies,
0
the gauge coupling approaches 0, and gauge field decouples. Higher loop corrections do not
affect our conclusion because they become unimportant when g → 0.
Whenβ > 0, g2 becomesgreateratlowenergies(asymptoticfreedom), andhigherloop
0
corrections becomes important. In order to find the minimum of A˜(2)+A˜(3), we define the
following quantities,
F = y yˆ −6g2Cˆψ,
2 a a
I = −itr(Y Y¯ )+itr(Y Y¯ ), (3.3)
ab a b b a
t = Y Y +Y Y +Y Y .
ijkl aij akl aik ajl ail ajk
Using
1 1 1
t t¯ijkl =tr(y yˆ y yˆ )+ tr(y yˆ )tr(y yˆ )− I I , (3.4)
ijkl a b a b a b a b ab ab
3 4 4
the Yukawa terms in A˜(3) can be written as a sum of perfect squares,
1 1 1 1
A˜(3) =− n g4(b +σβ )+ tr[F F ]+ t t¯ijkl + I I , (3.5)
V 1 0 2 2 ijkl ab ab
2 24 36 48
where
3 1 (cid:16) (cid:17) 2 2
b = β + tr[(Cˆψ)2] = C 34C −10Rψ −Rφ + tr[(Cψ)2]− tr[(Cφ)2]. (3.6)
1 1 G G
n 3 n n
V V V
If β > 0 and b < 0, A˜(2)+A˜(3) has a local minimum at2
0 1
β
F = I = t = 0, g2 = g2 ≡ − 0. (3.7)
2 ab ijkl m b
1
Notice t is proportional to the tree amplitude of 4 positive fermion,
ijkl
[12][34]
A(ψ (k ),ψ (k ),ψ (k ),ψ (k )) = t . (3.8)
i 1 j 2 k 3 l 4 ijkl
s
12
and the vanishing of t forbids these UHV(ultra-helicity-violating) amplitude3 at tree
ijkl
level. Actually we found the amplitude also vanishes at one loop if t = 0, and it is
ijkl
natural to expect it to vanish at all loops in conformal deformations of N = 4 SYM.
Now consider a theory with n fermions and n scalars, both in adjoint representation,
ψ φ
1 1
β = C (11−2n − n ),
0 G ψ φ
3 2
1
β = C2 (34−16n −7n ), (3.9)
1 3 G ψ φ
1
σ = C (102−20n −7n ).
G ψ φ
6
The pairs of (n ,n ) satisfying β > 0, b < 0, and the corresponding g2 N are
ψ φ 0 1 m
collected in Table 1. Notice for some choices of (n ,n ), g2 N is still much smaller than 1,
ψ φ m
and perturbation theory may be trusted.
2The stationary point of A˜is given by ∇ A˜=0 instead of ∂ A˜=0.
I I
3It can be easily checked that t also vanishes for supersymmetric Yang-Mills theories.
ijkl
– 4 –
(n ,n ) (1,6) (1,7) (1,8) (1,9) (1,10) (1,11) (1,12) (1,13)
ψ φ
g2 N 1 11 1 1 2 7 1 1
m 26 4 6 17 82 16 22
(n ,n ) (1,14) (1,15) (1,16) (1,17) (2,6) (2,7) (2,8) (2,9)
ψ φ
g2 N 1 1 1 1 1 7 1 1
m 31 46 76 166 22 6 10
(n ,n ) (2,10) (2,11) (2,12) (2,13) (3,6) (3,7) (3,8) (3,9)
ψ φ
g2 N 1 1 1 1 1 1 1 1
m 16 26 46 106 6 16 46
Table 1: (n ,n ) satisfying β > 0, b < 0, and the corresponding g2 N.
ψ φ 0 1 m
4 The Leigh-Strassler Theory
The Leigh-Strassler theory is a N = 1 superconformal deformation of the N = 4 SYM. Su-
perconformalsymmetryandunitaritygivesstrongconstraintstotheanomalousdimensions
of operators in supersymmetric theories, and a classification of supersymmetric deforma-
tions has been carried out in [18]. For N = 1 SYM it has been proved non-perturbatively
that the conformal fixed point is always stable by showing there is a positive a-function
aroundthefixedpoint[19]. Thea-functionwasalsocomputedperturbativelyine.g. [9,20–
22].
Less effort has been paid on the non-supersymmetric deformations of conformal field
theories. The non-supersymmetric theories have much more parameters than the super-
symmetric theories, which makes a direct study of RG flow unfeasible. And without super-
symmetry, the known unitarity bounds are not enough to decide whether the deformation
operators are relevant or irrelevant.
We will study the RG flow of non-supersymmetric theories using the a-function. From
the perspective of last section, the Leigh-Strassler theory is a gauge theory with four chiral
fermions six real scalars in adjoint representation of the SU(N) gauge group. Interestingly,
from (3.9) we find β = σ = b = 0, and
0 1
1 1 1
A˜(2)+A˜(3) = tr[F F ]+ t t¯ijkl + I I ≥ 0. (4.1)
2 2 ijkl ab ab
24 36 48
So A˜(2)+A˜(3) has a global minimum at
F = I = t = 0. (4.2)
2 ab ijkl
Eq. (4.2) is ’homogeneous’ in g and Y : if g and Y solves (4.2), |z|g and zY also
aij aij aij
solves (4.2) for arbitrary non-zero complex z. This implies unlike the theories in Table 1,
where g2 N is fixed, conformal fixed points may exist for arbitrary gauge couplings when
m
(n ,n ) = (4,6).
ψ φ
We will focus on the planar limit, then in terms of SU(N) matrix-valued fields, the
Yukawa interaction can be written as Y Tr(φIψAψB)+Y¯IBATr(φIψ¯ ψ¯ ) and the quar-
IAB A B
tic scalar coupling is 1λ Tr(φIφJφKφL).
4 IJKL
It is convenient to combine φI into 3 complex scalars φi, and to discriminate ψi and
ψ4. ψi are the super-partner of φi, while ψ4 is the super-partner of the gauge field. The
– 5 –
Yukawa interaction of the Leigh-Strassler theory is given by
(cid:34) (cid:35)
(cid:18) (cid:19)
1 (cid:88)
L =Tr κ(cid:15)+ φi qψjψk − ψkψj +h φiψiψi+gφ¯[ψi,ψ4] +c.c. (4.3)
Y ijk q i
i
where
1
(cid:15)+ = ((cid:15) +|(cid:15) |), (4.4)
ijk 2 ijk ijk
and h and q are two complex parameters.
The quartic scalar interactions are related to the Yukawa couplings Y by,
ijk
g2
L = − Tr([φi,φ¯]2)−Y Y¯lkmTr(φiφjTa)Tr(φ¯ φ¯Ta), (4.5)
φ i ijm k l
2
where Ta are the SU(N) generators.
Besides supersymmetry, the theory is also invariant under a U(1) transformation:
ψi → eiξψi, ψ4 → e−3iξψi, φi → e−2iξφi. (4.6)
In order for the theory to be conformal, κ, h and q must satisfy a condition. At 2 loops
and in the planar limit, this condition is
2g2 = |h|2+κ2(|q|2+|q|−2). (4.7)
The condition under which the theory is conformal up to three loops (four loops in planar
limit) was given in [23].
In the planar limit, (4.2) becomes
(F )B =Y Y¯IBC +Y Y¯ICB −6g2δB,
2 A IAC ICA A
1
I = (Y Y¯ICD −Y Y¯JCD) = 0, (4.8)
IJ JCD ICD
i
t =Y Y +Y Y .
ABCD IAB ICD IDA IBC
It can be verified that (4.8) holds4 and the Leigh-Strassler theory does lie in the the global
minimum of A˜(2)+A˜(3). However, it is possible that there are other (non-supersymmetric)
conformalfixedpointsintheneighborhoodoftheLeigh-Strasslerdeformation, thentheRG
flow may end up reaching these non-supersymmetric conformal fixed points, and supersym-
metry fails to emerge. in order to exclude this possibility we will examine the anomalous
dimensions of marginal operators in the next section.
5 Marginal Operators and Conformal Deformations
InthissectionwestudytheconformaldeformationsoftheLeigh-Strasslertheory. Thespace
of’allconformaldeformations’mayhavemultiplecomponents,andindifferentcomponents,
the space may have different dimensions. So to be more accurate we define the term
4Actually (4.2) hold even in the non-planar case for the Leigh-Strassler theory.
– 6 –
’sector’: the sector of a given conformal fixed point is the irreducible component containing
a neighborhood of the point in the space of conformal deformations. The Leigh-Strassler
deformationhas4realphysicalparameters,solocallyitistheonlyconformalfixedsubspace
if the enclosing sector also has 4 physical parameters.
Different fixed points in the same sector are physically equivalent if they are related by
SO(6) and U(4) redefinitions of scalars and fermions. So number of ’physical’ parameters
is the dimension of the quotient space of the sector by SO(6)×U(4).
Suppose L is the Lagrangian of the enclosing sector of the Leigh-Strassler theory,
cft
and g are the physical parameters,
a
L =L (g ,A ,UA(ω )ψA,OIJ(ω )φJ), (5.1)
cft cft a µ B i i
where ω ’s are parameters of U(4) (16 parameters) and SO(6) (15 parameters) redefinitions
i
UA and OIJ.
B
General deformations can be written as L , the Lagrangian of the Leigh-Strassler the-
0
ory, plus four types of dimension-4 operators,
∂L ∂L
L = L0 +δg cft +δω cft +cmO +zαZ . (5.2)
O cft a ∂g i ∂ω m α
a i
∂L
The first type, cft, corresponds to the variation of the Lagrangian when the parameters
∂ga
∂L
g changes. The number of cft is N , which is the number of physical parameters The
a ∂ga p
∂L
second type, cft, corresponds to the variation of the Lagrangian under the redefinition of
∂ωi
∂L
scalars and fermions. The number of cft is at most 31, but it is possible that a subgroup
∂ωi
of U(4) × SO(6), G , is preserved in the theory. In this case, the corresponding O
sym i
vanishes, so the number of O is the same as the number of generators of the quotient
i
group U(4) × SO(6)/G . For example, N = 4 SYM has a SU(4) symmetry, so the
sym
number of O is only 16. The γ -deformed SYM [24, 25] has a U(1)3 symmetry, and the
i i
number of O is 28. Adding these two types of operators to the Lagrangian does not break
i
the conformal symmetry, and the corresponding beta functions vanishes.
The third type, O , are operators with non-zero anomalous dimensions, or marginally
m
irrelevant operators. The last type, Z are operators which break conformal symmetry
α
but with vanishing anomalous dimensions. We will call operators with zero anomalous
dimensions protected operators, Z will be called accidentally protected operators, and the
α
number of Z will be denoted by N .
α acci
Each protected operator corresponds to a zero eigenvalue of the anomalous dimension
matrix ∇ βJ, so we have
I
N = Dim(Ker(∇ βJ))−N −Dim(U(4)×SO(6))+Dim(G ). (5.3)
p I acci sym
As an example, we consider the Yukawa couplings of γ deformed N = 4 SYM. There
i
are in all 34 protected operators, in which none is accidentally protected, and the theory
preservesaU(1)3symmetry. From(5.3)thetheoryhas6physicalparameters. Thecomplete
Lagrangian of this 6-parameter theory is not known yet, but it has been formulated for a
– 7 –
3-parameter sub-theory(when all γ ’s are real) [24]. The gravity dual of this 3-parameter
i
sub-theory [24] is a sub-theory of a 6+2 parameter5 deformation of AdS ×S5 [25]. This
5
indicates the 6+2 parameter deformation is the gravity dual of the enclosing sector of γ
i
deformed SYM.
TheLeigh-Strasslerdeformationhas46protectedYukawaoperators6. Amongthem30
operators correspond to the variation of the Lagrangian under SO(6) and U(4) rotations of
scalars and fermions. 4 protected operators corresponds to the variation of the Lagrangian
when the parameter of the theory, β and h change. The left 12 operators are7:
(cid:16) 1 κ
Oi =Tr κ(p2+ −1)φiψ4ψ4+ (q¯−q¯−1)2(φ1)4
1 p2 h¯
h¯ (cid:17)
+(φ1)2(q¯φ2φ3−q¯−1φ3φ2)+(q¯−q¯−1)φ1φ2φ1φ3− (φ2φ3)2 ,
κ
(cid:16) κ (5.4)
Oi =Tr φj{ψk,ψ4}+φk{ψj,ψ4}− (q¯−q¯−1)φi{ψi,ψ4},
2 h¯
+(φj)2[φ¯ ,φk]+(φk)2[φ¯ ,φj]+[φi,φ¯]{φj,φk}
j k i
κ (cid:17)
− (q¯−q¯−1)(φi)2([φj,φ¯ ]+[φk,φ¯ ]) ,
h¯ j k
Adding these operators to the Leigh-Strassler Lagrangian,
δL = a Oi +a¯ O¯i +b Oi +¯b O¯i, (5.5)
i 1 i 1 i 2 i 2
t and I are invariant, but F is not. For example,
ijkl ab 2
1 κ2|q−1/q|2
δ(F )1 = |b |2+|b |2+|b |2,
2 2 1 |h|2 1 2 3
(5.6)
1 (cid:18) κ2|q−1/q|2(cid:19)
δ(F )4 =|a |2+|a |2+|a |2+ 2− (|b |2+|b |2+|b |2).
2 2 4 1 2 3 |h|2 1 2 3
Apparently turning on any of these operators will change F , and in the end increase A˜.
2
So these operators are accidentally protected operators, and do not corresponds to new
conformal field theories. From (5.3), the number of parameters in the enclosing sector is
N = 4, so we have proved that as far as Yukawa couplings are concerned, locally the
p
Leigh-Strassler deformation is the only conformal deformation.
Last, let us emphasize that these 12 accidentally protected operators may not be pro-
tected by higher loop corrections. If the two loop anomalous dimensions turn out to be
negative, the Leigh-Strassler theory will be unstable even at weak coupling. However, it
will be guaranteed to be stable at weak couplings if we complete turn off the accidentally
5These 2 extra parameters corresponds to the variation of complex gauge coupling τ = 1 + iθ .
g2 8π2
6The anomalous dimensions of quartic scalar operators will be modified when these protected Yukawa
operators are added to the Lagrangian. This is why in (5.4), protected Yukawa operators also contain a
quartic scalar piece.
7In (5.4) Oi and Oi are both complex operators. Each of them corresponds to 2 real accidentally
1 2
protected operator which contain both chiral and anti-chiral fermions.
– 8 –
protected operators, for example, in the subspace of deformations described by
(cid:104) (cid:105)
L =Tr Y φiψjψk +Xiφ¯ψjψ4+Ziφ¯ψ4ψj +c.c.
Y ijk j i j i
(5.7)
1
L =λ Tr(φiφjφ¯ φ¯)+ λ Tr(φiφ¯ φjφ¯).
φ ijk¯¯l k l 2 ik¯j¯l k l
6 Emergent Supersymmetry
In the planar limit, the λ-dependent terms of A˜(4) in (3.1) is reduced to
3 3 1
A˜(4) =− g4λ − g2λ λ + D λ λ ,
λ 8 IIJJ 4 IJKL LKJI 8 MN MIJK KJIN
(6.1)
1
+B λ − λ λ λ ,
IJKL LKJI IJKL LKMN NMJI
6
in which D corresponds to a fermion bubble diagram,
IJ
D = Y Y¯JCD +Y Y¯ICD, (6.2)
IJ ICD JCD
and B corresponds to a fermion box diagram,
IJKL
B = Y Y¯ADY Y¯CB +Y¯ABY Y¯CDY . (6.3)
IJKL IAB J KCD L I JCB K LAD
A˜(4) is a polynomial of degree 3 in λ so it does not have a global minimum.
λ IJKL
Nevertheless, V may have local minimum or saddle points, corresponding to stable or
S
unstable fixed points of single trace scalar couplings, respectively.
Numerical tests shows that the anomalous dimension matrix of λ is still positive
IJKL
semi-definite. There are 6 protected scalar operators, and they are combinations of three
holomorphic and three anti-holomorphic operators.
1 1
OR = Tr(O +O¯ ), OI = Tr(O −O¯ ). (6.4)
i 2 i i i 2i i i
One anti-holomorphic operators is
(cid:20) (cid:21)
κ 1 1 1
O¯ = h3(q2+ )+κ3(q− )3 φ¯4+h2κ(q2+ −1)φ¯ (φ¯3+φ¯3)
1 h q2 q 1 q2 1 2 3
(cid:20) (cid:21) (cid:20) (cid:21)
1 1
+ h3 +κ3(q2−1)2 φ¯2φ¯ φ¯ + −h3q+κ3( −1)2 φ¯2φ¯ φ¯
q 1 2 3 q2 1 3 2
(6.5)
(cid:20) (cid:21)
1 1
+(q− ) −h3+κ3(q− ) φ¯ φ¯ φ¯ φ¯
1 2 1 3
q q
(cid:20) (cid:21)
1 1 h 1
−hκ2(q− )(q2+ −1)φ¯2φ¯2+ h3−κ3(q− ) φ¯ φ¯ φ¯ φ¯ .
q q2 2 3 κ q 2 3 2 3
The other two anti-holomorphic operators can be obtained form O¯ using the Z sym-
1 3
metry. The holomorphic operators are the Hermitian conjugate of anti-holomorphic oper-
ators.
If we add these protected operators to the Leigh-Strassler Lagrangian,
L = L +z O +z¯O¯ . (6.6)
LS i i i i
– 9 –