On invariants and scalar chiral correlation functions in N=1 superconformal field theories

0.2 MB·English

by  Holger Knuth

arXiv:1010.2740v2
LQP:10101400
ON INVARIANTS AND SCALAR CHIRAL CORRELATION FUNCTIONS IN
= 1 SUPERCONFORMAL FIELD THEORIES
N
HOLGER KNUTH
Abstract. A general expression for the four-point function with vanishing total R-charge of anti-
chiral and chiral superfields in N = 1 superconformal theories is given. It is obtained by applying
the exponential of a simple universal nilpotent differential operator to an arbitrary function of two
1 cross ratios. To achieve this the nilpotent superconformal invariants according to Park are focused.
1 Severaldependenciesbetweentheseinvariantsarepresented,sothateightnilpotentinvariantsand27
0 monomials oftheseinvariantsofdegreed>1areleft beinglinearly independent. Itisanalyzed,how
2 terms within the four-point function of general scalar superfields cancel in order to fulfill the chiral
restrictions.
n
a
J
1. Introduction In the eighties the two authors of [12–14] con-
0
centrated on a group theoretical approach to ex-
2
In theearly beginningsof supersymmetry[1–5]
tended conformal supersymmetry.
four-dimensional = 1 superconformalfield the-
h] ories came into tNhe focus of several researchers. A later series of papers started also with re-
t sults for chiral superfield in = 1 [15–17], but
- But due to the mainly perturbative approach N
p it focused afterwards the analysis of theories with
to supersymmetric field theories, in which non-
e extended superconformal symmetry on analytic
h trivialtheoriescouldnotsustainconformalinvari-
superspace(e.g. [18–23]). Thisapproach ispurely
[ ance, the mainstream attached to super-Poincar´e
on-shell. It was shown, that there are no nilpo-
2 symmetry.
tent invariants of up to four points of analytic
v Still insights in the conformal invariance of
0 superspace [24,25].
supersymmetric non-Abelian gauge theories at
4 Another line of work concentrated first on cor-
7 renormalization groupfixedpointswithvanishing
relation functions of only chiral superfields for
2 β-function ( [6] and more recently [7–9]) brought
. = 1 [26,27]. While the two point function
0 new interest to a non-perturbative analysis of su- N
is a pure contact term, the three point function
1 perconformal field theories in the mid-90s. The
0 is only consistent with the Ward identities, if the
veryrecentstudyofnewsupergravitymodelswith
1 total R-charge is one. Later papers then mainly
: thepotential toresolvesomeproblemsofminimal
v dealt with = 4 [28–32].
supersymmetric standard models [10,11] could N
i In a more general approach implications of su-
X possiblyagainrenewattention onthistopic. Here
perconformal symmetry on correlation functions
r big emphasis is given to the underlying supercon-
a of arbitrary quasi-primary fields were derived by
formal symmetry.
Park[33,34]. Forthatpurposeallinvariantsofthe
Chiral superfields gathered most attention not
-extended superconformal group for all 1
only because of their prominentrole in supersym- N N ≥
were constructed on ordinary super Minkowski
metry through the construction of F-Term La-
space. Itisstated, thatcorrelation functionswith
grangians and as suitable multiplets containing
vanishing R-charge are functions of these invari-
elementary particles, butalso for thesake of their
ants. For the case of non-vanishing R-charge the
relative simplicity. This paper focuses on chiral
correlation functions are nilpotent and dependon
and anti-chiral superfields as well, as it comes to
a larger set of invariants, which are not required
correlation functions in the second half.
to be invariant under the U(1) symmetry. Here
Y
only the former case is discussed.
Section 2 takes a look at the coordinates of
superspace and variables, which are suitable for
Key words and phrases: Superconformal symmetry;
the construction of invariants and expression of
Correlation functions; Chiral superfields.
n-point functions because of their homogeneous
PACS 2010: 11.30.Pb; 11.25.Hf
MSC 2010: 81T60. transformation properties.
1
2 Holger Knuth
Insection3thethree-andfour-pointinvariants and the commutation relations containing M ,
µν
for = 1 from [34] are given. It is argued, that which express the spinoral, vectorial and tenso-
N
there are only 10 independent invariants of four rial transformation properties of the generators.
points: twocrossratiosandeightnilpotentinvari- With this the general element of the translational
ants. This part is central to this paper because supergroup is given by
it shows, that in N = 1 supersymmetry fewer in- (2.2) g(z) = ei(xµPµ+θαQα+Q¯α˙θ¯α˙) ,
variantsareneededtoexpressanysuperconformal
four-point function than thought hitherto ( [34] where the parameters zM = xµ,θα,θ¯α˙ are
argues it to be maximally 16). the coordinates of superspace. θα and θ¯α˙ are
(cid:0) (cid:1)
Section 4 shortly introduces scalar chiral and Graßmann-valued spinors. The actual reason for
anti-chiral superfields, before their correlation theresultsinthispapertobeonlyvalidfor = 1
N
functionswithvanishingR-chargearediscussedin is the property
section 5. While two- and three-point functions 1
do not need a long discussion, the discussion of (2.3) θαθβ = ǫαβθθ˜,
−2
four-point functions is arranged around the proof
of a theorem giving their general form. (2.4) θ¯α˙θ¯β˙ = 1ǫ¯α˙β˙θ˜¯θ¯
2
The theorem gives the four-point function as
with the anti-symmetric 2 2 matrices ǫαβ and
an arbitrary function of two superconformalcross ǫ¯α˙β˙ with ǫ = ǫ21 = ǫ¯ =×ǫ¯2˙1˙ = 1. The exterior
ratios containing all model specific information, 12 1˙2˙
derivative on superspace is given by
to which a universal differential operator is ap-
plied. Its proof starts with the general form of ∂
(2.5) d = dxµ+idθσµθ¯ iθσµdθ¯
scalar four-point functions from [34] and so gives − ∂xµ
an insight in the elimination of terms needed, so (cid:0)+dθαD dθ¯α˙D¯ (cid:1)
α α˙
−
that the four-point function only depends on the
with the covariant derivatives1
correct set of chiral- and anti-chiral coordinates.
∂ ∂
While it was already known, that this function (2.6) D = iσµθ¯ ,
∂θ − ∂xµ
can be written in terms of two non-nilpotent in-
∂ ∂
variants, this theorem answers the question, how (2.7) D¯ = +iθσµ .
these chiral scalar four-point functions are ex- −∂θ¯ ∂xµ
pressed in terms of the set of invariants chosen For the discussion of chiral and anti-chiral super-
in section 3. fields it is convenient to use chiral and anti-chiral
At last a short calculation derives another way coordinates, which are respectively defined by
to write rational superconformal four-point func- (2.8) x µ = xµ iθσµθ¯,
+
tions as a power series. This allows to directly −
(2.9) x µ = xµ+iθσµθ¯.
transfer results from global conformal field the- −
ory to global superconformal field theory, which Thecovariantderivativesforthesecoordinatesare
is the reason, why we prefer this expression to ∂
(2.10)D¯ = , D = ∂ 2iσµθ¯ ∂ ,
those known so far. + −∂θ¯ + ∂θ − ∂x−µ
∂
(2.11) D = , D¯ = ∂ +2iθσµ ∂ .
− ∂θ − −∂θ¯ ∂x+µ
2. Superspace
As we will see in the following sections the cor-
Supersymmetry is generated by operators Qα relation functions are expressed in terms of the
and their hermitian conjugates Q† = Q¯ . They following intervals
α α˙
are two-component spinors like the fermionic (2.12) xµ = x µ x µ 2iθ σµθ¯ ,
component fields, as they relate these fields to ¯ij i− − j+ − j i
bosonic fields and vice versa. Together with Pµ = (2.13) θiαj = θiα−θjα ,
(H, P)andMµν theyfulfillthefour-dimensional (2.14) θ¯α˙ = θ¯α˙ θ¯α˙ ,
=−1 supersymmetry algebra. The only non- ij i − j
N
vanishing (anti-)commutators are 1The spinoral indices will be omitted in most of the
paper and a tilde is used to indicate lower indices of the
(2.1) Qα,Q¯α˙ = 2σαµα˙Pµ spinors.
(cid:8) (cid:9)
On invariantsandscalar chiralcorrelation functionsin N =1SCFT 3
which transform homogeneously under supercon- and either X˜ or X˜ , because
1+ 1−
formal transformations (cf. [34]). Writing the in-
(3.8) X˜ = X˜ 4iΘ¯ Θ .
terval (2.12) as hermitian 2 2 matrices by con- 1− 1+ − 1 1
tractionwithPaulimatrices,×x¯ijαα˙ = x¯µijσµαα˙ and With the help of this equation one has
x˜α˙α = xµσ˜α˙α, their inverses are
¯ij ¯ij µ Θ X Θ¯
1 1+ 1
(3.9) I = 4i .
(2.15) x −1 = x˜¯ij , x˜−1 = x¯ij . 3PF − X12+
¯ij x 2 ¯ij x 2
¯ij ¯ij
3.2. Invariants of four points. The construc-
3. Superconformal invariants tion of four-point invariants leads to both non-
nilpotent and nilpotent invariants.
In preparation of the discussion of the correla-
tionfunctionstheinvariantsofuptofourpointsin
3.2.1. Cross ratios. The cross ratios,
superspacehavetobeconstructed. Thiswasdone
together with a detailed analysis of superconfor- x2 x2
(3.10) r¯s t¯u ,
mal symmetry and correlation functions in [34]. x2 x2
r¯u t¯s
Heretheresults aresummarizedand relations are
found between invariants of four points. These reduce for vanishing Graßmann variables to the
reduce the supposed number of independent in- ordinary conformal cross ratios.
variants, which have to be considered in the ex- Defining analogously to (3.3)
pression of the four-point function, from 16 to 10.
x2 x2
Eight of them are nilpotent and form 36 linear (3.11)X2 = ¯i4 , X2 = ¯4i
1(i−1)+ x2 x2 1(i−1)− x2 x2
independent monomials including those of degree ¯i1 ¯14 ¯41 ¯1i
0 and 1. with i = 1,2, one gets the nilpotent invariant of
three points – ignoring the added constant 1 –
3.1. Invariant of three points. There are no
with the corresponding indices replaced:
invariants of two points, but one of three points.
This three point invariant can be expressed in X2
1(i)+
terms of (3.12) J = .
1(i) X2
(3.1) X˜ = x −1x x −1 , 1(i)−
1+ ¯13 ¯23 ¯21
−
(3.2) X˜ = x −1x x −1 . They relate the cross ratios with each other.
1− ¯12 ¯32 ¯31 There are six different cross ratios and their in-
The squares of the four-vectors corresponding to verses (cf. eq. (3.10)).
X˜1+ and X˜1− are needed: Picking out the two cross ratios,
(3.3) X12+ = x2¯2x12¯2x32¯13 , X12− = x2¯3x12¯3x22¯12 . (3.13) I1 = xx2¯12¯124xx2¯32¯342 , I2 = xx2¯12¯143xx2¯22¯234 ,
With these the three-point invariant is
the other cross ratios are
(3.4) I = J 1
3PF 1− (3.14) = x2¯32x2¯41 = J−1J ,
(3.5) with J = X12+ . I3 x2¯31x2¯42 1 1(1)I2
I is nilpotent beca1use theX1s2−quares of the in- (3.15) I4 = xx2¯22¯241xx2¯32¯314 = J1I1I2−1 ,
te3PrvFals, x2 , become symmetric in their indices, if (3.16) = x2¯23x2¯41 = J −1 ,
the Graßm¯ijann variables are set to zero, and so J I5 x2¯21x2¯43 1I1I2
is one in this case. 1 (3.17) I6 = xx2¯12¯123xx2¯42¯432 = J1J1−(11)J1(2)I1I2−1 .
Anotherpossibility toexpressthisinvariantisthe
There are no other non-nilpotent invariants,
use of two Graßmann spinors,
which cannot be expressed by , and nilpo-
1 2
(3.6) Θ1 = i θ˜¯21x¯21−1 θ˜¯31x¯31−1 , tent invariants, because the nuImbeIr of indepen-
−
dent non-nilpotent invariants has to be the same
(cid:16) (cid:17)
(3.7) Θ¯ = i x −1θ˜ x −1θ˜ ,
1 ¯13 13 ¯12 12 as the number of ordinary conformal invariants.
−
(cid:16) (cid:17)
4 Holger Knuth
3.2.2. Nilpotent invariants. There are many pos- The following set I of all linear independent
siblesetsofeightindependentnilpotentinvariants monomials in the nilpotent invariants containing
one can choose. Here the choice is only 36 elements is chosen here:
(3.18) I1ij = Θ˜¯1(j)X˜1(1)+Θ˜1(i) , (3.30) I0,1 ∈ {1} ,
− X2 X2 I I ,
1(1)+ 1(1)− 1,1...8 ∈ { ijk}
Θ˜¯q X˜ Θ˜ I2,1...18 ∈ {I1ijI1kl,I1iiI212,I1iiI221,
1(j) 1(2)− 1(i)
(3.19) I = , I I ,I I ,I I ,
2ij 111 222 122 211 i12 221
− X2 X2 }
1(1)+ 1(1)− I I2 I ,I2 I ,I2 I ,
3,1...8 ∈ { 111 i22 122 i11 112 j21
which apart from X12(q1)+ and X12(1)− (eq. (3.11)) I1221Ij12} ,
uses the following matrices and Graßmannian I = I2 I2
4,1 111 122
spinors:
with i,k,j,l = 1,2. They are sorted with the first
(3.20) X˜1(1)+ = x¯14−1x¯24x¯21−1 , index being the degree.
−
(3.21) X˜ = x −1x x −1 , Thissetisasubsetofminimalsizeofthemono-
1(2)− ¯13 ¯43 ¯41
mials of nilpotent invariants from eqns. (3.18)
(3.22) Θ˜ = i x˜−1θ¯ x˜−1θ¯ ,
1(i−1) ¯41 41− ¯i1 i1 and (3.19), so that any nilpotent invariant can
(3.23) Θ˜¯1(i−1) = i(cid:0)θ1ix˜−¯1i1−θ14x˜−¯141(cid:1) . bemeeenxtpsrewsistehdcaoseffiacileinnetas,r wcohmicbhinaarteiofnunocftioitnsseol-f
Third powers of each(cid:0) of these spino(cid:1)rs vanish, non-nilpotent invariants, e.g. the superconformal
which restricts the number of possible monomials
cross ratios.
of the nilpotent invariants correspondingly. The
In the following the relations between the
variables introduced for the three point invariant
monomials, which have been left outin thechoice
in section 3.1 are related to these variables by
of the subset, and the chosen ones are given using
properties of the Pauli matrices and eqns. (2.3)-
(3.24) Θ = Θ Θ ,
1 1(1) 1(2)
− (2.4). While both equations are applied to show
(3.25) Θ¯ = Θ¯ Θ¯ ,
1 1(1) 1(2)
−
(3.26)X˜1+ = X˜1(1)+ −X˜1(2)− −4iΘ¯1(2)Θ1(1) . (3.31) I I = Xˆ1µ(1)+Xˆ1(2)−µI I
1jk 2jk Xˆ2 1jk 1jk
It is convenient to divide eqns. (3.20) and (3.21) 1(1)+
byanormalizationsimilartotheoneinthedefini-
for j,k =1,2, only either one or the other can be
tion of the nilpotent invariants (3.18) and (3.19):
used–togetherwiththesymmetryofthecontrac-
X˜ tions of Xµ with the Pauli-matrices hidden in
(3.27) Xˆ˜ = 1(1)+ , 1(2)−
1(1)+ 1 the definition (3.19) – to get
X2 X2 4
1(1)+ 1(1)−
Xˆ2
(3.28) Xˆ˜1(2)− = (cid:16) X˜1(2)− (cid:17)1 . (3.32) I2jkI2mn = Xˆ12(2)−I1jkI1mn
X2 X2 4 1(1)+
1(1)+ 1(1)−
for j,k,m,n = 1,2 with j = m k = n.
(cid:16) (cid:17)
Again with the corresponding four-vectors one ∨
I I , I I , I I and I I are
211 112 211 121 112 222 121 222
finds J in Xˆ2 :
1(1) 1(1)+ equal to the expressions
(3.29) Xˆ2 = J = 1 2iI +6I2 . (3.33)
1(1)+ 1(1) − 111 111
Also J , Xˆµ Xqˆ and Xˆ2 can be ex- I211I112 = 2Xˆ1µ(1)+Xˆ1(2)−µI111I112 −I111I212 ,
1(2) 1(1)+ 1(2)−µ 1(2)− I I = 2Xˆµ Xˆ I I I I ,
pressed in terms of the ten invariants (3.13), 211 121 1(1)+ 1(2)−µ 111 121 − 111 221
(3.18) and (3.19), which generate all four-point I I = 2Xˆµ Xˆ I I I I ,
invariants. Butheretheserather longexpressions 112 222 1(1)+ 1(2)−µ 112 122 − 212 122
are not needed. I121I222 = 2Xˆ1µ(1)+Xˆ1(2)−µI121I122 −I221I122 .
On invariantsandscalar chiralcorrelation functionsin N =1SCFT 5
A bit longer are the relations giving I I , conditionsthensimplymean,thatchiralandanti-
121 212
chiral fields do not depend on θ¯ and θ, respec-
(3.34)I I = I I I I I I
121 212 − 112 221− 111 222− 122 211 tively. They have the following expansions in θ
+2Xˆ1µ(1)+Xˆ1(2)−µ(I111I122+I112I121) , and θ¯:
and I211I222, (4.3) Φ xµ,θ =φ xµ +√2θψ˜ xµ
+ + +
(3.35) I211I222 = −I212I221 (cid:0) (cid:1) +(cid:0)θθ˜(cid:1)m xµ+ , (cid:0) (cid:1)
+Xˆ12(2)−(I111I122+I112I121) . (4.4) Φ¯ xµ,θ¯ =φ∗ xµ (cid:0) √(cid:1)2θ˜¯ψ¯ xµ
− − − −
In [34] also nilpotent invariants with lowest or- (cid:0) (cid:1) (cid:0)θ˜¯θ¯m(cid:1)∗ xµ . (cid:0) (cid:1)
der in Graßmann variables θθ¯ 2 are constructed. − −
This was necessary, as the construction was made Fromthesuperconformaltrans(cid:0)form(cid:1)ationsofthese
(cid:0) (cid:1)
also for extended supersymmetry > 1, which superfields (cf. [34]) it follows, that scale dimen-
N
is not discussed here. For = 1, for which eqns. sions and R-charges of these fields are related as
N
(2.3)-(2.4) are valid, these invariants can be ex-
1
pressed by the nilpotent invariants above:
(4.5) κ= η ,
±3
(3.36) I2
1ij
where + and are valid for anti-chiral and chiral
1
= 4Xˆ12(1)+Θ1(i)σµΘ¯1(j)Θ1(i)σµΘ¯1(j) , superfields, re−spectively.
(3.37) I I
1ii 112
1
= Xˆ2 Θ σµΘ¯ Θ σ Θ¯ , 5. Correlation functions
4 1(1)+ 1(i) 1(i) 1(1) µ 1(2)
(3.38) I I Only the consequences of the superconformal
1ii 121
1 symmetry and the restrictions of the superfields
= Xˆ2 Θ σµΘ¯ Θ σ Θ¯ ,
4 1(1)+ 1(i) 1(i) 1(2) µ 1(1) are discussed here without any reference to a spe-
cific model. Non-vanishing correlation functions
(3.39) I I +I I
111 122 112 121
of the component fields always have a vanishing
1
= 2Xˆ12(1)+Θ1(1)σµΘ¯1(1)Θ1(2)σµΘ¯1(2) , total R-charge, which is the sum of the R-charges
of the fields therein. So the total R-charge of a
1
= Xˆ2 Θ σµΘ¯ Θ σ Θ¯ correlation function of superfields has to vanish
2 1(1)+ 1(1) 1(2) 1(2) µ 1(1)
for it to be non-nilpotent. For non-vanishing to-
with i,j = 1,2.
tal R-charge and thus nilpotent correlation func-
tions of scalar chiral superfields, the Ward iden-
4. Chiral superfields
tities seem to be rather restrictive: It was shown
Superfields are operator-valued distributions in [26], that the total R-charge of the three point
on superspace. They form representations of the function has to be 1.
superconformalgroupand haveas quantum num- Here the nilpotent case will not be discussed.
bers the Lorentz spin, (j1,j2), the scale dimen- Therefore we have for the R-charges κi of the su-
sion, η, and the R-charge, κ. Here only scalar perfields in a n-point function
chiral/anti-chiral superfields are discussed, i.e.
n
j1 = j2 = 0 and the restrictions (5.1) κi = 0.
(4.1) D¯ Φ xµ,θ,θ¯ = 0 i=1
+ + X
for chiral fields, Φ, an(cid:0)d (cid:1) Ingeneralthescalarn-pointfunctioninsupercon-
formalfieldtheoryisafunctionofallinvariantsof
(4.2) D Φ¯ xµ,θ,θ¯ = 0
− − n points times a factor dueto the superconformal
for anti-chiral fields,(cid:0) Φ¯. He(cid:1)re chiral and anti- transformations of the n fields (cf. [34]):
chiral coordinates and the corresponding covari-
F (n-point invariants)
ant derivatives (eqns. (2.10) and (2.11)) are al- (5.2) S ...S = ,
ready used, which is very convenient, as these h 1 ni n x2 ∆lm
l,m=1;l6=m ¯lm
Q
6 Holger Knuth
where and a constant C:
n
1 1
(5.3) ∆ = η (5.4) Φ¯ xµ ,θ¯ Φ xµ ,θ = C .
lm −2(n 1)(n 2) i 1 1− 1 2 2+ 2 (x2 )η
− − i=1 ¯12
X (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11)
1 3
From (2.12) one easily verifies the conditions on
+ (η +η )+ (κ κ ) .
l m l m
2(n 2) 2n − the two-point function given by the restrictions
−
Here we are interested in chiral scalar super- (4.1) and (4.2),
fields. As these depend only on the chiral vari-
(5.5) D Φ¯ Φ = 0 Φ¯ D¯ Φ = 0 .
ables, one immediately sees, that there is no way 1− 1 2 1 2+ 2
to construct a three point invariant with only The th(cid:10)ree-point f(cid:11)unction is(cid:10)also deter(cid:11)mined up to
half the Graßmann variables. Therefore the three a constant just like the ordinary conformal three-
point function can be easily written down in the point function:
next section, as it was already done in e.g. [26].
A four point function with vanishing total R- Φ¯1 xµ1−,θ¯1 Φ2 xµ2+,θ2 Φ3 xµ3+,θ3
charge depends on two chiral and two anti-chiral C
variables (here (x1−,θ¯1), (x2−,θ¯2), (x3+,θ¯3) and (5.6(cid:10)) (cid:0) (cid:1) (cid:0) (cid:1) =(cid:0)x2η2x2η3(cid:1)(cid:11).
(x ,θ¯ )), as will be seen in section 5.2. This ¯12 ¯13
4+ 4
leads to the problem, that there is only one su- Note, that η1 = η2+η3 for these threepoint func-
perconformal cross ratio given by eq. (3.10) tions because of eqns. (4.5) and (5.1).
depending only on these four variables, namely As the strategy for the four-point function will
. But there have to be two independent non- be to look at all the invariants and then imple-
2
I
nilpotent superconformal four-point invariants, ment the chirality conditions, the following short
because there are two conformal four-point in- calculationshows,howthechiralthreepointfunc-
variants, on which the four-point functions of the tion fits into the form of the most general three
component fields depend. One possibility is the point function (cf. eq. (5.2)),
construction of a trace invariant as the second
Φ¯ xµ ,θ¯ Φ xµ ,θ Φ xµ ,θ
non-nilpotent invariant (cf. [26]). 1 1− 1 2 2+ 2 3 3+ 3
2,Hearnedtwthoessuepterocfonnfiolprmotaelntcroinsvsarraiatniotss, II1fraonmd (5.7(cid:10)) (cid:0) (cid:1) =(cid:0) 3f3P(cid:1)F (I(cid:0)3PxF2)∆lm(cid:1)(cid:11),
I l,m=1;l6=m ¯lm
eq. (3.30) will be used to get an expression for
the four-point functions. This will simplify con- which contains an arbQitrary function of the three-
clusions from properties of global conformal field point invariant, I3PF. The restrictions (4.1) and
theories to global superconformal field theories. (4.2) and so the differential equations,
With the help of the chirality conditions ap-
(5.8)D Φ¯ Φ Φ = 0 , D¯ Φ¯ Φ Φ = 0 ,
plied to the four-point function, the dependence 1− 1 2 3 2+ 1 2 3
on all these invariants can bereduced to a depen- (5.9)D¯3+(cid:10)Φ¯1Φ2Φ3(cid:11) = 0, (cid:10) (cid:11)
denceonlyontwocrossratioswithafixeduniver-
have to b(cid:10)e satisfied(cid:11). The solution gets obvious,
sal differential operator applied to the whole rest
when the denominator is rewritten with the help
of the correlation function. This rest contains the
of the identities relating the ∆ with each other:
model specific information. lm
For rational four-point functions this differen- 1
(5.10)
tial operator can be applied to their power series. 3 x2 ∆lm
The coefficients of this power series turn out to l,m=1;l6=m ¯lm
be the same as those of the conformal four-point Q J1∆21
= .
function of the scalar fields, which are the lowest x2η2x2η3
¯12 ¯13
components of the chiral and anti-chiral fields.
The derivatives only vanish, if the whole does not
5.1. Two and three-point functions. As depend on I any more. So with eq. (3.4) the
3PF
there are no two-point invariants, the two-point function f (I ) is
3PF 3PF
function is solely determined by the superconfor-
mal transformations properties of the two fields (5.11) f3PF (I3PF) = (I3PF +1)−∆21 .
On invariantsandscalar chiralcorrelation functionsin N =1SCFT 7
5.2. Four-point functions. First of all the gen- the conformal four-point function. The cross ra-
eral form of the scalar four-point function given tios, and , reduce to conformal cross ratios
1 2
by eq. (5.2) for n = 4 can be rewritten with the and aIll I vIanish. Thus the function f is the
i,j 0,1
help of , , J , J and J : function appearing in the scalar four-point func-
1 2 1 1(1) 1(2)
I I
tion of conformal field theory.
(5.12) S ...S =
h 1 4i In order to see, what consequences the restric-
1Λ34 2Λ24J1∆43−∆31J1(1)−∆43−∆41 tive conditions defining chiral and anti-chiral su-
I I
x 2 Σ1 x 2 Σ2 x 2 Σ3 perfields have on the scalar four-point functions,
J ∆43 ¯12 ¯23 ¯24 eq. (5.18), and to get a general expression of the
1(2) x 2 x 2 x 2
(cid:18) ¯21 (cid:19) (cid:18) ¯32 (cid:19) (cid:18) ¯42 (cid:19) chiral/anti-chiral scalar four-point function with
x¯232 Ξ F ( 1, 2, I111,...,I222) vanishing total R-charge, the following four dif-
I I
(cid:18)x¯122x¯132(cid:19) x¯132(η3−η2)x¯142η4x¯232η2 ferential equations have to be solved:
with (5.20) D Φ¯Φ¯ΦΦ = 0 ,
1−
(5.13)Λ = ∆ +∆ D Φ¯Φ¯ΦΦ = 0 ,
lm lm ml 2−(cid:10) (cid:11)
1 4 1 D¯3+(cid:10)Φ¯Φ¯ΦΦ(cid:11) = 0 ,
= η + (η +η ) ,
−6 i 2 l m D¯ Φ¯Φ¯ΦΦ = 0 .
i=1 4+(cid:10) (cid:11)
X
1 3
A four-point func(cid:10)tion wit(cid:11)h only one anti-chiral
(5.14) Σ = ∆ +∆ +∆ = η κ ,
1 21 31 41 1 1
2 − 2 and three chiral fields (or vice versa) with van-
1 3
ishing total R-charge can be excluded with the
(5.15) Σ = ∆ +∆ +∆ = η κ ,
2 31 32 34 3 3
2 − 2 corresponding differential equations already with
1 3
lowest order calculations.
(5.16) Σ = ∆ +∆ +∆ = η κ ,
3 41 42 43 2 4− 2 4 Using z = xµ ,θ¯ and z = xµ ,θ , the
i− i− i i+ i+ i
1
following theorem states the solution of the dif-
(5.17) Ξ = (η +η η η ) ,
1 2 3 4 (cid:0) (cid:1) (cid:0) (cid:1)
2 − − ferential equations.
whereeq. (5.1)isusedtocalculateΣ ,Σ andΣ .
1 2 3 Theorem. All chiral/anti-chiral scalar four-
Because the invariant prefactors pulled out can
point functions with vanishing total R-charge are
be expressed in terms of the invariants, which are
of the form
the arguments of F, we can define a new function
f4PF, so that (5.21) Φ¯1(z1−)Φ¯2(z2−)Φ3(z3+)Φ4(z4+)
(5.18) (cid:10) =eD f0,1(I1,I2) (cid:11)
x 2 Σ1 x 2 Σ2 x 2 Σ3 x¯132(η3−η2)x¯142η4x¯232η2
hS1...S4i=(cid:18)x¯1¯2212(cid:19) (cid:18)x¯2¯3322(cid:19) (cid:18)x¯2¯4422(cid:19) wpainthsiothneozferfoth howrdiethr rceosepffiecctietnot,thfe0,1ni,ilpoofttehnet einx--
4PF
x¯232 Ξ f4PF (I1, I2, I111,...,I222) . variants (eq. (5.19)) and a fixed, universal differ-
(cid:18)x¯122x¯132(cid:19) x¯132(η3−η2)x¯142η4x¯232η2 ential operator
For four-point functions of anti-chiral and chiral ∂
(5.22) = 4i T ( , , I ,...,I ) .
superfields the four exponents Σ1, Σ2, Σ3 and Ξ D I1 I2 111 222 I1∂ 1
vanish, as is seen below leading to eq. (5.23). I
The fixed function T is nilpotent and will be
In general the function f has the expansion
4PF
given in the course of the proof in eq. (5.34).
4 ni The exponential with the derivatives has to be
(5.19) f = f ( , )I
4PF i,j I1 I2 i,j understood in form of its Taylor expansion with
Xi=0Xj=1 onlyfinitelymanytermsbecauseofthenilpotency
with respect to the nilpotent invariants (3.30), of T.
where the coefficients are functions of the two Sketch of the proof: The starting point is the gen-
cross ratios from eq. (3.13). eral scalar four point function. The chiral/anti-
If the Graßmann variables are set to zero, the chiral scalar four-point function has to be of the
superconformal four-point function passes into same form, eq. (5.18).
8 Holger Knuth
However, scaledimensionsandR-chargesofthe of thenilpotent invariants, which simplifiescalcu-
chiral and anti-chiral superfields fulfill the eqns. lations in lowest order. I is a combination of all
Σ
(4.5) and (5.1). Consequently Σ = Σ = Σ = eight nilpotent invariants
1 2 3
Ξ = 0 (eqns. (5.14)-(5.17)). Thus the four-point Θ˜ X˜ Θ˜¯
function reduces to (5.28) I = 1 1+ 1
Σ
(5.23) X2 X2
1(1)+ 1(1)−
f ( , , I ,...,I )
Φ¯1Φ¯2Φ3Φ4 = 4PxF¯132I(η13−Iη22)x¯114121η4x¯232η2222 = +−qII111+II112+II121−+II122
T(cid:10)he derivat(cid:11)ives in the differential equations 211− 212− 221 222
+4i(2I2 2I I
(5.20)appliedtothedenominatorofthis formare 112 − 111 112
2I I +I I +I I )
zero. Thusforthedifferentialequations tobesat- 112 122 111 122 112 121
−
isfied the derivatives of the function f4PF have to −4I1211I222−8I1212I121 ,
vanish. These derivatives can be evaluated using
which is I with the indices 4 replaced by 3 and
111
the expansion (5.19):
is simply related to the three-point invariant in
3 ni ∂f eq. (3.9).
(5.24) Df4PF = i,j (D 1)Ii,j Eq. (5.26) shows exactly, what restrictions fol-
∂ I
1
i=0 j=1 I low from the lowest non-vanishing (i.e. first) or-
XX
4 nk der of the differential equations (5.20). If one
+ fk,lDIk,l looks beyond first order of Graßmann variables,
k=1l=1 one simplifies matters even more going over to a
XX
with D D ,D ,D¯ ,D¯ . Thesumover i newset ofthreenilpotent invariants replacingthe
1− 2− 3+ 4+
ends at∈i= 3 because the product of I and the three appearing in eq. (5.27):
4,1
(cid:8) (cid:9)
derivative of is zero due to their nilpotency. 2
I1 (5.29) T = I4 I 2i I4 I2
Thissuggeststolookatthelowestapriorinon- 222 J 222− J 222
vanishing order, θ resp. θ¯, because it clarifies the 1(1) 1(1)
= ρ +ρ +ρ
relation between the function f and the func- −p14 13 34
0,1 +2i ρ 2 ρ 2 ρ 2 ,
tions f : In this lowest order the term 14 13 34
1,l − −
I I2
(5.25) ∂∂f0,1 (DI1)+ 8 f1,lDI1,l (5.30) TΣ = I5 (cid:0)ΣJ1(1) −2iI52JΣ1(1) (cid:1)
I1 Xl=1 =−ρp13+ρ12+ρ23
has to vanish. +2i ρ 2 ρ 2 ρ 2 ,
13 12 23
− −
So the problem breaks down to the cancelation
(5.31)
(cid:0) (cid:1)
of the derivatives of the four-point invariants in
2
lowest order and one finds, that all f1,l are fully T212 = IJ6 I212+2iJI6 I2212
determined by f0,1. 1(1) 1(1)
To this order the differential equation is satis- =−pρ14−θ34x˜−¯341θ¯24−θ31x˜−¯211θ¯21
fied by
+θ x˜−1x˜ x˜−1θ¯
34 ¯34 ¯23 ¯21 21
∂
(5.26) f4PF = (cid:18)1+4iT′I1∂I1(cid:19)f0,1(I1,I2) +2i[ρ214 + θ34x˜−¯341θ¯24 2
+ θθ¯ 2 , − θ31x˜−¯211θ¯(cid:0)21 2+ θ34x˜−¯3(cid:1)41x˜¯23x˜−¯211θ¯21 2
O
where (cid:16)(cid:0) (cid:1) (cid:17) −(cid:0)2 θ34x˜−¯341θ¯(cid:1)24−(cid:0)θ31x˜−¯211θ¯21−ρ34 (cid:1)
(5.27) θ34x˜(cid:0)−¯341x˜¯23x˜−¯211θ¯21−2θ34x˜−¯341θ¯24ρ34(cid:1)]
T′ = I4 I222+ 1 IΣ I6 I212 +16θ34x˜−¯341x˜¯23x˜−¯211θ¯21θ31x˜−¯211θ¯21
J J − J
1(1) I5 1(1) 1(1) ρ θ x˜−1θ¯
34− 34 ¯34 24
is a fixepd function of fopur-point invapriants. The
with (cid:0) (cid:1)
factors with cross ratios (eqns. (3.14)-(3.17)) and
J1(1) (eq.(3.12)) just eliminate overall prefactors (5.32) ρij = θijx˜¯−ij1θ¯ij .
On invariantsandscalar chiralcorrelation functionsin N =1SCFT 9
They have especially simple expressions in terms As this is the expansion of the exponential in
of the variables θ , θ¯ and x˜ and differ only in eq. (5.21), this finishes the proof. The depen-
i j ¯ij
secondandhigherordersfromthoseineq. (5.27). dence on nilpotent invariants has been totally
As a side remark, in analogy T can be de- fixed and the superconformal four-point function
111
fined, which even is exactly again I , of scalar chiral and anti-chiral superfields is left
111
with no more degrees of freedom than the scalar
(5.33) T = I
111 111
conformal four point function.
I I2
= 111 2i 111 The four point function in eq. (5.21) is not
J1(1) − J1(1) explicitly given in terms of the chiral and anti-
= ρ +ρ +ρ chiral coordinates z , z , z and z . and
+−p2i14ρ 212 ρ 224 ρ 2 . depend on θ2 an1d−θ¯3.2−The3s+e terms4,+ofIc1ause,
14 12 24 D
− − cancel.
Consequently T′ is (cid:0)replaced by (cid:1) Rational four-point functions: In the remaining
(5.34) T = T +T T part of this section the case of rational four-point
222 Σ 212
−
functions is discussed, which will lead to an ex-
which still solves the differential equations to first
pressioncontaining onlyz ,z , z andz . It
order. Butnowfurthertermsinhigherordervan- 1− 2− 3+ 4+
was shown in [35], that in global conformal field
ish conveniently, because
theory all correlation functions are rational. As
(5.35) D 1 = 4i 1DT . a consequence of global conformal symmetry this
I − I
The term in (5.25) vanishes completely. Thus is also the case for supersymmetric theories with
there are two summands less in eq. (5.24). The this space-time symmetry.
first order result and eq. (5.35) can be plugged In rational four-point functions the function
in, so that f0,1 is a power series:
(5.36) Df4PF = 16I1∂∂ I1∂∂f0,1 TDT (5.38) f0,1(I1, I2)= f0,1,k1,k2I1k1I2k2 .
I1 (cid:18) I1 (cid:19) kX1,k2
3 ni ∂f 4 nk
i,j (4i DT)I + f DI The differential operator applied to this leads to
1 i,j k,l k,l
− ∂ 1 I (5.39)
i=2 j=1 I k=2l=1
XX XX eDf ( , )= f e4iT k1 k1 k2 .
Note, thatthecoefficientfunctionsfk,l withk 2 0,1 I1 I2 0,1,k1,k2 I1 I2
have been redefined corresponding to the step≥, in kX1,k2
which the nilpotent invariants were replaced. Thisisrewrittennowasafunctionofchiraland
Looking at the lowest order left here ( θθ¯ θ anti-chiral variables. Thus only θ , θ and x
34 21 ¯ij
resp. θθ¯ θ¯) and using the productrulefor I it with i = 1,2 and j = 3,4 may appear. Because of
(cid:0)2,l(cid:1)
gets clear, that the same derivatives of four-point the exponential function in (5.39) it is useful to
(cid:0) (cid:1)
invariants need to cancel and thus the result can express x , x and x by the ”allowed“ inter-
¯12 ¯34 ¯32
again only depend on T. vals times an exponential function with nilpotent
This way it is possible to proceed iteratively exponent. One finds, that
order by order. The resulting function f is
4PF (5.40) x2 = xµ xµ 2
∂ ¯12 ¯13− ¯23
(5.37) f4PF = (cid:20)1+4iTI1∂I1 ·(cid:0)e4i(cid:16)θ23x˜−¯121θ¯(cid:1)21−2i(θ23x˜−¯121θ¯21)2(cid:17) ,
−8T2 I1∂∂ I1∂∂ (5.41) x2¯34 = xµ¯13−xµ¯14 2
32 (cid:18) ∂I1 (cid:18)3 I1(cid:19)(cid:19) (cid:0)e4i(cid:16)θ43x˜−¯341θ¯(cid:1)13−2i(θ43x˜−¯341θ¯13)2(cid:17) ,
·
i
− 3 I1∂ (5.42) x2 =x2 e−4i(ρ23−2iρ232)
(cid:18) I1(cid:19) ¯32 ¯23
32 ∂ 4 using ρ from eq. (5.32).
+ f ( , ) 23
1 0,1 1 2
3 I ∂ I I This is inserted into the cross ratio . The
(cid:18) I1(cid:19) # I1
exponents in eqns. (5.40)-(5.42) multiplied with
+ θθ¯ 2 . k cancel most of the exponent in eq. (5.39). The
O 1
(cid:16) (cid:17)
(cid:0) (cid:1)
10 Holger Knuth
following four-point function remains: Especially interesting to transfer to supercon-
(5.43) Φ¯ (z )Φ¯ (z )Φ (z )Φ (z ) formal theories are the results for global con-
1 1− 2 2− 3 3+ 4 4+
formal invariance aiming at the rigorous con-
(cid:10) = Pk1,k2f0,1,k1,k2I1′k1I2k2 . (cid:11) struction of a non-trivial 4D quantum field the-
x¯132(η3−η2)x¯142η4x¯232η2 ory in the setting of Wightman axioms. Both
with
four-point functions of scalar chiral and anti-
(5.44) ′ = e4i(γ−2iγ2) x2¯14x2¯23 chiralsuperfieldsand four-pointfunctionsoftheir
I1 xµ xµ 2 xµ xµ 2 lowest component fields have the same coeffi-
¯13− ¯23 ¯13− ¯14 cients f of their respective power series
and 0,1,k1,k2
(cid:0) (cid:1) (cid:0) (cid:1) (cf. (5.43)). Pole bounds [35] and an exact par-
(5.45) tial wave expansion computed for the scalar four-
θ x x˜ x θ¯ point function [36] are also valid for lowest com-
γ = 34 ¯14 ¯13 ¯23 21 ponent fields of chiral and anti-chiral superfields
xµ xµ 2 xµ xµ 2
¯13− ¯23 ¯13− ¯14 in a global superconformal theory, i.e. they give
(cid:0)x¯132 θ34x(cid:1)¯23θ¯(cid:0)21+θ34x¯1(cid:1)4θ¯21−θ34x¯13θ¯21 . further restrictions on the coefficients f0,1,k1,k2.
− (cid:0) xµ¯13−xµ¯23 2 xµ¯13−xµ¯14 2 (cid:1) Cmoanlsjeuqsuteanstlyforthgelsoebaalresurpeseurcltosnffoorrmgalolbfaolucr-opnofoinrt-
Setting the Grassmann variables to zero, the ex-
(cid:0) (cid:1) (cid:0) (cid:1) functions.
ponent in (5.44) vanishes and thus I′ passes into
1
a conformal cross ratio just like I . Hence the co- Remark: In a very recent paper [37] the four-
2
efficients f are also those of the four-point point function of two chiral and two anti-chiral
0,1,k1,k2
functionofthelowest componentfields,whichare superfields all having the same scaling dimension
scalar global conformal fields. are given using a trace invariant. This trace in-
variant is given by
6. Conclusion
(6.1) tr x −1x x −1x = 1 ′−1+ −1
¯23 ¯13 ¯14 ¯24 −I1 I2
The proof of a general formula for the four-
point function of chiral and anti-chiral superfields with I1′ a(cid:0)nd I2 from eqns.(cid:1)(5.44) and (3.13), re-
spectively. If this invariant is used and the ra-
withvanishingtotalR-chargewassketchedinthis
tional four-point function is written as a power
paper, whilea longversion will beavailable in my
series as in eq. (5.43), the coefficients would be
Ph.D.-thesis. In this formula the exponential of
resorted in comparison to eq. (5.43) and the step
a universal nilpotent differential operator is ap-
from global conformal to global superconformal
plied to an arbitrary function of two supercon-
four-point function would be not so direct.
formal cross ratios. This function will depend on
the specific model. In the context of the proof
References
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N [2] V. V. Molotkov, S. G. Petrova, and D. Ts. Stoy-
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