1. ASPECTS OF THE CRITICAL RENORMALIZATION
GROUP IN HIGHER DIMENSIONS
R.M. SIMMS & J.A. GRACEY
UNIVERSITY OF LIVERPOOL, R.SIMMS@LIVERPOOL.AC.UK
ABSTRACT
The renormalization group can be used to understand conformal field theories beyond 2−dimensions. Working perturbatively at the Wilson-Fisher fixed point and utilizing known critical exponents in
2−dimensions, we can build a tower of connected theories through the dimensions. Studying higher dimensional theories could give an insight into physics beyond the Standard Model.
THE RENORMALIZATION GROUP
The idea of the Renormalization group (RG) results from comparing phenomena at different en-
ergy scales. In perturbation theory the d−dimensional β−function is
β(g) = µ
∂g
∂µ
= (d − D)g + Ag2
+ Bg3
+ Cg4
+ . . .
where D is the critical dimension. The non-trivial coupling where β(g∗
) = 0 is known as the
Wilson-Fisher (WF) fixed point. g∗
= 0 corresponds to the free field theory.
The RG functions evaluated at g∗
= 0 are termed critical exponents, they describe the behaviour
of physical quantities near continuous phase transitions. Other critical exponents can be found us-
ing scaling relations. An important property of RG flows are the IR (µ → 0) and UV (µ → ∞) limits.
In general, thermodynamic features of a system near a phase transition depend only on a small
number of variables, but are insensitive to details of the underlying microscopic properties of the
system. This coincidence of critical exponents for ostensibly quite different physical systems is
called universality - and is successfully explained by the RG.
Renormalization Group + 2−Dimensional CFT’s
Two theories are said to be in the same universality class if they have a common WF fixed
point which gives identical values of the critical exponents. Using the property of univer-
sality, along with exact 2−dimensional critical exponents, we can extract the behaviour
across several dimensions to build a tower of connected theories.
2−DIMENSIONAL CFT’S
As any analytic function mapping the complex plane to itself is conformal, the group of conformal
transformation in d = 2 is infinite. In a simple basis, conformal transformations are
z → w(z) =
az + b
cz + d
group abbreviated to SL(2)
Correlation functions are invariant under the SL(2) transformation
φ(reiθ
)φ(0) = r−2(h+¯h)
e−2iθ(h−¯h)
where h + ¯h = x is the scaling dimension of φ and h − ¯h is its spin. Scaling dimensions determine
critical exponents of the system. We can write a 2−dimensional CFT in Hilbert space in terms of
the Virasoro algebra
[Lm, Ln] = (m − n)Lm+n + (c/12)m(m2
− 1)δm,−n
where T(z) =
∞
n=−∞ z−n−2
Ln and ¯T(¯z) =
∞
n=−∞ z−n−2 ¯Ln. Realization of conformal sym-
metry in a given system is parametrized by a real number c, the coefficient of the trace anomaly.
Unitarity restricts the possible values of scaling dimensions to the simple list of rational num-
bers
c = 1 −
6
m(m + 1)
(m = 2, 3, 4, . . . ) h = hp,q(c) =
[(m + 1)p − mq]2
− 1
4m(m + 1)
for p = 1, 2, . . . , m − 1, q = 1, 2, . . . , p, c < 1 and h > 0.
Example: The Ising model corresponds to m = 3. This gives c = 1/2 and therefore h = 0, 1/16, 1/2.
We have a finite number of scaling dimensions which give exact critical exponents.
BUILDING A UNIVERSAL THEORY IN d−DIMENSIONS
d
Universal Theory
Wilson-Fisher fixed point
d = 2
CFT’s
L(d) =1
2
∂µφi∂µφi
+1
2
g1σφiφi + f(σ, φi)
L(d=6) =1
2
∂µφi∂µφi
+1
2
∂µσ∂µσ+1
2
g1σφiφi
+1
6
g2σ3
L(d=4) =1
2
∂µφi∂µφi
+1
2
g1σφiφi + 1
2
σ2
L(d=2) = 1
2
∂µφi∂µφi
+1
2
g1σφiφi − 1
2
σ
β(6)(g)
β(4)(g)
g1
g1
Fixed point
equivalent in
4 < d < 6
d = 6
d = 4
UV Safe
IR Safe
Example: φ4
Theory
Universality states that theories in different dimensions are connected through a non-trivial fixed
point. Perturbative fixed points in higher dimensional theories could be linked to non-perturbative
fixed points in a lower dimension. These may then be access perturbatively using a higher dimen-
sional Lagrangian.
One can extract the behaviour across several dimensions by building increasingly higher dimen-
sional connected Lagrangian’s. Requirements to build a (d + 2)−dimensional scalar Lagrangian
from a d−dimensional scalar Lagrangian:
• Same symmetry group in both theories.
• Lagrangian is renormalizable in (d + 2)−dimensions.
• At least one connecting interaction in both Lagrangian’s (all other interactions are called spec-
tator interactions).
• Check: Calculate critical exponents in the large N expansion to ensure both theories are in the
same universality class.
One can think of this as having a universal theory in d−dimensions with an infinite set of (local) operators. These operators obey a symmetry and become relevant in the RG sense in the critical
dimension, otherwise they are irrelevant. At fixed dimensions we are therefore able to write down an exact Lagrangian. IR properties of one theory could be regarded as being driven by the UV
behaviour of another, as operators which are UV irrelevant may become IR relevant and dominate IR dynamics.
Example: Connection of O(N) φ3
theory with O(N) φ4
theory in 4 < d < 6 at the WF fixed point via the large N expansion. O(N) φ4
theory connected with NLσM in 2 < d < 4, see [1, 2].
Towers of higher dimensional gauge theories can also be looked at. However, these require additional conditions such as the invariance of gauge symmetry at different dimensions. Examples include
higher dimensional QCD, see [4].
O(N) × O(m) LANDAU-GINZBURG-WILSON MODEL IN d = 6
The 4−dimensional Lagrangian for the O(N) × O(m) LGW model is
L(d=4)
=
1
2
∂µ
φia
∂µφia
+
1
2
σ2
+
1
2
Tab
Tab
+
1
2
g1σφia
φia
+
1
2
g2Tab
φia
φib
The properties of L(d=4)
have been studied extensively, see [3]. The 6−dimensional Lagrangian
can be built from this and has been renormalized to three loops
L(d=6)
=
1
2
∂µ
φia
∂µφia
+
1
2
∂µσ∂µ
σ +
1
2
∂µ
Tab
∂µTab
+
1
2
g1σφia
φib
+
1
6
g2σ3
+
1
2
g3Tab
φia
φib
+
1
2
g4σTab
Tab
+
1
6
g5Tab
Tac
Tbc
Motivations:
• Non-perturbative conformal boostrap proved that there exists an IR non-trivial fixed point,
see [5]. We may be able to access this perturbatively from the 6−dimensional theory.
• O(N) × O(m) LGW model in d = 3 offers a theoretical prediction of the phase diagram in
frustrated spin models with non-collinear order. There is a long standing debate whether the
system shows a 1st
or 2nd
order phase transition. Proving the existence of a non-trivial fixed
point would give strong evidence for a 2nd
order phase transition.
• Also want to achieve a greater understanding of the UV/IR duality and universality class as
a whole.
PERTURBATIVE CALCULATIONS AND THE LARGE N EXPANSION
There are various techniques that can be used to calculate critical exponents. We use the large N
expansion to calculate exponents directly at the WF fixed point. At g∗
= g∗
( , N) we expand with
respect to 1/N rather than . N is a dimensionless parameter in d−dimensions, unlike the coupling
constant in dimensionally regularised perturbation theory.
g∗
= a11
N
+
∞
i=2
∞
j=1
aij
j
Ni
The d−dimensional large N technique was developed by Vasiliev et al. for the O(N) NLσM
in d = 2 + 2 dimensions. Comparing the 6−dimensional critical exponents with the known
4−dimensional exponents gives us a non-trivial check on our calculation.
We can also calculate the conformal window for the LGW model in d = 6. The conformal window
is the range of N valules for which the non-trivial fixed point exists. To obtain this we solve
βi(gj) = 0 and det
∂βi
∂gj
= 0
REFERENCES
[1] J.A. Gracey. Phys.Rev. D92 (2015) no.2, 025012.
[2] L. Fei, S. Giombi & I.R. Klebanov, Phys. Rev. D90 (2014), 025018.
[3] P. Calabrese & P. Parruccini, Nucl.Phys. B679 (2004) 568-596.
[4] J.A. Gracey, Phys.Rev. D93 (2016) no.2, 025025.
[5] Y. Nakayama & T. Ohtsuki, Phys.Rev. D89 (2014) no.12, 126009.
CONCLUSIONS
• Looking at higher dimensional theories has links to physics beyond the Standard Model.
• Interest in higher dimensional QFT’s due in part to non-trivial fixed points in 4−dimensional
theories.
• Other application of the critical RG include looking at asymptotic safety in quantum gravity and
the AdS/CFT correspondence.
• One question is whether there is a deeper connection of the Tarasov construction of relating d−
and (d + 2)−dimensional Feynman integrals and the corresponding field theories.