a/c Theorem
Old and New Results
Hugh Osborn
!
January 26th 2015
Since Wilson we know that QFTs should be considered
as belonging to a space of QFTs with differing couplings
RG flow provides a topology on the space
of quantum field theories
•
• Is RG flow always between isolated fixed
points?
• Are fixed points CFTs?
• Is more exotic behaviour such as limit
cycles possible?
• Is RG flow gradient flow?
• Is Unitarity crucial for these properties?
Possible RG flows
UV, IR
fixed
points
UV fixed
point, IR
limit cycle
UV fixed
point, IR
chaos
Most of of our understanding of RG flow is
based on fixed points, but limit cycles can
exist in non unitary theories
Zamolodchikov c-theorem in 2 dimensions
provides strong constraints on RG flow
Based just on em tensor conservation and unitarity
Tµ⌫ ! Tzz , Tzz̄ = Tz̄z , Tz̄z̄
Let
C(µ x ) = 2z hTzz (x) Tzz (0)i
2 2
4
H(µ x ) = z x hTzz̄ (x) Tzz (0)i
2 2
2 2
G(µ x ) = x x hTzz̄ (x) Tzz̄ (0)i > 0
2 2
x = z z̄
@z̄ Tzz + @z Tz̄z = 0
2
t=
2 2
1
2
d
C̃ =
dt
ln µ2 x2
I
C̃ = C
@
C̃ =
I
@g
12G =
⇥=
8H
1
4 Tz̄z
=
I
J
<0
12G
3
GIJ
4
I
OI
Crucial properties
C̃ =
Virasoro central charge at fixed points
C̃(0) = cU V
C̃(1) = cIR
cU V > cIR
Zamolodchikov metric
GIJ = x2 x2 hOI (x) OJ (0)i
positive for unitary theories
Basic eqs for a-theorem
d I
g =
dt
I
(g)
Fixed point
t
I
increases to the IR
(g⇤ ) = 0
a-theorem (minimal)
aU V
aIR > 0
d
a-theorem (strong) ã(g) = GIJ (g)
dt
GIJ is a positive metric on couplings
I
(g)
J
(g)
ã(g) is stationary
at a fixed point
ã(g) = a(g) + O( )
gradient flow
@
ã(g)
=
T
(g)
IJ
@g I
J
(g)
T(IJ) = GIJ gradient flow requires integrability
conditions on beta functions
Cardy proposal in 4 dimensions
For a CFT on curved space
µ⌫
2
hTµ⌫ i = c Weyl tensor
a Euler density
For conformally flat spaces, e.g. a sphere, the Weyl
tensor vanishes
Free theories
c = 12nV + nS + 3nF
a=
1
3
62nV + nS +
11
2 nF
On flat space for a CFT
hTµ⌫ T ⇢ i / c
hTµ⌫ T
⇢
c>0
T↵ i / A c + B a + C
In simple versions of AdS/CFT
a=c
Is
?
a>0
First proper argument given by
Hofman and Maldacena in 2008
Look at energy flux at infinity
E(n) = lim
r!1
Z
1
1
dt ni T 0 i (t, rn)
h✏ij Tij E(n) ✏kl Tkl i ⇠ c 1 + t2 Y2 (n) + t4 Y4 (n)
0
Requiring positivity in all directions
restricts t2 , t4 to a triangular region, which implies a > 0
t4
(0, 1)
(
5 1
3, 6)
a=
pure
vector
pure scalar a = 1 c
3
31
18 c
( 53 , 13 )
pure fermion
a = 11
18 c
t2
scalars, vectors give the extreme values of a/c
for Susy
t4 = 0
Derivation of 4 dim RG eqs
Consider Weyl rescalings
Couplings
g (x)
I
µ⌫
µ⌫ (x)
! e2
µ⌫
µ⌫
W vacuum functional
E =R
Z
Z
µ⌫
I
2
W
=
W
µ⌫
gI
Z ⇣
2
c Weyl tensor
+
Z
@µ
µ⌫
1 µ⌫
2g R
Einstein
tensor
a Euler density
⌘
µ⌫
I
J
1
2 GIJ E @µ g @⌫ g + . . .
⇣
⌘
E µ⌫ wI @⌫ g I + . . .
Defines
as well as
GIJ , wI
c, a
There are crucial integrability conditions by commuting Weyl rescalings for different
@I a = GIJ
L wI =
J
J
L wI
@J wI + @I
J
wJ
If @g = 0 then the equation is equivalent to
µ⌫
hTµ⌫ i =
I
2
hOI i + c Weyl tensor
a Euler density
In two dimensions there are very similar eqs
2
Z
µ⌫
W
=
µ⌫
Z
+
I
Z
Z
⇣
@µ
gI
W
µ⌫
I
J
1
@µ g @⌫ g
2 GIJ
1
2c R
⇣
µ⌫
Consistency @I c = GIJ
L wI =
J
wI @⌫ g + . . .
I
J
L wI
@J wI + @I
J
⌘
wJ
This is equivalent to Zamolodchikov’s eqs
+ ...
⌘
Let
ã = a + wI
I
Then
@I ã = TIJ
I
@I ã = GIJ
ã = a
TIJ = GIJ + @I wJ
J
I
@J wI
J
at fixed points
Ambiguities
GIJ ⇠ GIJ + L DIJ
ã ⇠ ã + DIJ
I
J
wI ⇠ wI + DIJ
J
For irreversible RG flow require
GIJ
positive
In two dimensions
✓
@
+L
@t
◆
hOI (x) OJ (0)i = GIJ @
2 2
From this it follows that
GIJ ⇠ (x ) hOI (x) OJ (0)i > 0
2 2
equivalent to Zamolodchikov metric
(x)
In four dimensions at a fixed point
1
hOI (x) OJ (0)i = gIJ R 2 4
(x )
⇣
⌘
1
2 3 1
2 2
=
g
(@
)
ln
µ
x
IJ
3 ⇥ 44
x2
@
2 2 4
µ hOI (x) OJ (0)i / gIJ (@ ) (x)
@µ
GIJ / gIJ > 0
Away from a fixed point GIJ and gIJ differ
In four dimensions for general renormalisable
field theories
with gauge couplings g Yukawa couplings Y
quartic scalar couplings
using perturbation theory
2
dg
GIJ dg I dg J = nV 2
g
+ dY 2 + d
1 loop 2 loops
2
3 loops
Integrability gives constraints on beta functions
(1)
=
4
Y ,
(2)
Y
=
Y
3
Possible proof of a-theorem in 4 dimensions
Komargodski & Schwimmer 2011,
Luty, Polchinski & Rattazzi 2012
Based on constructing effective Lagrangians for the dilaton. This couples to the trace of the energy momentum tensor. In a CFT if conformal symmetry is spontaneously broken
then the dilaton is a physical Goldstone boson.
The construction assumes a lot of folklore
about effective Lagrangians and the role of
anomalies.
pµ p ⌫
h. . . T̃µ⌫ (p) . . . i ⇠ 2 h. . . ⌧ (p) . . . i
p
p2 !0
The pole determines a non zero amplitude
in a CFT for a massless dilation
⌧
!
By assumption the CFT is invariant under Weyl rescaling of the metric and a shift of the
dilaton field
W [e
2
µ⌫ , ⌧
+ ] = W[
µ⌫ , ⌧ ]
A trivial solution is one which depends only on
˜µ⌫ = e
2⌧
µ⌫
but W should be invariant under diffeomorphisms
˜µ⌫ ! R̃↵
W[
µ⌫ , ⌧ ]
= W [˜µ⌫ ] +
Z
⇣
4
d x
p
2
Lanomaly = ⌧ c Weyl tensor
⇣
4a E µ⌫ @µ ⌧ @⌫ ⌧
µ
Lanomaly
a Euler density
2
@ ⌧ @µ ⌧ r ⌧ +
1
2
µ
@ ⌧ @µ ⌧
This can be used to construct a low
energy effective action for ⌧
⌘
2
⌘
If there are couplings g to operators with
(4
)⌧
g
dimension
then g ! e
The trace anomaly generates contributions
involving ⌧ on flat space
µ
2
Lanomaly = a 4 @ ⌧ @µ ⌧ r ⌧
µ
2 (@ ⌧ @µ ⌧ )
2
A effective kinetic term is may be constructed
'
⌧
R̃
from
e =1
Lkinetic =
1 2
2f
e
2⌧
µ
@ ⌧ @µ ⌧ =
1
2
µ
@ '@µ '
f
On shell r2 ⌧ = @ µ ⌧ @ ⌧ or r2 ' = 0
µ
µ
2
2
2
Lanomaly ! 2a (@ ⌧ @µ ⌧ ) = 2a r ⌧ r ⌧
a
2 2
2 2
= 4 r ' r ' + ...
2f
This determines the leading low energy E ⌧ f
contribution to the dilaton scattering amplitude
a
A(s, t) ⇠ 4 h p3 , p4 |(r2 '2 )2 |p1 , p2 i
2f
2a 2
2
2
= 4 (s + t + u )
f
Other contributions to the effective action are less important or vanish
on shell
At a fixed point the dilaton is formal and
need not represent additional degrees of
freedom, a physical dilaton represents SSB
for scale invariance
Away from a fixed point such dilatons become massive
Assume Ldilaton is defined along RG flow
from UV to IR
Crucial step in KS and LPR is in the effective
a ! aU V aIR
dilaton action to take
Two arguments: Anomaly matching of UV and IR fixed points,
Removes the singular high energy behaviour of
the dilaton theory
Positivity of aU V aIR follows by assuming
an unsubtracted dispersion relation for
the forward dilaton-dilaton scattering amplitude A(s, 0) = A( s, 0)
4 2
A(s, 0) ⇠ (aU V aIR ) 4 s + O(s4 )
f
4 Z 1
f
ds
aU V aIR =
Im A(s, 0)
3
2⇡ 0 s
Im A(s, 0) = s (s)
0
Positivity obtains from positivity of
the absorptive part of dilaton-dilaton
forward scattering (unitarity)
Perturbatively we expect
1
(s) ⇠ 4 (g(s))2 s
f
or
1
(s) ⇠ 4 GIJ
f
I
(g(s))
J
(g(s)) s
but this is not straightforward to show
the analysis is significantly more intricate
than in two dimensions
It is not clear how to to carry through
perturbative calculations beyond lowest
order
Generalisations to higher dimensions
!
Weyl anomalies are present in any even
dimension
In six dimensions there are three terms
constructed from the Weyl tensor, as well
as the six dimensional Euler density with
coefficient a
No good argument for
a > 0
The KS argument requires analysis of
3 to 3 dilaton scattering and positivity
of aU V aIR is problematic
3
Perturbatively the metric is negative in
,
but this is rather unphysical
AdS/CFT
!
This provides an alternative route to an
a-theorem if a CFT has an AdS dual
Identify the radial direction away from the boundary with the RG scale
Construct a scalar from the metric which
is monotonic under radial evolution subject
to a positivity condition on the bulk energy momentum tensor
This works in any dimension
In odd dimensions an analogue of
a
can be found by considering contributions
to the partition function on a sphere
There is an analogue of a weak version
of the a theorem, but no strong version
in which a is stationary at a fixed point
The theorem is doubtless true for unitary
theories, at least in even dimensions, but the derivation, and the necessary assumptions, is still rather murky
In general beta functions are not unique
if
@ ¯ Z @ ! @ ¯0 @
0
¯0 = ¯Z̄
Z = Z̄Z
0
but Z, Z̄ are not unique
Z̄ =
leads to
I
Z̄!
⇠
I
=Z
Z = !Z
+ (!g)I
⇠
+!
is the anomalous dimension matrix for
Conventionally
is hermitian but this
is not essential
Fortin et al showed that at three loops
there were solutions of
I
= (Sg)
I
which appeared to generate limit cycles
but
BI =
gives
I
Tµµ = 0
(Sg)I = 0
and hence CFT
but the anomalous dimension matrix
is then non hermitian and might have non real eigenvalues