an a theorem in 6D? - uni-regensburg.demaa29312/conference/...the water-peptide interactions E hp...

an a-theorem in 6D?

Benjamín Grinsteinwith

Andreas Stergiou, David Stone and Ming Zong

CONFORMAL SYMMETRY IN SIX-DIMENSIONAL FIELD THEORIESRegensburg, July 14-16, 2014

Prologue

• Introduction (briefest)• c-theorem (2D), very thin

• original (Zamolodchikov)• a la Osborn

• a-theorem (4D), very minimal• Cardy’s proposal• a la Osborn

• Results in (6D), very rough• consistency conditions• sign of metric in loop expansion

• Epilogue

2

Introduction

3

Counting degrees of freedom

Textbook heat capacity (constant V) of diatomic gas

UVIR

4He

The temperature dependence of the heat capacity of hydration water near biosurfaces

Fig. 2: Deviations of the total potential energy Etot of thehydration water (open symbols) and of the water-water inter-actions energy Ehh+Ehb (closed symbols) from the bulkenergy Eb.

a sum of 3 components: energies of interaction withpeptide (Ehp), with bulk water molecules (Ehb) and withother water molecules in the hydration shell (Ehh). Thedeviations of Etot from the average interaction energy ofbulk water, Eb are shown in fig. 2 (open symbols). Atlow temperatures, the Etot near the hydrophilic peptideis close to Eb, whereas near the hydrophobic peptide itis slightly below Eb (by just ∼ 0.002Eb). Upon heating,Etot increases faster than Eb, making the potential energyof hydration water higher than that of the bulk due tothe unavoidable missing neighbor effect near any surface.When considering water-water interactions only, Ehh+Ehb, it is more negative near the hydrophobic peptide(closed symbols in fig. 2).Upon heating, the slope of the temperature dependences

of Etot shown in fig. 3 (upper panel) decreases, indicatinga reduction of the specific heat of hydration water. Wehave calculated Chp as

Chp = d [(Ehh+Ehb)/2+Ehp)] /dT + pV αp+3R, (1)

using numerical differentiation of the potential energy with±3-points smoothing (fig. 4). Interestingly, the tempera-ture dependence of Chp in both hydration shells is found

to be similar to that in the bulk. The Chp of hydra-

tion water exceeds the bulk value Cbp by ≈ 1.1 J/(molK)near hydrophilic peptide and by ≈ 5.3 J/(molK) nearhydrophobic peptide (see the inset of fig. 4). The lattervalue indicates that the surface of the hydrophobic peptideis close to that of the lysozyme and ribonuclease [17]. Thedifference Chp −Cbp obtained here for a biosurface is consid-erably smaller than that found in hydration shells of alka-nes and other non-polar solvents [19,20]. It is to be notedthat the water-peptide interactions practically do not

Fig. 3: The total potential energy Etot of water molecules inthe hydration shell (upper panel) and the potential energy ofthe water-peptide interactions Ehp (lower panel) as a functionof temperature. Vertical dotted and dashed lines denote themidpoints of the percolation transition of hydration water nearhydrophobic and hydrophilic peptides, respectively.

Fig. 4: The temperature dependence of the heat capacity Cpof water in the bulk and in the hydration shells of the twopeptides. The differences Chp −Cbp and their average values areshown in the inset by symbols and dashed lines, respectively.

contribute to the Cp of hydration water, because Ehp doesnot change notably with temperature (fig. 3, lower panel).The Cp data shown in fig. 4 were used for the esti-

mation of the constant volume heat capacity Cv =Cp−Tαp(dp/dT )V . The thermal expansivity of hydration

36001-p3

EPL, 90 (2010) 36001 doi: 10.1209/0295-5075/90/36001

4

A.B. Zamolodchikov, Pis'ma Zh. Eksp. Teor. Fiz., 43 (1986), 565 (Sov. Phys. JETP Lett., 43 (1986), 730)

Review: c-theorem

In 2D-QFT, c(g) ≥ 0 exists such that

1. Monotonic flow:

Equality only at fixed (critical) points.

2. Stationary at fixed points:

3. At fixed point (a CFT) c equals the central charge of the CFT

dc

dt= �i@ic 0

�i(g⇤) = 0 ! @ic|g⇤ = 0

#3 suggests c bares some relation to d.o.f, then #1 is in accord with intuition

(hereafter )@i =@

@gi

5

Z’s proof:

Oi = @iL

z = x

1 + ix

2

z = x

1 � ix

2

⇥ = �iOi

Central charge (#3):

Define and let

then std RGE:

give monotonicity (#1):

#2 is more complicated by Z’s methods (conformal pert theory) -- leave for later, simpler with Weyl CCs

T = Tzz, ⇥ = Tzz, where (trace anomaly):

C(g) = 2z4hT (x)T (0)i��x

2=⇤�2

Hi

(g) = z2x2hT (x)Oi

(0)i��x

2=⇤�2

Gij

(g) = x4hOi

(x)Oj

(0)i��x

2=⇤�2

c(g) = C + 4�iHi � 6�i�jGij

12�

i@iC = �3�iHi + �i�j@jHi + �j(@j�i)Hi

�j@jHi + (@i�j)Hj �Hi = �2�jGij + �j�k@kGij + �j(@i�

k)Gjk + �j(@j�k)Gik

�i@ic = �12�i�jGij

hT (x)T (0)i��g⇤

= z

�4c(g⇤)/2

6

Weyl Consistency ConditionsH. Osborn, Nuclear Physics B363 (1991) 486-526

2D QFT in curved background, �µ⌫(x)

Trace is now

�µ⌫Tµ⌫ = �iOi +

12�cR

View this as local RGE. First recast global RGE eW =

Z[d�]e�S0

W [e�2�(x)�µ⌫

, gi(e��(x)µ)] = W [�µ⌫

, gi(µ)] + local terms

✓µ

@

@µ+ 2

Zdv �µ⌫ �

��µ⌫

◆W = 0

To pick up operators Oi (rather than integrals): gi ! g

i(x)

✓�Zdv �(x)�

i �

�g

i+ 2

Zdv �(x)�

µ⌫ �

��

µ⌫

◆W =

Zdv �(x)

�dim 2 terms, from �µ⌫(x) and g

i(x)

�

Or, in integral form

need gi(x)Weyl !

dv = d

2x

p�

7

Weyl: or

Classify all dim 2 monomials (“operators”), diff invariant:

where�W

� ⌘ 2

Zdv ��µ⌫ �

��µ⌫��

� ⌘Zdv �(x)�i �

�g

iand

��W = (�W� ��

�)W = �Zdv �

�12�cR� 1

2�ij�µ⌫@µg

i@⌫gj�+

Zdv @µ�wi@

µgi

�c,�ij , wi• “beta” functions for local terms:

• Some freedom (choices made to simplify consistency conditions):

• Directly related to RGE of correlators (flat space, x-independent couplings). For example, 2-derivatives w.r.t. coupling

Note: correlator positive, but χij not obviously positive

Zdv @µ�wi@

µgi = �Zdv �

�wiD

2gi + @jwi�µ⌫@µg

i@⌫gj�

�ik�

jl

✓µ

@

@µ+ �m @

@gm

◆+ �i

k�jl + �ik�

jl

�hOi(x)Oj(0)i = �ij@

2@2�(2)(x)

�µ⌫

! e�2�(x)�µ⌫

��µ⌫ = �2�(x)�µ⌫

8

Good candidate for c-theorem! Need-positivity of “metric” χij

-relation to central charge-note that #2 is immediate!

Now, consistency conditions for Weyl transformations, a la Wess-Zumino

��W = (�W� ��

�)W = �Zdv �

�12�cR� 1

2�ij�µ⌫@µg

i@⌫gj�+

Zdv @µ�wi@

µgi

[��,��0 ]W = 0

recall:

@i�c = �ij�j � L�wi L�wi = �j@jwi + @i�

jwjwhere

�c = �c + wi�iRewrite in terms of @i�c = �ij�

j + (@iwj � @jwi)�j

dc

dt= �i@i�c = �ij�

i�j

Consistency condition:

9

Then straightforward to relate to Z’s work:

and positivity of metric (up to freedom to redefine) follows from unitarity in Gij

Seems like roundabout way to recover Z’s result.Would be independent if argument for positivity of χij were different/distinct/new.However, appears to be best avenue to generalize to >2 dims.

Freedom (finite part in counterterms, i.e., choice of subtraction scheme)

W 0 �W =

Zdv

�12bR� 1

2cij@µgi@µgj

� �0c � �c = cij�

i�j

�0ij � �ij = L�cij

(consistency conditions are invariant under these)

C = 3(�c + cij�i�j) = 3�0

c

Gij = �ij + L�cij = �0ij

10

J.L. Cardy, Phys. Lett., B215 (1988), p. 749

Cardy proposed generalization: 2D central charge ☛ conformal anomaly in curved background ☛ coefficient of R(as we just saw)

In 2n-dim take instead coefficient of Euler density (in 2D R is Euler density).

Cardy gives no proof. Makes interesting observation, in QCD:

a-theorem

a-theoremviolation:

asymptoticfreedomviolation:

UV: IR:

Nf > N (1)f ⌘ 11

2 Nc +q

( 112 Nc)2N2c + 62(N2

c � 1) + 1

Nf > N (2)f ⌘ 11

2 Nc

�b(g) = N2f � 1lim

g!0�b(g) = 62(N2

c � 1) + 11NcNf

Perturbative a-theorem in 4D

11

I. Jack & H. Osborn, Nucl. Phys. B343 (1990) 647-688; H. Osborn (above)

Note: non-perturbative: Zohar’s talk (?), first talk today!!

Repeat 2D argument in 4D. What is new?

1. Many more terms:

��W = (�W� ��

�)W = �Zdv �T +

Zdv @µ�Zµ

JHEP11(2013)195

with arbitrary functions b, bij , then

δβΦ = βi∂ib = Lβb, δwi = −∂ib+ bijβj (2.6)

δχij = βk∂kbij + ∂iβk bjk + ∂jβ

k bik = Lβbij , (2.7)

where Lβ is the Lie derivative along the beta-function vector. Nevertheless, the consis-

tency condition is invariant under this arbitrariness.7 Osborn then establishes that there

is a choice of the arbitrariness so that the corresponding χij is positive-definite, essentially

equal to Zamolodchikov’s metric Gij = (x2)2 ⟨[Oi(x)][Oj(0)]⟩. With that choice βφ be-

comes Zamolodchikov’s c-function C. As a final remark let us point out here that possible

dimension-one vector operators are neglected in the treatment of Osborn — such operators

were considered in [19].

2.2 The 4d case

In four dimensions the elegance of the two-dimensional case is obfuscated by the fact that

there exist four curvature invariants (that conserve parity) and quite a few terms that

involve derivatives on the couplings. The terms that account for the trace anomaly may

be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1

2χaij∇2gi∇2gj +

1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg

i∂µgj∂νgk∂νgl, (2.10)

and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)

+ Sij∂µgi∇2gj +

1

2Tijk∂νg

i∂νgj∂µgk, (2.12)

up to terms with vanishing divergence. Definitions of the various curvatures can be found

in (A.3). Gµν is the Einstein tensor and the various coefficients are functions of the cou-

plings.

Here one finds six consistency conditions (which can be further decomposed). Two of

them are particularly interesting. First, there is a consistency condition like (2.5) involving

βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)

The consistency condition (2.13) can lead to an extension of Zamolodchikov’s result to 4d

if the metric χgij can be shown to be positive-definite. Of course, just like in 2d, there is an

arbitrariness in the definition of χgij as well as in other coefficients in (2.10) and (2.12). To

7The arbitrariness we are discussing here is analogous to the arbitrariness that affects the coefficient of

!R in the 4d trace anomaly at the fixed point. In 2d we see that outside the fixed point βΦ has a degree

of arbitrariness. Of course when βi = 0 the well-defined βΦ is the central charge of the corresponding CFT

(up to normalization).

– 6 –

where

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

and


11




1. Many more terms:

��W = (�W� ��

�)W = �Zdv �T +

Zdv @µ�Zµ

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

where

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

and

are all beta functions�a,�b, . . . , Tijk

(apologize for confusing notation...)


11




1. Many more terms:

��W = (�W� ��

�)W = �Zdv �T +

Zdv @µ�Zµ

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

where

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

and

Trace anomaly in curved background:I = square of Weyl tensorE4 = Euler density

12

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

2. Many consistency conditions: 6 at operator level, 8 for coefficients (i) a-theorem candidate

(ii) Coefficient of R2:

JHEP11(2013)195

get an a-theorem it suffices to show that there is a choice of the arbitrariness so that χgij

is positive-definite. This relies on the fact that (2.13) is invariant under the arbitrariness.

The other consistency condition we would like to draw attention to is

βc =1

4(∂ib

′ − χei )β

i.

This shows that the coefficient of R2 in the trace anomaly is generally non-zero outside the

fixed point. It also motivates the use of the term “vanishing anomalies” for contributions

to the trace anomaly like R2 in d = 4: these are anomalies that are present along the flow

but vanish at the fixed point.

In our treatment so far we have neglected relevant operators with classical scaling

dimension three or two that may be present in a four-dimensional theory. Osborn has

considered such operators in [3], and has shown that the condition (2.13) is actually un-

affected by their presence, except for a shift of βi due to the presence of dimension-three

vector operators. This shift played an important role in [16], where it was calculated at

three loops in the most general renormalizable 4d QFT, and was used to show that at the

perturbative level scale implies conformal invariance in unitary renormalizable 4d QFTs.

3 The 6d case

As we saw in the previous section the elegance of the consistency conditions rapidly dis-

appears in the jump from 2d to 4d. Nevertheless, a consistency condition similar to (2.5)

remains, and it is interesting to see if this is an accident or if such a consistency condition

can be obtained in higher (even) dimensions. This is the main motivation behind this work,

and the treatment of the highly nontrivial 6d case gives us valuable intuition that actually

applies to all even dimensions. We postpone the discussion of the general even-d case until

the next section, and we turn now to the consistency conditions in d = 6. Appendices A

and B contain information on conventions, basis choices, as well as the terms that appear

in the trace anomaly in 6d away from the fixed point.

3.1 Basis of curvature tensors

It is clear from the complexity of the 4d case that the situation in 6d is significantly more

challenging. As a first step we have to classify the curvature tensors that can be used in

the anomaly terms. Of course terms without curvatures also need to be considered.

To begin, note that for the various contributions to (∆Wσ − ∆β

σ)W we are only con-

strained by diffeomorphism invariance and power counting. Let us look at a consequence of

this in 4d: in anomaly terms with one power of curvature one cannot involve the Riemann

tensor (without contracting its indices). Indeed, the Riemann tensor has four free indices,

for which we would need four derivatives on one or more couplings. This would result in

a term with mass dimension six. Therefore, in 4d, one can only include curvature tensors

with up to two free indices, and those are R, γµνR, and Rµν .8 Since the variation of the

4d Euler density in d = 4 is δσ(√γE4) = −8

√γGµν∇µ∂νσ, it is preferable to include the

8Incidentally, using the same argument one sees that ∇λGµν , where Gµν is the Einstein tensor, can also

not be included in the anomaly terms.

– 7 –

3. Vanishing anomalies: vanish at fixed points but not on flow.

12

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

2. Many consistency conditions: 6 at operator level, 8 for coefficients (i) a-theorem candidate

�(p�E4) = �8Gµ⌫rµ@⌫�

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

⇒

(ii) Coefficient of R2:

JHEP11(2013)195

get an a-theorem it suffices to show that there is a choice of the arbitrariness so that χgij

is positive-definite. This relies on the fact that (2.13) is invariant under the arbitrariness.

The other consistency condition we would like to draw attention to is

βc =1

4(∂ib

′ − χei )β

i.

This shows that the coefficient of R2 in the trace anomaly is generally non-zero outside the

fixed point. It also motivates the use of the term “vanishing anomalies” for contributions

to the trace anomaly like R2 in d = 4: these are anomalies that are present along the flow

but vanish at the fixed point.

In our treatment so far we have neglected relevant operators with classical scaling

dimension three or two that may be present in a four-dimensional theory. Osborn has

considered such operators in [3], and has shown that the condition (2.13) is actually un-

affected by their presence, except for a shift of βi due to the presence of dimension-three

vector operators. This shift played an important role in [16], where it was calculated at

three loops in the most general renormalizable 4d QFT, and was used to show that at the

perturbative level scale implies conformal invariance in unitary renormalizable 4d QFTs.

3 The 6d case

As we saw in the previous section the elegance of the consistency conditions rapidly dis-

appears in the jump from 2d to 4d. Nevertheless, a consistency condition similar to (2.5)

remains, and it is interesting to see if this is an accident or if such a consistency condition

can be obtained in higher (even) dimensions. This is the main motivation behind this work,

and the treatment of the highly nontrivial 6d case gives us valuable intuition that actually

applies to all even dimensions. We postpone the discussion of the general even-d case until

the next section, and we turn now to the consistency conditions in d = 6. Appendices A

and B contain information on conventions, basis choices, as well as the terms that appear

in the trace anomaly in 6d away from the fixed point.

3.1 Basis of curvature tensors

It is clear from the complexity of the 4d case that the situation in 6d is significantly more

challenging. As a first step we have to classify the curvature tensors that can be used in

the anomaly terms. Of course terms without curvatures also need to be considered.

To begin, note that for the various contributions to (∆Wσ − ∆β

σ)W we are only con-

strained by diffeomorphism invariance and power counting. Let us look at a consequence of

this in 4d: in anomaly terms with one power of curvature one cannot involve the Riemann

tensor (without contracting its indices). Indeed, the Riemann tensor has four free indices,

for which we would need four derivatives on one or more couplings. This would result in

a term with mass dimension six. Therefore, in 4d, one can only include curvature tensors

with up to two free indices, and those are R, γµνR, and Rµν .8 Since the variation of the

4d Euler density in d = 4 is δσ(√γE4) = −8

√γGµν∇µ∂νσ, it is preferable to include the

8Incidentally, using the same argument one sees that ∇λGµν , where Gµν is the Einstein tensor, can also

not be included in the anomaly terms.

– 7 –

3. Vanishing anomalies: vanish at fixed points but not on flow.

13

Not new but worth mentioning: as before, arbitrariness-in all coefficients, but-not all independently-a few can be set to zero: “trivial anomalies.” Primary example: -most importantly, precisely as before:

498 H. Osborn / Weyl consistency conditions

This may be decomposed into the separate equations

a ai& - x$3’ = -qwi , (3.10a)

2x; + x;p’ = -qq. ) (3.10b)

8p, - x#3’ =To(2d + r/,p’) , (3.1Oc)

4 a,p, + (/yG + x;)p’ =Yp( aid + y - 4) ) (3.10d)

/y$ + 2x; + Aij =2&sij, Aij=2a;PkX~j+pkXkbij, (3.10e)

2( 4 +x;) + nij + pk(2&j, - XGk) ‘2-g sjj -x; - 2u,,, i) + Vii) ) h x,Tk = Xt;, k - xk(ij) (3.10f)

xf(i, j) - fx$,k + ak~‘x~~~~ + ,&;klP’ = fppT’jk + ai ajP’SklT (3.W

where PP is again the Lie derivative defined by the vector field pi. These equations are not independent. Using Xtjij3jpk + Aij@ = aJx,“,pjpk> it is clear that eqs. (3.1Of), (3.10b) and (3.7) are sufficient for the compatibility of (3.1Od) with (3.10~). Furthermore, contracting (3.1Og) with pi, antisymmetrising on j, k and combining with (3.10e) gives

a[j (XbkP”) =yp(S[ijl - [’ a rPkSj]k - iPkTk[ij]) )

so that (3.8) is necessary for the integrability of (3.10a). In the four-dimensional case the potential arbitrariness in W is given by, with a

similar notation to (3.2),

6W= duB.9, / B=(a,b,C,ei,fij,gij,a;j,bijk, ‘ijkl) * (3.11)

This gives

‘(P,,Pb,Pc,X~,X~,XC,X~)=~~(a,b,c,ei,frj,gij,aij),

6,-&k = Ppbijk i- 2 ai ajp’ iZlk ,

Gxhk! = 9@Cijkl + ai aj,‘n bkl,,, + ak vrn bij,, ,

6Wi = -8aib +gijp, 6d=4c+eiPi,

Sl( = -2e, - aij@, Sy = -2e, -ai(ejpj) +fijpj,

SFj = -4e(,, jj + 2fij - gij - bijkpk,

6S,, = gij + 2aij + 2 aipk akj + bkijpk ,

qjk = 2gk(i.j) - gij,k + 2bijk + 2 a@’ bij, + 2Cijk#‘.

(3.12)

�g0ij � �g

ij = L�cij

�0b � �b = cij�

i�j

r2R

14

Positive Metric?

Perturbatively

�g = �g(0) +g2

16⇡2�g(1) +

✓g2

16⇡2

◆2

�g(2) + · · ·

then: if �g(0)ij > 0 we also have �g

ij > 0 for sufficiently small g2

16⇡2

What theory? (which computation?)

4D: yang mills + spinors + scalars with Yukawa couplings and scalar self-interactions

Result: �g(0)ij > 0 for all couplings. Copying form Jack & Osborn:

I. Jack, H. Osbom / Renormalisable field theories 669

vergences and taking into account the various d-dependent factors in (2.6) weobtain

A~2~~= n~g2 (~s(C — ~R)F(16~2c)2

—4(11C— 4R)(2G’2vv~v~+Hv2— (VUv~)2— 2V2V~’vff— 2v2v2)

+ kc(51C — 20R)(2G’2”v~v~— (V~Vc,)2)+ ~c(29C — 12R)Hv2

~ ~c(23C_4R)v2v2). (5.8)

An important consistency check is that the double poles are in accord with therenormalization group equation (2.8).In addition we have calculated the M-dependent terms in (4.23). At one loop

A~~ 16~2s (Htr(M2) + 2tr(D~MD~’2M)), (5.9)

while at two loops

A~/ 12 2(2(12 llc)g2Htr(t2M2) + (12—5s)g2tr(t2D~MD~’2M)(16w c)

+(12— 11c)2ga’2gtr(t2MD~M)_24ea~ga’2gtr(t2M2)). (5.10)

Again the double poles are in accord with the renormalization group equation(4.26) since

‘~= —6t2(g2/16ir2). (5.11)

From eqs. (5.5) and (5.8) we obtain, apart from (2.9), to two-loop order

1 2

1~a= —~(—~(2nv+nF)+ ~nv(C— ~R)h), h =

~= —2~=16~~g2(4+~(51C_20R)h)~

~ 16~~g2(2+4(29C_12R)h)~ Xe=O,

~b ~ 3(4+~(11C—4R)h),16~g

Xc = 16~r2g4(—8— ~(341C — 76R)h), (5.12)

Yang Mills

I. Jack, H. Osbom / Renormalisable field theories 679

The result for /3~3)is in accord with eq. (6.27) given U1 and e,~from eq. (6.29).In addition

T( 4,)(3) = (16~2)3 ~ (— ~gklmngiktpgjmnp~i~j + ~gj~1~gjk/mVjj( 4,)).

From eq. (6.26) this allows the calculation of the lowest-order, four-loop, contribu-tion to /3”,

1 12/3~(4,) = 2 ~ -~ — ~ — ~pqmn~pqkt~kmri~tnrj

(16~)

+grpqlgrrnnkg/rnn~gkpqj)4,i4,j, N,J = giktrngjktrn. (6.33)

This agrees with previous results [11,12] for a single component field.From eq. (6.30) we can also determine the lowest contribution to the metric on

the space of scalar couplings, as in eqs. (3.17a,b),

1 1 1 1g(3) = —6 W~

3~= _________ — (6 34)— (16~f 72 ‘~‘ ‘ (16~2)3

216g1.

Since W1 a a1(g~g~),eq. (3.17a) becomes 8a113,, = XfjI3~to this order. At four loops

then 813,, = WJI3f so /3b remains zero. To the next order we expect corrections toxij and from an analysis of the relevant diagrams it is clear that there is a uniquepossibility X~h

1hja ~ and that therefore W~= -~aJ(~7~gfg~).In thiscase these terms do not contribute, in association with the one loop /3k’, to /3~tofive-loop order and we then obtain,

8/3b = (16~2)5~(gijklgijmngpqkmgpq1n — ~ (6.35)

Also from eqs. (6.30) and (3.20) we obtain

1 1 1 1a(3) — —6 y(3)= — (636)Xjj (16~r2)3144 ‘~‘ (16~r2)3

432g1g1.

Hence eqs. (3.21a,b) give

8/3c = — (162)6 ~ (6gijkjgktrnflgrnflpqg,prsgjqrs

+ l2gijk/~k/mngmrpqgj,pqginr, — N,jgimklgjmpqgktpq). (6.37)

Scalar quartics

I. Jack, H Osbom / Renormalisable field theories 681

discussion of the previous section is extended to this case these additional termsgive rise to modifications akin to eq. (6.24) arising from the essential arbitrarinessunder F,—* UtF,U for UtU= 1.At one loop as part of the necessary counterterms we find a contribution

— 16~2c(2tr(D~MD’2M)+ 4R tr(M2) + ~Fjy’2D~F,~i), (7.3)

forD~M= a,.,M + [At, MI. Using

D~M-*F,D~4,,+D~F,4,,, D~F,=a~F,+ [4M] +A~,1F~,

this can be cast into the required renormalised form leading to the one-loopcontribution to y,1 in (7.2) and

- 16~ tr(F[,D~F,]), /3~=- ___

/3~(4,)= - 16~2tr(D~F,.D’2~)4,,4,1. (7.4)

At two loops the essential counterterms, for the purposes of this paper, aregiven by

~= ~(~tr(F~F,)+ 2G’2v tr(D~F,D~F,)

+Htr(D~F,D’2F,)— tr(D2F~D2F1)). (7.5)

This expression, apart from the F-term, is similar in form to (6.30); both aredetermined by the finiteness requirements of (3.9e, f) neglecting higher-order0(p)-terms. From (7.5) the lowest-order contribution to the metric for the Yukawacouplings is

1 4 a 1 1h .~~(

2) . h = 2 — tr(h,h,) , W~2~~h = h . — 2 — tr(F2) , (7.6)(16~2) 3 9F (16~2) 6

for any hermitian matrix h, of the same dimension as F,. It is easy to see that eq.(3.17a) now becomes

a(7.7)

Yukawa LYuk = �i �i

I. Jack, H Osbom / Renormalisable field theories 681

discussion of the previous section is extended to this case these additional termsgive rise to modifications akin to eq. (6.24) arising from the essential arbitrarinessunder F,—* UtF,U for UtU= 1.At one loop as part of the necessary counterterms we find a contribution

— 16~2c(2tr(D~MD’2M)+ 4R tr(M2) + ~Fjy’2D~F,~i), (7.3)

forD~M= a,.,M + [At, MI. Using

D~M-*F,D~4,,+D~F,4,,, D~F,=a~F,+ [4M] +A~,1F~,

this can be cast into the required renormalised form leading to the one-loopcontribution to y,1 in (7.2) and

- 16~ tr(F[,D~F,]), /3~=- ___

/3~(4,)= - 16~2tr(D~F,.D’2~)4,,4,1. (7.4)

At two loops the essential counterterms, for the purposes of this paper, aregiven by

~= ~(~tr(F~F,)+ 2G’2v tr(D~F,D~F,)

+Htr(D~F,D’2F,)— tr(D2F~D2F1)). (7.5)

This expression, apart from the F-term, is similar in form to (6.30); both aredetermined by the finiteness requirements of (3.9e, f) neglecting higher-order0(p)-terms. From (7.5) the lowest-order contribution to the metric for the Yukawacouplings is

1 4 a 1 1h .~~(

2) . h = 2 — tr(h,h,) , W~2~~h = h . — 2 — tr(F2) , (7.6)(16~2) 3 9F (16~2) 6

for any hermitian matrix h, of the same dimension as F,. It is easy to see that eq.(3.17a) now becomes

a(7.7)

15

6D consistency conditionsBG, Andreas Stergiou and David Stone, JHEP11(2013)195

JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –

Both in 2D and 4D obtained candidate form CCs

Is this generic? Can we obtain the same result in 6D?

Answer here, details later: yes and no!

(Change notation again, my apologies! , , )�b(g) ! a(g) �gij ! H1

ij wi ! H1i

15


JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –




(Change notation again, my apologies! , , )

JHEP11(2013)195

which can be brought to the form

∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)

The consistency condition (3.3) has a new feature compared to the 2d and 4d cases, i.e.

that the function a contains the coefficients b1 and b3 of the vanishing anomalies L1 and L3

respectively. This is of no consequence as far as the value of a at the fixed point is concerned:

there a = a, for b1 = b3 = 0 at the fixed point. This fact is actually made explicit by

three consistency conditions. More specifically, from terms proportional to (∂κσ∇λ∂µσ′ −∂κσ′∇λ∂µσ)∇κGλµ, (σ∂µσ′ − σ′∂µσ)∇νH

µν4 , and (σ∂µσ′ − σ′∂µσ)∇νH

µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)

respectively. From similar consistency conditions we can verify that b2, b4, b5 and b6 are

also zero at the fixed point, as expected since they are coefficients of vanishing anomalies.

3.4 Possibility for an a-theorem in 6d

The consistency condition (3.3) has the potential to lead to a result similar to that of

Zamolodchikov in 2d. Indeed, contracting with the beta function it follows that (3.3)

impliesda

dt= −H1

ijβiβj . (3.5)

Note that here the conditions (3.4) allow us to absorb the b1 and b3 contributions in a to

a shift of H1i . Of course what is missing is a proof of the positive-definiteness of H1

ij .

It is important to point out that the consistency condition (3.3) is actually stronger

than (3.5). Indeed, (3.3) also contains information about the possibility of a gradient flow

interpretation of the RG flow. For that, it has to be that ∂[iH1j] = 0, in which case a is the

“potential” whose gradient produces the RG flow.

Let us now concentrate on a technical but important point. It turns out that the

tensor Hµν1 , which appears in δσ(

√γE6) = 6

√γHµν

1 ∇µ∂νσ, is divergenceless. A similar

statements holds in two, δσ(√γR) = 2

√γ γµν∇µ∂νσ, and four dimensions, δσ(

√γE4) =

−8√γGµν∇µ∂νσ. This is actually crucial for the coefficient of the Euler term to be involved

in a consistency condition like (3.3), which has the chance to lead to an a-theorem. This

is not so easy to see in 2d and 4d, but it is clear in 6d.

Indeed, consider, for example, the consistency condition arising from terms propor-

tional to (σ∂µσ′ − σ′∂µσ)Hµν4 . It reads

∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

yes:

�b(g) ! a(g) �gij ! H1

ij wi ! H1i

15


JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –





JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

yes: no:

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

�b(g) ! a(g) �gij ! H1

ij wi ! H1i

15


JHEP11(2013)195












2.2 The 4d case




be written as

∆Wσ W = ∆β

σW −∫

d4x√γ σT +

∫

d4x√γ ∂µσZ

µ, (2.8)

where

T = βaI + βbE4 +1

9βcR

2 +1

3χei∂µg

i∂µR+1

6χfij∂µg

i∂µgjR+1

2χgij∂µg

i∂νgjGµν (2.9)

+1


1

2χbijk∂µg

i∂µgj∇2gk +1

4χcijkl∂µg


and

Zµ = Gµνwi∂νg

i +1

3∂µ(b′R) +

1

3RYi∂

µgi (2.11)

+ ∂µ

(

Ui∇2gi +1

2Vij∂νg

i∂νgj)


1

2Tijk∂νg




plings.



βb:

∂iβb =1

8χgijβ

j +1

8∂[iwj]β

j , βb = βb +1

8wiβ

i. (2.13)








– 6 –





JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

yes: no:

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

yes again: among full set of CCs:

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

⇒ b1 and b3 are vanishing anomalies

�b(g) ! a(g) �gij ! H1

ij wi ! H1i

16

Some aspects of the 6D CCs:

• Much larger basis of operators

The choice is arbitrary (provided complete), but there are “better/worse” choices.• Bianchi identities!!• In 4D, the a-theorem CC was in

So it makes sense to look at the variation of Euler in 6D:

Include this (H1) in the basis in operators with two derivatives on couplings (togetherwith all other independent dim-4 curvature operators).More on this, and generalization to higher 2nD, later.

• Trivial anomalies: recall, completely scheme dependent, can be removed.Of 17 terms cubic in curvature, 6 are trivial anomalies, J1, ..., J6

• Vanishing anomalies: recall, vanish at critical points (but not on RG flow)Of 17 terms cubic in curvature, 7 are vanishing anomalies, L1, ..., L7 (b1, ..., b7 of previous page are coefficients of L1, ..., L7 in Weyl variation)It is possible to choose

• 17-6-7=4; 4=3+1, where 3 = # of Weyl invariants and 1= Euler density

JHEP11(2013)195

Einstein tensor instead of the Ricci tensor. This choice produces the consistency condi-

tions in a convenient form, but it is not essential. Indeed, the consistency conditions in a

specific basis can be recast to the form obtained in any other basis by a redefinition of the

coefficients of the various anomaly terms.

In 6d a similar choice is dictated by the fact that the Weyl variation of the 6d Euler

density is

δσ(√γE6) = 12

√γ (3E4γ

µν − 2RRµν + 4RµκR

κν + 4RκλRκµλν − 2R µ

κλρ Rκλρν)∇µ∂νσ,

where E4 is given in even d > 2 by E4 = 2(d−2)(d−3)(R

κλµνRκλµν − 4RκλRκλ + R2). The

tensors quadratic in curvature that we have to consider can be found in (A.3); the ten-

sor Hµν1 is chosen so that in d = 6 the variation of the Euler density is δσ(

√γE6) =

6√γHµν

1 ∇µ∂νσ. As far as terms quadratic in curvature are concerned, we also have to

include the terms (A.4), which are basically derivatives of the terms in (A.3). Terms lin-

ear in curvature include (A.1) and (A.2). In writing down the various curvature tensors

one has to identify a complete but not over-complete basis, a problem complicated by the

symmetries of the Riemann tensor and the Bianchi identities.

As far as scalar terms cubic in curvature are concerned [7, 8], the situation is slightly

more subtle. We have to include the terms in (A.5), but among them there are trivial

anomalies, i.e. the terms J1,...,6 whose coefficient can be varied at will by a choice of local

counterterms. These are not genuine anomalies, but they nevertheless appear in the trace

anomaly, even at the fixed point. The well-known example is the term !R in 4d. In (A.5)

there are also vanishing anomalies, i.e. curvature terms that have to be included outside

the fixed point, but that do not satisfy the consistency conditions at the fixed point and

thus their coefficient has to be set to zero there. These are the terms L1,...,7 in (A.5). As

we already mentioned there is only one such term in 4d, namely R2. Here, the form of

L1,...,6 is chosen based on the fact that these are the terms that shift the coefficients of the

trivial anomalies at the fixed point, i.e. δσ∫

d6x√γ L1,...,6 =

∫

d6x√γ σJ1,...,6.

While J1,...,6 can be included in the basis of terms cubic in curvature, there is a

more convenient choice based on the fact that in order to show that δσ∫

d6x√γ L1,...,6 =

∫

d6x√γ σJ1,...,6 one has to integrate by parts. But since total derivatives can be neglected

in our considerations (for σ can be taken to have local support), this implies that we don’t

have to include the trivial anomalies J1,...,6 in (∆Wσ −∆β

σ)W , so long as we include terms

arising from δσ∫

d6x√γ z1,...,6L1,...,6 before any integrations by parts. (Here z1,...,6 are ar-

bitrary functions of the couplings.) Consequently, terms cubic in curvature that we need to

consider are the three terms I1,2,3 that lead to Weyl-invariant densities, the 6d Euler term

E6, and the seven vanishing anomalies L1,...,7. As we explained, this relies on the ability

to discard total derivatives.

3.2 Contributions to the anomaly

Now that we have a complete basis of curvature tensors we are ready to write down the

most general anomaly functional (∆Wσ −∆β

σ)W . It takes the form

∆Wσ W = ∆β

σW +65∑

p=1

∫

d6x√γ σTp +

30∑

q=1

∫

d6x√γ ∂µσZ

µq ,

– 8 –�(p�E4) = �8Gµ⌫rµ@⌫�

JHEP11(2013)195






density is

δσ(√γE6) = 12

√γ (3E4γ








√γE6) =

6√γHµν

















d6x√γ L1,...,6 =

∫

d6x√γ σJ1,...,6.



d6x√γ L1,...,6 =

∫















∆Wσ W = ∆β

σW +65∑

p=1

∫

d6x√γ σTp +

30∑

q=1

∫

d6x√γ ∂µσZ

µq ,

– 8 –

⌘ 6p�Hµ⌫

1 rµ@⌫�

JHEP11(2013)195






density is

δσ(√γE6) = 12

√γ (3E4γ








√γE6) =

6√γHµν

















d6x√γ L1,...,6 =

∫

d6x√γ σJ1,...,6.



d6x√γ L1,...,6 =

∫















∆Wσ W = ∆β

σW +65∑

p=1

∫

d6x√γ σTp +

30∑

q=1

∫

d6x√γ ∂µσZ

µq ,

– 8 –

17

• Total: 41 consistency conditions (some further decomposable), eg, from coefficient of , obtain

which can be rewritten as

The one of central interest (for a-theorem) is from , which gives

• Arbitrariness: as before, from freedom in finite part of counterterms.

(same operators as in non-derivative terms in anomaly, but arbitrary coefficients). As

consequence, one example: from W’-W =

one has

which leaves invariant our candidate a-theroem:

JHEP11(2013)195

where the Tp and Zµq are dimension-six and dimension-five terms respectively, that can

involve curvatures as well as derivatives on the couplings gi (see appendix B).

Much like Osborn did in d = 2, 4 we split the anomaly contributions into terms with

σ and ∂µσ. This splitting may seem mysterious — as we could also introduce terms of

the form∫

d6x√γ!σV , for example — but it is used here in order to get the consistency

conditions in a most convenient form. We can obtain the desired form of the consistency

conditions even without the splitting, if we carefully choose the coefficients of the various

terms in the anomaly. This can be seen by integrating by parts to rewrite the Zµq terms

in the form of the Tp terms, which would lead to some new Tp terms but also to shifts of

coefficients of existing Tp terms.

Let us illustrate this point more clearly in the 2d case. Suppose that instead of (2.4)

we started with the equivalent

∆Wσ W = ∆β

σW −∫

d2x√γ σ

(

1

2βΦR−

1

2χ′ij∂µg

i∂µgj + wi!gi)

. (3.1)

After an integration by parts of the !gi term this amounts simply to the definition χij =

χ′ij + 2∂(iwj) in (2.4). This can also be seen by computing the Weyl consistency condition

from (3.1) directly. We get

∂iβΦ = (χ′

ij + 2∂iwj)βj = (χ′

ij + ∂(iwj))βj + ∂[iwj]β

j . (3.2)

Clearly, (3.2) is equivalent to (2.5) with the proper definition of χij .

3.3 Some consistency conditions

Here we include some consistency conditions and we comment on the most interesting ones.

A Mathematica file with all the consistency conditions is included with our submission.

Just like in 2d and 4d we obtain consistency conditions simply by the requirement

[∆Wσ − ∆β

σ,∆Wσ′ − ∆β

σ′ ]W = 0. In our case we find a total of forty one consistency condi-

tions.9 For example, consistency requires that terms proportional to ∂µσ!2σ′ − ∂µ σ′!2σ

add up to zero, which leads to

∂µ(4b11 − 3Aiβi) + (4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i ) ∂µg

i + 6Ai ∂µβi −A′

ijβj ∂µg

i = 0,

which implies that

∂i(4b11 − 3Ajβj) + 4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i + 6Aj ∂iβ

j = A′ijβ

j .

Among the forty one consistency conditions in 6d the most interesting is the one similar

to (2.5), obtained from terms proportional to (σ∂µσ′ − σ′∂µσ)Hµν1 . It reads

∂µ

(

6a+ b1 −1

15b3

)

+H1i ∂µβ

i + βi∂iH1j ∂µg

j −H1ijβ

i ∂µgj = 0,

9Some of these consistency conditions can be further decomposed as a result of the variety of ways with

which spacetime derivatives can act on couplings.

– 9 –

JHEP11(2013)195





the form∫









∆Wσ W = ∆β

σW −∫

d2x√γ σ

(

1

2βΦR−

1

2χ′ij∂µg

i∂µgj + wi!gi)

. (3.1)




∂iβΦ = (χ′



j . (3.2)






[∆Wσ − ∆β

σ,∆Wσ′ − ∆β




∂µ(4b11 − 3Aiβi) + (4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i ) ∂µg


ijβj ∂µg

i = 0,

which implies that

∂i(4b11 − 3Ajβj) + 4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i + 6Aj ∂iβ

j = A′ijβ

j .



∂µ

(

6a+ b1 −1

15b3

)

+H1i ∂µβ


j −H1ijβ

i ∂µgj = 0,



– 9 –

JHEP11(2013)195





the form∫









∆Wσ W = ∆β

σW −∫

d2x√γ σ

(

1

2βΦR−

1

2χ′ij∂µg

i∂µgj + wi!gi)

. (3.1)




∂iβΦ = (χ′



j . (3.2)






[∆Wσ − ∆β

σ,∆Wσ′ − ∆β




∂µ(4b11 − 3Aiβi) + (4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i ) ∂µg


ijβj ∂µg

i = 0,

which implies that

∂i(4b11 − 3Ajβj) + 4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i + 6Aj ∂iβ

j = A′ijβ

j .



∂µ

(

6a+ b1 −1

15b3

)

+H1i ∂µβ


j −H1ijβ

i ∂µgj = 0,



– 9 –

JHEP11(2013)195





the form∫









∆Wσ W = ∆β

σW −∫

d2x√γ σ

(

1

2βΦR−

1

2χ′ij∂µg

i∂µgj + wi!gi)

. (3.1)




∂iβΦ = (χ′



j . (3.2)






[∆Wσ − ∆β

σ,∆Wσ′ − ∆β




∂µ(4b11 − 3Aiβi) + (4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i ) ∂µg


ijβj ∂µg

i = 0,

which implies that

∂i(4b11 − 3Ajβj) + 4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i + 6Aj ∂iβ

j = A′ijβ

j .



∂µ

(

6a+ b1 −1

15b3

)

+H1i ∂µβ


j −H1ijβ

i ∂µgj = 0,



– 9 –

JHEP11(2013)195





the form∫









∆Wσ W = ∆β

σW −∫

d2x√γ σ

(

1

2βΦR−

1

2χ′ij∂µg

i∂µgj + wi!gi)

. (3.1)




∂iβΦ = (χ′



j . (3.2)






[∆Wσ − ∆β

σ,∆Wσ′ − ∆β




∂µ(4b11 − 3Aiβi) + (4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i ) ∂µg


ijβj ∂µg

i = 0,

which implies that

∂i(4b11 − 3Ajβj) + 4Ai + 2G4

i + 5H5i + 2H6

i − 2I4i + 6Aj ∂iβ

j = A′ijβ

j .



∂µ

(

6a+ b1 −1

15b3

)

+H1i ∂µβ


j −H1ijβ

i ∂µgj = 0,



– 9 –

W

0 �W =

Zd

6x

p�

65X

p=1

T 0p

JHEP11(2013)195

The contribution 16I

7i does not allow (3.6) as a candidate for the generalization of Zamolod-

chikov’s result.10 This contribution in fact arises from the term T18 = I7i ∂µg

i∇νHµν4 . Were

∇νHµν1 non-vanishing, we would not be able to find a consistency condition like (3.3). It

can be verified by explicit computations that Hµν1 is the only divergenceless symmetric

two-index tensor quadratic in curvature. It is thus a generalization of the Einstein tensor.

As we will see in the next section Lovelock has constructed all such generalizations a long

time ago [10], something that will allow us to argue for a consistency condition similar

to (3.3) in all even d.

3.5 Arbitrariness

Just like the coefficient of !R in the four-dimensional trace anomaly, the various coefficients

in Tp and Zµq are affected by the choice of additive, quantum-field-independent countert-

erms. Indeed, calculations in curved space and with x-dependent couplings will result in

infinities that will need to be renormalized via counterterms whose finite part is arbitrary.

Therefore, different subtraction schemes will result in different coefficients for Tp and Zµq .

The most general addition to the generating functional of our theory is

δW = −65∑

p=1

∫

d6x√γ Xp,

where the Xp terms have the same form as the Tp terms but with arbitrary coefficients.

There is no arbitrariness introduced by terms Xµq similar in form to the Z

µq terms, for

those are total derivatives. Now, the consistency conditions are invariant under the shift

W → W + δW , although the coefficients in the consistency conditions will shift. Let us

see how this works for (3.3).

The relevant terms are∫

d6x√γ (zaE6 + zb1L1 + zb3L3 −

1

2zH

1

ij ∂µgi∂νg

jHµν1 ) ⊂ δW (3.7)

and one can verify that their inclusion leads to shifts

δa = Lβza, δb1 = Lβzb1 , δb3 = Lβzb3 , (3.8)

δH1i = −6∂i

(

za +1

6zb1 −

1

90zb3

)

+ zH1

ij βj , δH1ij = Lβz

H1

ij , (3.9)

under which (3.3) is invariant. Note that a, which is of course well-defined at the fixed

point, is arbitrary along the flow, while H1i and H1

ij have a degree of arbitrariness even at

the fixed point. Also note that the shifts (3.9) cannot be used to set the corresponding

coefficients to zero, except for H1i if H1

i = ∂iX for some X.

This observation leads to an important point, which we have already emphasized:

regarding the a-theorem, one should not be able to prove that the metric H1ij is positive-

definite in all generality. Instead, one ought to be able to show that there is a choice for

10Of course this can also be seen from the fact that b1 becomes zero at fixed points, and so it cannot

possibly be monotonically-decreasing along an RG flow.

– 11 –

JHEP11(2013)195




i∇νHµν4 . Were







3.5 Arbitrariness







δW = −65∑

p=1

∫

d6x√γ Xp,



µq terms, for






1

2zH

1

ij ∂µgi∂νg

jHµν1 ) ⊂ δW (3.7)



δH1i = −6∂i

(

za +1

6zb1 −

1

90zb3

)

+ zH1


H1

ij , (3.9)












– 11 –

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

18

Is there an a-theorem in 6D?BG, A. Stergiou, D. Stone & M. Zhong, arXiv:1406.3626

• Test possibility using loop expansion (weak coupling expansion) (perturbation theory) • Use renormalizable theory• Only 6D possibility: scalar 𝜙3 theories • On the plus side:

• These are asymptotically free: have gaussian UV fixed point• Relatively simple• Metric requires only two loop calculation (compare with 3-loop in 4D)

• On the anti-plus side:• Potential is unbounded • No other fixed points known

• Interest? There are non-lagrangian 6D CFTs (from brane constructions)• It may be possible to flow among them• It may be that scalar 𝜙3 theories flows to one of these once non-perturbative

• Method• Arbitrary background and spacetime dependent couplings• dim-reg in spacetime (no FT into momentum space)• zeta-function/dim-reg, extract -poles in Green functions

Jack, I. Nucl.Phys. B274 (1986) 139 Kodaira, J Phys.Rev. D33 (1986) 2882

http://arxiv.org/abs/arXiv:1406.3626

http://arxiv.org/abs/arXiv:1406.3626

• Text

We do not take the most general Lagrangian:

2

in perturbation theory. We discuss the implications ofthis result in section IV.

II. WEYL CONSISTENCY CONDITIONS

In general, the classical symmetries of a theory may bebroken for its renormalized Green functions. The form ofthis “anomaly” is constrained by the algebra of the sym-metry group: for an infinitesimal transformation gener-ated by ∆a acting on the generating functional of renor-malized Green functions Γ, we have

[

∆a,∆b]

Γ = ifabc∆cΓ, (1)

where fabc are the structure constants of the symmetrygroup. These are the so-called Wess–Zumino consistencyconditions [10].It is useful to study a QFT on a curved background

with spacetime-dependent couplings so that the metricγµν(x) and couplings gI(x) act as sources for the stress-energy tensor and the operators (labelled by I) in theLagrangian, respectively. We only consider the case ofdimensionless couplings, so that in perturbation theoryall the interaction terms in the Lagrangian are nearlymarginal. We introduce their infinitesimal local Weyltransformations as

∆σγµν(x) = 2σ(x)γµν(x) ,

∆σgI(x) = σ(x)βI (x) ,

(2)

where βI(x) is the beta function of the associated cou-pling and depends on x only through gI(x). The groupof Weyl transformations is Abelian and has only a singlegenerator. Thus, Eq. (1) becomes

[∆σ,∆σ′ ]Γ = 0, (3)

where it is understood that Γ = Γ[γµν , gI ], indicating thedependence on the metric and couplings as backgroundfields.The response of Γ to Weyl rescaling produces the Weyl

anomaly

∆σΓ[γµν , gI ] =

∫

ddx√γ σ∑

i

(aiAi[γµν ]

+ biBi[γµν , gI ] + ciCi[g

I ])

,

(4)

where d is the dimension of spacetime (presumed evenhere), and i is a counting index. The form of Eq. (4)is fixed by general diffeomorphism invariance and powercounting. Ai, Bi and Ci are functions of the metric andcouplings, and by dimensional analysis must include dspacetime derivatives. The Ai do not contain any deriva-tives on couplings and are therefore of d/2-th order incurvature, the Ci are functions of d derivatives on thecouplings, and, finally, the Bi are functions of both cur-vature and derivatives of the couplings. The coefficients

ai, bi and ci are all functions of the couplings only. Inparticular, the Ai contain the Euler term in d dimensionswith coefficient (−1)d/2a, so that at fixed points a > 0.

Now, the consistency conditions from Eq. (3) imposeintegrability relations on the terms in Eq. (4). The rela-tion of interest involves the coefficient of the Euler termin Eq. (4) and coefficients of terms in the Bi involvinga generalization of the Einstein tensor to d dimensionsfound by Lovelock [11]. In even dimensions, it was shownthat an integrability relation exists [12] involving a suchthat [13]

∂I a = 16 (χIJ + ∂IwJ − ∂JwI)β

J , (5)


da

d logµ= 1

6χIJβIβJ , (6)

where µ is the renormalization scale. Here χIJ and wI

are tensors in the space of couplings and they appearin the coefficients of the Bi terms ∂µgI∂νgJHµν and∇µ∂νgIHµν in Eq. (4), where Hµν is the Lovelock tensorin d dimensions, and a is a scalar in the space of couplingswith a = a at the critical points [14]. Both quantities maybe related to correlation functions of the stress-energytensor, its trace, and the operators in the QFT. SinceβI = 0 at the critical points, a is stationary with re-spect to variations of scale there, and in fact becomesa. Moreover, Eq. (6) that a satisfies is very similar tothat found for the analogous quantity in two dimensionsin [1]. This suggests a as the analog of Zamolodchikov’smonotonically-decreasing function in two dimensions.While the consistency conditions impose this integra-

bility relation, a strong version of the a-theorem must es-tablish that the “metric” χIJ is positive-definite, whichthen proves that da/d logµ > 0. To compute χIJ , othermethods must be used.

III. RESULTS FROM THE EFFECTIVE

POTENTIAL

To compute χIJ in six dimensions, we work with theconformally-coupled scalar field theory [15] on a curvedbackground with Lagrangian

L = 12∂µφi∂νφ

iγµν + 15Rφiφ

i + 13!gijkφ

iφjφk, (7)

with the fields, spacetime metric, and couplings all im-plicitly functions of spacetime. The generic coupling con-stants gI are here specifically gijk with the label I = (ijk).At the classical level the term ∂µgI∂νgJHµν clearly doesnot show up, so χIJ = 0 at the classical level. To findthe first (quantum) contributions to χIJ , we can computethe effective potential in a curved background with theloop expansion to two loop order or, equivalently, secondorder in ! [16].

2-loop beta functions and anomalous dimensions (new!):

3

The six-dimensional two-loop effective potential can becomputed using heat kernel methods in dimensional reg-ularization [3, 17, 18]. From this computation we deter-mine the one- and two-loop anomalous dimensions of theelementary fields φi and the beta functions for the cou-plings gijk:

γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.

Here we have used diagrammatic notation to indicate thecorresponding contraction of the couplings, e.g.,

= giklgjkl, (12)

and permutations of the free indices in the wavefunction-renormalization corrections to the beta function are un-derstood. For example,

= gijlglmngkmn + permutations. (13)

Eq. (10) generalizes the single field result of [19] (seealso [17, 18, 20, 21]) to the multi-field case. The firstcontribution to (11) is non-planar. For the seeminglyasymmetric vertex corrections in (11) (the second andthird terms) a symmetrization is understood; for exam-ple,

∼ + +

where “∼” means “the left-hand side stands for the right-hand side.”Our main result is the two-loop expression for the “met-

ric” in theory space:

χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)

With this result and the one-loop beta function (10) wecan use the consistency condition (5) to compute a at

three loops, using w(2)I ∼ gI [22]. We find [23]

a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)

The three-loop contribution to the coefficient of the Eulerterm a is

a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)

In the single-coupling case both a and a increase in theflow out of the trivial fixed point.One may wonder if the results in (14) and (15) depend

on the renormalization scheme we used to compute thetwo loop effective potential [24]. Actually, Eq. (5) (andthus Eq. (6)) is invariant under the choice of renormaliza-tion scheme. The individual terms are, however, scheme-dependent. The corresponding arbitrariness is of theform δa = zIJβIβJ and δχIJ = βK∂KzIJ + zKJ∂IβK +zIK∂JβK , where zIJ is an arbitrary regular symmetricfunction of the couplings. Since the arbitrariness in a van-ishes (quadratically) when fixed points are approached,it cannot change the nature of the flow in the vicinity offixed points.

IV. DISCUSSION

Using the result of our computation, Eq. (14), in theevolution equation (6), or equivalently, the explicit formof a in (15), it is apparent that in perturbation theory thequantity a in Eq. (6) actually increases as one decreasesthe renormalization scale. This is contrary to intuitiondeveloped in d = 2, 4 dimensions where a seems to countthe degrees of freedom in a QFT.This result should be taken with two comments in

mind. Firstly, that the result is a perturbative one, andwe cannot say anything about non-perturbative regimesof six-dimensional QFTs. And secondly, that there are noknown perturbative critical points other than the single,trivial one at gijk = 0, so in this context renormalizationgroup flows do not connect pairs of critical points [25].However, it is still true that, with Eq. (6) identical ind = 2, 4, and 6 dimensions, the strong version of the a-theorem holds perturbatively in d = 2, 4 but not in d = 6.We do not know the reason for this difference. One

possibility may be the unstable nature of the theory weare considering. After all, a cubic potential is unboundedfrom below. However, the state with ⟨φi(x)⟩ = 0 is per-turbatively stable and our computations are valid only inthe perturbative regime. Moreover, the analogous case infour dimensions, the inverted quartic potential, is also un-stable, but does satisfy a perturbative a-theorem (sincethe metric in theory space, χIJ , is perturbatively posi-tive in four dimensions, independently of the sign of thequartic couplings). Another possibility is that a flow be-tween critical points is required for an a-theorem to hold,but the only perturbatively-accessible critical point in theclass of theories in Eq. (7) is the Gaussian fixed point atgijk = 0. But, again comparing to known cases, a pertur-bative strong a-theorem holds for scalar theories in fourdimensions, in spite of only having a Gaussian fixed pointat the origin of coupling-constant space.

3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




Here, diagrams are contractions of couplings, eg

3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




Permutations are understood Symmetrization is understood:

20

We find

Alternatively,

3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




increases towards the IR since the coupling is asymptotically free.

3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




We can also compute a, which itself increases towards the IR

This gives that increases as it flows towards the IR.a

Arbitrariness: can change

This is arbitrarily small when arbitrarily close to fixed point: flow away from fixed pointis always increasing

3


γ(1) =1

64π3

1

12, (8)

γ(2) =1

(64π3)21

18

(

−11

24

)

, (9)

β(1) = −1

64π3

(

−1

12

)

, (10)

β(2) = −1

(64π3)21

2

(

−7

36

+1

2−

1

9(11)

+11

216

)

.


= giklgjkl, (12)




∼ + +



χ(2)IJ = −

1

(64π3)21

3240δIJ . (14)



a(3) =1

(64π3)31

77760

(

−1

4

)

. (15)


a(3) =1

(64π3)37

388800

(

−1

4

)

. (16)



IV. DISCUSSION




H1IJ I = (ijk)

a0 = a+ zIJ�I�J

21

• Is this an artifact of “bad theory,” since it has unbounded potential?• The vacuum at the origin of field space is perturbatively stable• In 4D the perturbative calculation makes equal sense in

• Why not −a? • Because it is negative at fixed points. In fact,

• Maybe this is the wrong candidate for an a-theorem? • Let’s agree “a-theorem” means the quantity is coefficient of Euler in conformal

anomaly (i.e., at fixed points). • Our candidate does that• If there were an IR fixed point it would give• Can’t undo this

• Maybe a-theorem is only for difference between fixed points (a la Komargodski-Schwimmer). Once non-perturbative may decrease as it flows to IR.• Would be interesting to study flows around known non-Lagrangian CFTs

• Maybe this is the wrong candidate for a c-theorem?• Almost by definition!

��4 ! ��4

aUV < aIR

(�W� ��

�)W =

Zd

2nx

p�� [(�1)naE2n + · · · ]

a

22

Epilogue: 2nDRecall ��(

p�E6) ⌘ 6

p�Hµ⌫

1 rµ@⌫�

In 6D there are several independent 2-index symmetric R2 tensors, this is but one.And this is different from 4D.

This one is special because

A similar statement holds in 2D and 4D

rµHµ⌫1 = 0

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

��(p�E4) = �8

p�Gµ⌫rµ@⌫�

But in 2D and 4D there is no other choice.

Being divergence-free is crucial for a-theorem-like CC. For example, one hasfrom

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

JHEP11(2013)195


∂ia =1

6H1

ijβj +

1

6∂[iH1

j]βj , a = a+

1

6b1 +

1

90b3 +

1

6H1

i βi. (3.3)







µν3 we find

b7 =1

8Fiβ

i, (3.4a)

3b1 − 8b7 = −1

4(∂ib14 + I7

i )βi, (3.4b)

12b1 − b3 − 16b7 = −(∂ib13 + I6i )β

i, (3.4c)






impliesda

dt= −H1

ijβiβj . (3.5)



ij .







√γE6) = 6

√γHµν




√γE4) =






∂ib1 =1

12

(

H4ij +

1

2Fij

)

βj +1

12∂[iH4

j]βj +

1

6I7i , b1 = −b1 +

2

3b7 +

1

12H4

i βi. (3.6)

– 10 –

The spoiler is from

JHEP11(2013)195




i∇νHµν4 . Were







3.5 Arbitrariness







δW = −65∑

p=1

∫

d6x√γ Xp,



µq terms, for






1

2zH

1

ij ∂µgi∂νg

jHµν1 ) ⊂ δW (3.7)



δH1i = −6∂i

(

za +1

6zb1 −

1

90zb3

)

+ zH1


H1

ij , (3.9)












– 11 –

To generalize to 2nD we need a 2-index symmetric, divergence free tensor of order Rn-1

23

Lovelock tensor: is unique and is the variation of Euler!

JHEP11(2013)195

the arbitrariness (3.7) such that H1ij is positive-definite. That specific choice then gives us

the quantity a whose flow is monotonic, through the dependence of δH1i on zH

1

ij . Recall

that in 2d arbitrariness similar to the one described here was used by Osborn to rederive

Zamolodchikov’s c-theorem (see [3] for details).

4 Consistency conditions in even spacetime dimensions

In this section we identify the ingredients that allow us to conclude that a consistency

condition like (3.3) appears in all even spacetime dimensions. Of course non-trivial CFTs

in d > 6 are not known, but it is still interesting to consider the generalization of our results.

According to the classification of [9], for a CFT in any even spacetime dimension lifted

to curved space the conformal anomaly consists of a unique Euler term (type-A anomaly),

a number of terms that lead to locally Weyl invariant densities (type-B anomalies), as

well as a number of trivial anomalies. Outside the fixed point we also have a number of

vanishing anomalies. As for the trivial anomalies, these can always be accounted for by

terms with d/2− 1 powers of curvature.

Now, in any even spacetime dimension, d = 2n, it is easy to see that the Weyl variation

of the Euler density√γE2n, where

E2n =1

2nRi1j1k1l1 · · ·Rinjnknlnϵ

i1j1...injnϵk1l1...knln ,

gives

δσ(√γE2n) =

√γHµν∇µ∂νσ,

for some symmetric tensorHµν with n−1 powers of the curvature. As Lovelock showed [10],

this tensor Hµν is the unique tensor with the properties of the Einstein tensor — in par-

ticular, it is the only two-index symmetric tensor with n− 1 powers of the curvature that

is divergenceless:

∇νHµν = 0.

Regarding the consistency condition similar to (3.3), this observation allows us to conclude

that the only relevant terms among the various contributions to the anomaly (∆Wσ −∆β

σ)W

are

∫

d2nx√γ σ

[

(−1)naE2n +∑

p

bpLp +1

2Hij ∂µg

i∂νgj Hµν

]

+

∫

d2nx√γ ∂µσHi ∂νg

iHµν ,

where Lp are some vanishing anomalies. A consistency condition similar to (3.3) is thus eas-

ily found, and is of course invariant under arbitrariness generated by contributions similar

to (3.7).

A relation of the metric Hij to a positive-definite metric is currently only known in

2d [3]. A similar relation in higher even d is lacking, but its possible existence would

immediately imply the generalization of Zamolodchikov’s result. To summarize, in any

even spacetime dimension one can find a scalar quantity a such that

∂ia = Hijβj + ∂[iHj]β

j . (4.1)

– 12 –

JHEP11(2013)195



1

ij . Recall















E2n =1



gives

δσ(√γE2n) =





is divergenceless:

∇νHµν = 0.



σ)W

are

∫

d2nx√γ σ

[

(−1)naE2n +∑

p

bpLp +1

2Hij ∂µg

i∂νgj Hµν

]

+

∫


iHµν ,



to (3.7).






j . (4.1)

– 12 –

JHEP11(2013)195



1

ij . Recall















E2n =1



gives

δσ(√γE2n) =





is divergenceless:

∇νHµν = 0.



σ)W

are

∫

d2nx√γ σ

[

(−1)naE2n +∑

p

bpLp +1

2Hij ∂µg

i∂νgj Hµν

]

+

∫


iHµν ,



to (3.7).






j . (4.1)

– 12 –

This “explains” why coefficient of Euler is special (could not see this from 2D or 4D analysis)

JHEP11(2013)195



1

ij . Recall















E2n =1



gives

δσ(√γE2n) =





is divergenceless:

∇νHµν = 0.



σ)W

are

∫

d2nx√γ σ

[

(−1)naE2n +∑

p

bpLp +1

2Hij ∂µg

i∂νgj Hµν

]

+

∫


iHµν ,



to (3.7).






j . (4.1)

– 12 –

Without full 2nD analysis we see that the terms in the anomaly

JHEP11(2013)195



1

ij . Recall















E2n =1



gives

δσ(√γE2n) =





is divergenceless:

∇νHµν = 0.



σ)W

are

∫

d2nx√γ σ

[

(−1)naE2n +∑

p

bpLp +1

2Hij ∂µg

i∂νgj Hµν

]

+

∫


iHµν ,



to (3.7).






j . (4.1)

– 12 –

give rise to

JHEP11(2013)195

The quantity a becomes the coefficient of the Euler term in the trace anomaly at the fixed

point, but more generally it includes a linear combination of the bps and a term Hiβi. The

relation (4.1) immediately implies that

da

dt= −Hijβ

iβj ,

which, if Hij can be related to a positive-definite metric via the arbitrariness δHij = LβzHijwith zHij an arbitrary symmetric tensor, is the generalization of the 2d c-theorem.

Acknowledgments

We have relied heavily on Mathematica and the package xAct. We would like to thank

Aneesh Manohar, John McGreevy, and especially Ken Intriligator for helpful discussions.

This work was supported in part by the US Department of Energy under contract DE-

SC0009919.

A Conventions and definitions

Throughout this paper we follow the conventions of Misner, Thorne and Wheeler [6] for

the Riemann tensor. For the Weyl variation of the metric we choose

γµν → e−2σγµν .

Infinitesimally, then, δσγµν = −2σγµν and so δσγµν = 2σγµν (we do not use δσ for an

infinitesimal σ, since no confusion can arise).

It is important to classify the curvature terms of various mass dimensions. These will

be used subsequently to construct all possible terms that can appear in (∆Wσ −∆β

σ)W . In

two and four spacetime dimensions this is very easy, but in six it becomes a rather cum-

bersome problem, plagued by complications due to the large number of monomials and the

identities of the Riemann tensor.

A complete basis B2 of dimension-two curvature terms that can be used in ∆Wσ W is

given by the Ricci scalar, the Einstein tensor, and the Riemann tensor,

1

d− 1R, Gµν , Rκλµν , (A.1)

where we define the Einstein tensor as

Gµν =2

d− 2

(

Rµν −1

2γµνR

)

(d ≥ 3),

where Rµν is the Ricci tensor. Taking a derivative leads to three dimension-three terms,

but, by diffeomorphism invariance and simple power counting, only two can be used in

∆Wσ W , namely

1

d− 1∂µR and ∇κGµν . (A.2)

These form the basis B3.

– 13 –

and hence to

The End

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