PROGRESS TOWARDS AN a-THEOREM
with Hugh Osborn, Colin Poole
21st May 2014, MASS2014, CP3, Odense
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
1
Introduction to a theorem
2
Derivation of a-theorem (weakly coupled case)
3
A non-supersymmetric example
4
Supersymmetry (Wess-Zumino model)
5
Supersymmetry (gauged case)
6
Finte supersymmetric theories
7
Scale invariance versus conformal invariance
8
Conclusions
See review by Nakayama (arXiv:1302.0884) for comprehensive
list of references
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
There are two versions of the a-theorem:
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
There are two versions of the a-theorem:
The weak a-theorem: There is a function a(g) defined at
fixed points of the theory such that aUV − aIR > 0. Cardy;
Komargodski and Schwimmer; Luty, Polchinski, Ratazzi
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
There are two versions of the a-theorem:
The weak a-theorem: There is a function a(g) defined at
fixed points of the theory such that aUV − aIR > 0. Cardy;
Komargodski and Schwimmer; Luty, Polchinski, Ratazzi
The strong a-theorem: There is a function a(g) which has
monotonic behaviour under RG flow.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
There are two versions of the a-theorem:
The weak a-theorem: There is a function a(g) defined at
fixed points of the theory such that aUV − aIR > 0. Cardy;
Komargodski and Schwimmer; Luty, Polchinski, Ratazzi
The strong a-theorem: There is a function a(g) which has
monotonic behaviour under RG flow.
The a-theorem, if valid, can give important constraints on the
behaviour of RG trajectories.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The a-theorem is the generalisation to four dimensions of the
Zamolodchikov c-theorem for two dimensions. Consider a
theory in four dimensions with couplings {g I }.
There are two versions of the a-theorem:
The weak a-theorem: There is a function a(g) defined at
fixed points of the theory such that aUV − aIR > 0. Cardy;
Komargodski and Schwimmer; Luty, Polchinski, Ratazzi
The strong a-theorem: There is a function a(g) which has
monotonic behaviour under RG flow.
The a-theorem, if valid, can give important constraints on the
behaviour of RG trajectories.
We shall define a function a such that
∂I a = TIJ β J (g)
so that
µ
d
a = β I ∂I a = GIJ β I β J ,
dµ
with Hugh Osborn, Colin Poole
GIJ = T(IJ) .
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The strong a-theorem then amounts to the positive definiteness
of GIJ . This is easily checked to lowest order and thus is valid
perturbatively to all orders. But no non-perturbative proof at
present.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The strong a-theorem then amounts to the positive definiteness
of GIJ . This is easily checked to lowest order and thus is valid
perturbatively to all orders. But no non-perturbative proof at
present.
In two dimensions the c-function is derived from the coefficient
in the trace anomaly
η µν Tµν = −cR . . .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
The a-theorem
The strong a-theorem then amounts to the positive definiteness
of GIJ . This is easily checked to lowest order and thus is valid
perturbatively to all orders. But no non-perturbative proof at
present.
In two dimensions the c-function is derived from the coefficient
in the trace anomaly
η µν Tµν = −cR . . .
In four dimensions the trace anomaly involves 3 curvature
invariants F , G, R 2 . It turns out that only the coefficient of G,
usually denoted a, is viable as a 4-dimensional equivalent for c.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Derivation
We extend to curved space and x dependent couplings g I (x)
and consider infinitesimal local Weyl rescalings
δσ γµν = 2σ γµν
which are implemented by the operator
Z
δ
4
I δ
∆σ = d x σ 2γµν
+β
δγµν
δg I
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Derivation
We extend to curved space and x dependent couplings g I (x)
and consider infinitesimal local Weyl rescalings
δσ γµν = 2σ γµν
which are implemented by the operator
Z
δ
4
I δ
∆σ = d x σ 2γµν
+β
δγµν
δg I
Acting on the vacuum energy functional W [γµν , g I ], we get
R
√
1
∆σ 16π 2 W = − d 4 x −γ σ − C F + 14 A G + 72
B R2
+E µν GIJ ∂µ g I ∂ν g J − 21 AIJ ∇2 g I ∇2 g J
R
√
− 2 d 4 x −γ ∂µ σ E µν WI ∂ν g I
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PROGRESS TOWARDS AN a-THEOREM
Derivation
where the curvature terms, apart from the Ricci scalar R, are
1
µνσρ αβγδ R µναβ R σργδ
F = C µνσρ Cµνσρ , G =
4
1
E µν =
R µν − γ µν R
2
µν
so that G is the Euler density and E is the Einstein tensor.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Derivation
where the curvature terms, apart from the Ricci scalar R, are
1
µνσρ αβγδ R µναβ R σργδ
F = C µνσρ Cµνσρ , G =
4
1
E µν =
R µν − γ µν R
2
µν
so that G is the Euler density and E is the Einstein tensor.
Group of local Weyl transformations is abelian:
∆σ , ∆σ 0 = 0
Using
δσ F = − 4σ F ,
δσ G = −4σ G + 8 E µν ∇µ ∇ν , etc
implies consistency conditions
∂I a =
a = A + WI β I ,
TIJ β J (g)
TIJ = GIJ + ∂I WJ − ∂J WI
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Derivation
The action of δσ on W also encodes the conformal anomaly for
“simple” theories given by
1
1
1
η µν Tµν = β I OI + L(CF − AG −
BR 2 − D∇2 R)
4
72
6
where L = (16π 2 )−1 and OI is the operator “dual” to g I (e.g. φ4
for λφ4 ) .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
Consider general gauge theory with nψ , nχ 2-component chiral
fermions ψ, χ (opposite chirality) and nφ complex scalar fields
φ, with Lagrangian
1
L=
− tr[Fµν F µν ] − D φ̄i · Dφi − ψ̄ iσ · D ψ − χ̄ i ψ̄ · D χ
4
1
−χ̄ Y i φi ψ − ψ̄ Ȳi φ̄i χ − λij kl φ̄i φ̄j φk φl
4
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
Consider general gauge theory with nψ , nχ 2-component chiral
fermions ψ, χ (opposite chirality) and nφ complex scalar fields
φ, with Lagrangian
1
L=
− tr[Fµν F µν ] − D φ̄i · Dφi − ψ̄ iσ · D ψ − χ̄ i ψ̄ · D χ
4
1
−χ̄ Y i φi ψ − ψ̄ Ȳi φ̄i χ − λij kl φ̄i φ̄j φk φl
4
At low orders TIJ = GIJ and we have
(1)
βg =
∂g a(2) =
Lb0 g 3 ,
a(2) = g 4 b0 L2 ,
1
(1) (1)
(1)
Ggg βg , Ggg = L,
4
(1)
with other components of GIJ zero.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
Consider general gauge theory with nψ , nχ 2-component chiral
fermions ψ, χ (opposite chirality) and nφ complex scalar fields
φ, with Lagrangian
1
L=
− tr[Fµν F µν ] − D φ̄i · Dφi − ψ̄ iσ · D ψ − χ̄ i ψ̄ · D χ
4
1
−χ̄ Y i φi ψ − ψ̄ Ȳi φ̄i χ − λij kl φ̄i φ̄j φk φl
4
At low orders TIJ = GIJ and we have
(1)
βg =
∂g a(2) =
Lb0 g 3 ,
a(2) = g 4 b0 L2 ,
1
(1) (1)
(1)
Ggg βg , Ggg = L,
4
(1)
(2)
with other components of GIJ zero. GY Ȳ has been computed
directly (see diagrams on next slide) and we have
2)
∂Y a(3) = GY Ȳ β (1)Ȳ
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
In general a and higher-loop contributions to GIJ can also be
directly computed but easier to use "trial and error".
Antipin,Gillioz,Molgaard, Sannino;
Antipin,Gillioz,Krog,Molgaard, Sannino
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
In general a and higher-loop contributions to GIJ can also be
directly computed but easier to use "trial and error".
Antipin,Gillioz,Molgaard, Sannino;
Antipin,Gillioz,Krog,Molgaard, Sannino Note that we expect
(1) (2)
(2) (1)
∂g a(3) = Ggg βg + Ggg βg
(2)
and this in fact "predicts" the Yukawa contribution to βg .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
In general a and higher-loop contributions to GIJ can also be
directly computed but easier to use "trial and error".
Antipin,Gillioz,Molgaard, Sannino;
Antipin,Gillioz,Krog,Molgaard, Sannino Note that we expect
(1) (2)
(2) (1)
∂g a(3) = Ggg βg + Ggg βg
(2)
and this in fact "predicts" the Yukawa contribution to βg . Can
(3)
similarly use the 2-loop results to check βg Gracey, Jones,
Pickering (as noticed by Antipin et al in the SM case).
Complete prediction in general impossible without knowledge of
(3)
Ggg .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
In general a and higher-loop contributions to GIJ can also be
directly computed but easier to use "trial and error".
Antipin,Gillioz,Molgaard, Sannino;
Antipin,Gillioz,Krog,Molgaard, Sannino Note that we expect
(1) (2)
(2) (1)
∂g a(3) = Ggg βg + Ggg βg
(2)
and this in fact "predicts" the Yukawa contribution to βg . Can
(3)
similarly use the 2-loop results to check βg Gracey, Jones,
Pickering (as noticed by Antipin et al in the SM case).
Complete prediction in general impossible without knowledge of
(3)
Ggg . The next slide shows (for the nongauged case) a(4) and
(3)
the corresponding GY Ȳ .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
A non-supersymmetric example
Only constraints:
2(β̄ + γ̄) = 4ᾱ +
1
6
= 2ᾱ + δ̄ +
1
= η̄ + ¯
2
Interesting that a is almost completely determined, GIJ less so.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The Wess-Zumino model has chiral fields Φi and a
superpotential
Z
W [Φ] = d 4 xd 2 θY ijk Φi Φj Φk + h.c
The only couplings in this case are the Yukawa couplings so
g I = (Y ijk , Ȳijk )
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The Wess-Zumino model has chiral fields Φi and a
superpotential
Z
W [Φ] = d 4 xd 2 θY ijk Φi Φj Φk + h.c
The only couplings in this case are the Yukawa couplings so
g I = (Y ijk , Ȳijk )
The β-functions are given (according to the
non-renormalisation theorem) by
βY = Y ljk γ l i +Y ilk γ l j +Y ijl γ l k ,
with Hugh Osborn, Colin Poole
βȲ = γi l Ȳljk +γj l Ȳilk +γk l Ȳijl
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The one and two loop anomalous dimensions are
1
γ = Lγ1 + C2 L2 γ2
2
where
γ1
with Hugh Osborn, Colin Poole
γ2
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The three-loop anomalous dimension is
γ3 = L3 (AγA + BγB + CγC + DγD )
where
γA
γB
γC
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
and γD is the non-planar graph. (Values of A − D given later.)
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
A proposed all-orders result for the supersymmetric a-function
is
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ
[Barnes, Intriligator, Wecht, Wright; Kutasov and Schwimmer;
Osborn, Freedman] where
Y ◦ Ȳ = Y ijk Ȳijk
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
A proposed all-orders result for the supersymmetric a-function
is
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ
[Barnes, Intriligator, Wecht, Wright; Kutasov and Schwimmer;
Osborn, Freedman] where
Y ◦ Ȳ = Y ijk Ȳijk
This will satisfy the consistency condition if
(Ȳ Λ) = γ − γ 2 + Θ ◦ βȲ
for some Θ (sufficient but not necessary). We’ll call this the
“Λ-equation”. The diagram on the next slide shows how it works
up to two loops.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The “Λ-equation” requires
1
C2 = − ,
2
(as we saw on the last slide) and at three loops
1
1
A − 2B − C = − ,
2
2
which is satisfied by DRED results
1
A=− ,
4
1
B=− ,
8
with Hugh Osborn, Colin Poole
C = 1,
D=
3
ζ(3).
2
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the Wess-Zumino model
The “Λ-equation” requires
1
C2 = − ,
2
(as we saw on the last slide) and at three loops
1
1
A − 2B − C = − ,
2
2
which is satisfied by DRED results
1
A=− ,
4
1
B=− ,
8
C = 1,
D=
3
ζ(3).
2
In fact is satisfied up to four loops (in this non-gauge case).
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the gauged case
Yukawa β function still given by non-renormalisation theorem.
Gauge β-function given by
"
#
−1
Q
−
2n
tr[γC
]
R
V
βgNSVZ = g 3 L
1 − 2CG g 2 L
where
Q = TR − 3CG ,
TR δAB = tr(RA RB ),
CG = fACD fBCD .
Jones, DRT; Novikov, Shifman, Vainstein, Zakharov
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–the gauged case
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–The gauged case
Proposed a-function is now
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–The gauged case
Proposed a-function is now
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ +Ξ(g)βg
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–The gauged case
Proposed a-function is now
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ +Ξ(g)βg
and this works if
(Ȳ Λ) = γ − γ 2 + Θ ◦ βȲ
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–The gauged case
Proposed a-function is now
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ +Ξ(g)βg
and this works if
(Ȳ Λ) = γ − γ 2 + Θ ◦ βȲ +2Lg 3 nV−1 Ξ(g)CR .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Supersymmetry–The gauged case
Proposed a-function is now
a=
1
12
nC −
1
tr(γ 2 ) +
2
1
3
tr(γ 3 ) + Λ ◦ βȲ + βY ◦ H ◦ βȲ +Ξ(g)βg
and this works if
(Ȳ Λ) = γ − γ 2 + Θ ◦ βȲ +2Lg 3 nV−1 Ξ(g)CR .
for some Θ, Ξ. Can be verified up to three loops.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
3 loops: γ (1) = 0 ; γ (3) = 0 but can redefine Y ijk so that
γ 0(3) = 0.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
3 loops: γ (1) = 0 ; γ (3) = 0 but can redefine Y ijk so that
γ 0(3) = 0.
Higher orders? Old general arguments but not completely
explicit.Lucchesi,Piguet,Sibold;
Ermushev,Kazakov,Tarasov
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
3 loops: γ (1) = 0 ; γ (3) = 0 but can redefine Y ijk so that
γ 0(3) = 0.
Higher orders? Old general arguments but not completely
explicit.Lucchesi,Piguet,Sibold;
Ermushev,Kazakov,Tarasov
Explicit proof by assuming "Λ equation".
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
3 loops: γ (1) = 0 ; γ (3) = 0 but can redefine Y ijk so that
γ 0(3) = 0.
Higher orders? Old general arguments but not completely
explicit.Lucchesi,Piguet,Sibold;
Ermushev,Kazakov,Tarasov
Explicit proof by assuming "Λ equation".
Choosing
δY ijk = Λijk ⇒ δγ = (Ȳ Λ) + βY , βg terms
to lowest order.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Finite supersymmetric theories
Historically great interest in possible finite N = 1 SUSY
theories.
For N = 1 SUSY gauge theories, 1-loop finiteness ⇒
2-loop finiteness (for softly broken theories too).
3 loops: γ (1) = 0 ; γ (3) = 0 but can redefine Y ijk so that
γ 0(3) = 0.
Higher orders? Old general arguments but not completely
explicit.Lucchesi,Piguet,Sibold;
Ermushev,Kazakov,Tarasov
Explicit proof by assuming "Λ equation".
Choosing
δY ijk = Λijk ⇒ δγ = (Ȳ Λ) + βY , βg terms
to lowest order. Use
3 (Ȳ Λ) = γ − γ 2 + βY , βg terms
to set γ = 0 order by order.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale Invariance vs Conformal Invariance
Complications when there is a continuous symmetry. In
non-SUSY case, have symmetry under
φi → δφi =
λij kl →
ωi j φj ,
ω ∈ SO(nφ )
(ωλ)ij kl = ωi m λmj kl + ωj m λim kl − λij ml ωm k − λij km ωm l
(similarly for ψ, χ)
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale Invariance vs Conformal Invariance
Complications when there is a continuous symmetry. In
non-SUSY case, have symmetry under
φi → δφi =
λij kl →
ωi j φj ,
ω ∈ SO(nφ )
(ωλ)ij kl = ωi m λmj kl + ωj m λim kl − λij ml ωm k − λij km ωm l
(similarly for ψ, χ) and correspondingly there is a possible
modification
∂I a =
µν
η Tµν =
I
B =
TIJ B J (g),
∗
I
B OI ,
β I − (vg)I .
v is zero up to two loops but a non-zero contribution at 3 loops.
Fortin, Grinstein, Stergiou Can show that * determines this
non-zero 3-loop v .
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale invariance vs conformal invariance
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale invariance vs conformal invariance
Conformal invariance requires η µν Tµν = 0 and hence
B I = 0.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale invariance vs conformal invariance
Conformal invariance requires η µν Tµν = 0 and hence
B I = 0.
Scale invariance only requires β I = 0 (fixed point).
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale invariance vs conformal invariance
Conformal invariance requires η µν Tµν = 0 and hence
B I = 0.
Scale invariance only requires β I = 0 (fixed point).
Fortin et al claim that (vg)I = 0 in all examples they
consider.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Scale invariance vs conformal invariance
Conformal invariance requires η µν Tµν = 0 and hence
B I = 0.
Scale invariance only requires β I = 0 (fixed point).
Fortin et al claim that (vg)I = 0 in all examples they
consider.
So scale invariance implies conformal invariance (for a
unitary QFT)?
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Conclusions
Strong a theorem provides powerful set of constraints on
RG functions–possible use in predicting/checking new
higher-order results?
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Conclusions
Strong a theorem provides powerful set of constraints on
RG functions–possible use in predicting/checking new
higher-order results?
Need a full proof of the "Λ" equation–or at least more
evidence.
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Conclusions
Strong a theorem provides powerful set of constraints on
RG functions–possible use in predicting/checking new
higher-order results?
Need a full proof of the "Λ" equation–or at least more
evidence.
Weak a theorem–really proved?
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM
Conclusions
Strong a theorem provides powerful set of constraints on
RG functions–possible use in predicting/checking new
higher-order results?
Need a full proof of the "Λ" equation–or at least more
evidence.
Weak a theorem–really proved?
Full resolution of scale/conformal invariance issue?
with Hugh Osborn, Colin Poole
PROGRESS TOWARDS AN a-THEOREM