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(1)

One-loop -function for an infinite-parameter family of

gauge theories

Kirill Krasnov

(Nottingham)

(2)

This talk is about a calculation

But it is worth putting it into context

(3)

Any field theory is renormalisable once all terms compatible with symmetries are added

into the Lagrangian

Weinberg

(4)

If we could follow the RG flow in the infinite-

dimensional space of couplings, the flow could take us to “nice” UV fixed points

The idea of asymptotic safety

In the context of renormalisable theories, realised for

asymptotically-free theories

(5)

It is generally believed that “nice” UV fixed points of non-renormalisable theories (such as gravity) are of

non-perturbative nature

Cannot say anything using perturbative methods Have to use Wilsonian non-perturbative

interpretation of the RG flow

Most of the available results are of this nature, but

their interpretation is not without difficulty

(6)

In this talk, let me see how far one can go staying

perturbative

(7)

Perturbative renormalisation:

Divergences appear because propagators are generalised

functions, and their products (or products of their derivatives) are ill-defined

Can make sense of such products of generalised functions, but ambiguities arise

These are of the form of a dependence on some arbitrary energy scale, and come in typical logarithmic form

The dependence on the (log of) this energy scale is the same as the dependence on the logarithm of the energy, and this allows to track the energy logarithms

Perturbative RG flow resums the energy logarithms

In practice one computes the “divergencies” to extract the RG flow

(8)

In principle, no problem in applying this programme to a non-renormalisable theory

In practice, has to deal with higher-dimension operators, and computations become challenging, to say the least

It seems that to compute even at one-loop, one needs to start with the most general Lagrangian at the tree level

This is clearly impossible in practice

(9)

However, and this is the crucial insight:

In perturbative calculations, the higher-dimensional operators (typically) do not affect the RG flow of the

lower-dimensional ones

(10)

Example:

Imagine one adds an F^3 term to the tree-level Yang-Mills Lagrangian

Effect on the one-loop flow of the g 2 in L

YM

= 1

4g

2

(F

µ⌫

)

2

Answer: this operator does not affect the flow, the flow is

unaware of a possible presence of this operator in the Lagrangian But g affects the flow of the coupling constant for F^3

Lower-dimensional operators are relevant for higher- dimensional ones, but not the other way around

@

@ log(µ)

✓ 1 4g

2

= 11C

2

6(4⇡)

2

(11)

Philosophy:

Can work with a number of operators, and what has not been added to the tree-level Lagrangian

will not affect the flow of what is included

So, can trust the computed (one-loop) RG flow, at least in the regime where the perturbation theory is

valid (i.e. when one can neglect the second loop) One does not need all operators to do meaningful

computations

Only applies to perturbative RG flow.

Functional RG behaves completely differently!

(12)

The gravity example:

One loop renormalisable: the arising divergences can be absorbed into the metric redefinition, as well as renormalisation of the EH GR one-loop counterterm

Christensen and Duff ’80

1

1

180(4⇡)

2

Z

212(R

µ⌫⇢

)

2

2088⇤

2

Or

@

@ log(µ)

✓ ⇤

8⇡G

= 58⇤

2

5(4⇡)

2

On-shell EH action (modulo volume)

(13)

⇤ cannot run

@(⇤G)

@ log(µ) = 29

5⇡ (⇤G)

2

small, becomes even smaller in the UV (⇤G)

Interpretation:

This result must be insensitive to adding (W eyl)

3

required at two loops Asymptotic freedom, G weakens in the UV

Can we check this by an explicit calculation?

Leads to fourth order differential operator,

challenging but not impossible

(14)

Gravity as theory of connections

Formalism that describes geometry using a connection, not metric as the main variable

g = @A F = @A

Both metric and the curvature are derivatives of the connection

Field equations @

2

A = A

second order PDE’s on the connection On-shell

Ricci = @

2

g = @

3

A = @A = g W eyl = @

2

g = @

3

A = @A = F

Requires non-zero

cosmological constant

(15)

Linearised Lagrangian is of the form L

(2)

= (@A)

2

Allows to incorporate (W eyl)

3

as (@A)

3

type term More generally can consider

L =

X

1 n=2

g

n

(@A)

n

+ lower in derivatives terms

An infinite number of interactions interesting from the metric viewpoint

This formalism does not lead to an increase in the order of

the operators arising - still second order

(16)

“Deformations” of Yang-Mills theory

To gain intuition and develop technology, can play similar games with the usual Yang-Mills theory

Consider

L =

X

1 n=2

g

n

F

n

= M

4

f (F/M

2

) where is an arbitrary Lorentz and

gauge-invariant function of the curvature f F = dA + (1/2)[A, A]

Field equations d

A

✓ @f

@F

= 0

second-order PDE’s on the connection

M some energy scale Just an effective field theory

Lagrangian, but without derivatives of the curvature

Euler-Heisenberg

(17)

Calculations become possible and mimic what happens in

“deformations” of General Relativity when takes

self-dual part of the curvature Or, using spinor notations

F = (F

µ⌫a

)

sd

⌘ 1

2 F

µ⌫a

+ 1

4 ✏

µ⌫

F

a

a

Lie algebra index

F = F

ABa

A, B = 1, 2

spinor indices (unprimed)

L = M

4

f (F

ABa

/M

2

)

Linearisation around an arbitrary background

L

(2)

= 1

2 (f

00

)

ab ABCD

(dA)

aAB

(dA)

bCD

+ M

2

(f

0

)

a AB

[[A, A]]

aAB

(f00)ab ABCD @2f

@FABa @FCDb (f

0)a AB @f

@FABa

Second-order, but non-Laplace type

(dA)aAB ⌘ d(AA0Aa AB) 0 [[A, A]]aAB ⌘ fabcAbAA0AcBA0

self-duality makes things simpler

(18)

But the problem can be converted into a Laplace-type one using the first order-formalism

B

aAB

auxiliary field

Legendre transform of f (F

ABa

/M

2

) Linearised Lagrangian

L = B

a AB

F

ABa

M

4

V (B

ABa

/M

2

).

V (B

aAB

/M

2

)

L

(2)

= 2b

a AB

(da)

aAB

+ B

a AB

[[a, a]]

aAB

1

2 (V

00

)

abABCD

b

a AB

b

b CD

After gauge-fixing can write as L

(2)

= b a

✓ V

00

\d

\d B

◆ ✓ b a

\d - Dirac operator

(19)

The square of the arising operator is of Laplace-type

✓ V

00

\d

\d B

2

=

✓ \d

2

+ (V

00

)

2

V

00

\d + \dB

\dV

00

+ B \d \d

2

+ B

2

Can now compute the divergent parts of the regularised determinant in the usual way, using the heat kernel

\d

2

= + F

Result: after a (local) field redefinition, this family of “deformed”

non-renormalisable Yang-Mills theories is one-loop renormalisable

like GR at one-loop!

(20)

The arising renormalisation group flow

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ = f (x), (2)

where f is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature xaAB := FABa /M2. The result of our calculation is

f (x) = 1 (4⇡)2

1 6

hxabABxbaAB 3((f00) 1)abABAB(f0)bcM NxcaM N (3)

+3((f00) 1)abAMBN (f0)bcBC((f00) 1)cdCN DM (f0)daD Ai .

Here (f0)aAB and (f00)abABCD are the matrices of the first and second derivatives of the function f (f0)aAB := @f

@xaAB , (f00)abABCD := @2f

@xaAB@xbCD , (4)

and (f00) 1 is the matrix inverse to f00. We also use the notation

xabAB := CaebxeAB, (f0)abAB := Caeb(f0)eAB, where Cabc are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

fYM(x) = 1

4g2 (xaAB)2, (5)

and thus

(fYM0 )abAB = 1

2g2 xabAB, ((fYM00 ) 1)abABCD = 2g2 abA(C|B|D). A simple computation then gives

fYM = 11C2

6(4⇡)2 (xaAB)2, (6)

where C2 is the quadratic Casimir. This gives the correct running of 1/4g2.

When both the Lagrangian (1) and the -function (3) are expanded in powers of xaAB, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F3 interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ =

f

(x), (2)

where

f

is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature x

aAB

:= F

ABa

/M

2

. The result of our calculation is

f

(x) = 1 (4⇡)

2

1 6

h x

abAB

x

baAB

3((f

00

)

1

)

abABAB

(f

0

)

bcM N

x

caM N

(3)

+3((f

00

)

1

)

abAM BN

(f

0

)

bcB C

((f

00

)

1

)

cdCN DM

(f

0

)

daD A

i .

Here (f

0

)

aAB

and (f

00

)

abABCD

are the matrices of the first and second derivatives of the function f (f

0

)

aAB

:= @f

@x

aAB

, (f

00

)

abABCD

:= @

2

f

@x

aAB

@x

bCD

, (4)

and (f

00

)

1

is the matrix inverse to f

00

. We also use the notation

x

abAB

:= C

aeb

x

eAB

, (f

0

)

abAB

:= C

aeb

(f

0

)

eAB

, where C

abc

are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

f

YM

(x) = 1

4g

2

(x

aAB

)

2

, (5)

and thus

(f

YM0

)

abAB

= 1

2g

2

x

abAB

, ((f

YM00

)

1

)

abABCD

= 2g

2 ab

A(C

|B|D)

. A simple computation then gives

fYM

= 11C

2

6(4⇡)

2

(x

aAB

)

2

, (6)

where C

2

is the quadratic Casimir. This gives the correct running of 1/4g

2

.

When both the Lagrangian (1) and the -function (3) are expanded in powers of x

aAB

, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F

3

interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

xabAB ⌘ f aebxeAB

xaAB ⌘ FABa /M2

(f0)abAB ⌘ faeb(f0)eAB

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ = f(x), (2)

where f is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature xaAB := FABa /M2. The result of our calculation is

f(x) = 1 (4⇡)2

1 6

hxabABxbaAB 3((f00) 1)abABAB(f0)bcM NxcaM N (3)

+3((f00) 1)abAMBN(f0)bcBC((f00) 1)cdCNDM(f0)daD Ai .

Here (f0)aAB and (f00)abABCD are the matrices of the first and second derivatives of the function f (f0)aAB := @f

@xaAB , (f00)abABCD := @2f

@xaAB@xbCD , (4)

and (f00) 1 is the matrix inverse to f00. We also use the notation

xabAB := CaebxeAB, (f0)abAB := Caeb(f0)eAB, where Cabc are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

fYM(x) = 1

4g2 (xaAB)2, (5)

and thus

(fYM0 )abAB = 1

2g2 xabAB, ((fYM00 ) 1)abABCD = 2g2 abA(C|B|D). A simple computation then gives

fYM = 11C2

6(4⇡)2 (xaAB)2, (6)

where C2 is the quadratic Casimir. This gives the correct running of 1/4g2.

When both the Lagrangian (1) and the -function (3) are expanded in powers of xaAB, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F3 interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

what flows at one loop is the function f

1501.00849

(21)

The YM example

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ = f(x), (2)

where f is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature xaAB := FABa /M2. The result of our calculation is

f(x) = 1 (4⇡)2

1 6

hxabABxbaAB 3((f00) 1)abABAB(f0)bcM NxcaM N (3)

+3((f00) 1)abAMBN(f0)bcBC((f00) 1)cdCN DM(f0)daD Ai .

Here (f0)aAB and (f00)abABCD are the matrices of the first and second derivatives of the function f (f0)aAB := @f

@xaAB , (f00)abABCD := @2f

@xaAB@xbCD , (4)

and (f00) 1 is the matrix inverse to f00. We also use the notation

xabAB := CaebxeAB, (f0)abAB := Caeb(f0)eAB, where Cabc are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

fYM(x) = 1

4g2 (xaAB)2, (5)

and thus

(fYM0 )abAB = 1

2g2 xabAB, ((fYM00 ) 1)abABCD = 2g2 abA(C|B|D). A simple computation then gives

fYM = 11C2

6(4⇡)2 (xaAB)2, (6)

where C2 is the quadratic Casimir. This gives the correct running of 1/4g2.

When both the Lagrangian (1) and the -function (3) are expanded in powers of xaAB, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F3 interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

Then

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ = f(x), (2)

where f is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature xaAB := FABa /M2. The result of our calculation is

f(x) = 1 (4⇡)2

1 6

hxabABxbaAB 3((f00) 1)abABAB(f0)bcM NxcaM N (3)

+3((f00) 1)abAMBN (f0)bcB C((f00) 1)cdCN DM(f0)daD Ai .

Here (f0)aAB and (f00)abABCD are the matrices of the first and second derivatives of the function f (f0)aAB := @f

@xaAB , (f00)abABCD := @2f

@xaAB@xbCD , (4)

and (f00) 1 is the matrix inverse to f00. We also use the notation

xabAB := CaebxeAB, (f0)abAB := Caeb(f0)eAB, where Cabc are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

fYM(x) = 1

4g2 (xaAB)2, (5)

and thus

(fYM0 )abAB = 1

2g2 xabAB, ((fYM00 ) 1)abABCD = 2g2 abA(C|B|D). A simple computation then gives

fYM = 11C2

6(4⇡)2 (xaAB)2, (6)

where C2 is the quadratic Casimir. This gives the correct running of 1/4g2.

When both the Lagrangian (1) and the -function (3) are expanded in powers of xaAB, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F 3 interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

We compute one-loop divergences using the background field method. We work in Riemannian signature. The one-loop renormalisability of (1) means that, after a local field redefinition, the re- maining one-loop divergences are taken care of by counter terms of the type already contained in (1).

Thus, the one-loop running of the whole infinite set of the coupling constants can be encoded as the running of the dimensionless function f

@f (x)

@ log µ = f (x), (2)

where f is some gauge- and Lorentz-invariant function of the dimensionless self-dual part of the curvature xaAB := FABa /M2. The result of our calculation is

f (x) = 1 (4⇡)2

1 6

hxabABxbaAB 3((f00) 1)abABAB(f0)bcM NxcaM N (3)

+3((f00) 1)abAMBN (f0)bcB C((f00) 1)cdCN DM(f0)daD Ai .

Here (f0)aAB and (f00)abABCD are the matrices of the first and second derivatives of the function f (f0)aAB := @f

@xaAB , (f00)abABCD := @2f

@xaAB@xbCD , (4)

and (f00) 1 is the matrix inverse to f00. We also use the notation

xabAB := CaebxeAB, (f0)abAB := Caeb(f0)eAB, where Cabc are the Lie algebra structure constants.

Note that the result (3) is a homogeneity degree zero function in f , as it should be. As a further check, we note that for Yang-Mills

fYM(x) = 1

4g2 (xaAB)2, (5)

and thus

(fYM0 )abAB = 1

2g2 xabAB, ((fYM00 ) 1)abABCD = 2g2 abA(C|B|D). A simple computation then gives

fYM = 11C2

6(4⇡)2 (xaAB)2, (6)

where C2 is the quadratic Casimir. This gives the correct running of 1/4g2.

When both the Lagrangian (1) and the -function (3) are expanded in powers of xaAB, one can read o↵ an infinite set of beta-functions for the couplings stored in f . In the main text we carry out this exercise for the first non-trivial coupling parameterising the F3 interaction. We reproduce the result obtained in [1] by a di↵erent method. Some further comments on the interpretation of (3) are contained in the Discussion section.

Let us give an outline of how (3) is computed. All details are given in the main text. The straightforward application of the background field method to (1) runs into a difficulty. The problem is that the second-order operator that arises by linearising (1) is not of Laplace-type, even after gauge-fixing. We alleviate the problem by passing to the first-order formulation, by ”integrating in”

an auxiliary field. The linearisation of the resulting Lagrangian then gives a first-order operator that turns out to be of Dirac-type. Its square is of Laplace-type, which makes the well-developed heat-kernel

2

and

(22)

More involved example

L = 1

4g

2

(F

ABa

)

2

+ ↵

3!g

2

M

2

f

abc

F

AaB

F

Bb C

F

Cc A

in agreement with our philosophy, the flow of is unchanged g

2

For the flow of we obtain ↵

@(↵g)

@ log(µ) = ↵g

3

C

2

(4⇡)

2

It is that tells how to change g

2

grows in the UV as could be expected from a non-renormalisable interaction

(23)

Discussion

Can do meaningful (one-loop) calculations even in non-renormalisable theories

Results can be trusted (when perturbation theory can be) because what has not been included at the tree level should not affect the flow of what has been included

With some tricks, even the problem for higher-derivative operators can be reduced to Laplace-type operators

The calculation for L = GR + X

1

n=3

g

n

(W eyl)

n

using such tricks is in progress

Interpretation is clean, RG flow for

“observable” on-shell couplings

(24)

What can one hope to learn this way?

Such perturbative caclulations for non-renormalisable theories have been hardly ever done: surely surprises await us

It is not impossible that there are some sufficiently large families of non-renormalisable theories (but smaller than the general EFT) that are renormalisable, in the sense that if one is in, one stays in

Some yet unknown UV fixed points may be of perturbative

nature and thus discoverable by doing one-loop computations for sufficiently large families of theories

The computed RG flow for “deformations” of YM is still to be understood

Then one can compute

(25)

Perturbative (one-loop) RG flow calculations with non-renormalisable theories can be done, and results

can be trusted (in appropriate regimes) Take home message

Thank you!

The one-loop renormalisability of GR is just the tip of an iceberg

Potentially there is new physics discoverable this way

References

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• Utbildningsnivåerna i Sveriges FA-regioner varierar kraftigt. I Stockholm har 46 procent av de sysselsatta eftergymnasial utbildning, medan samma andel i Dorotea endast

På många små orter i gles- och landsbygder, där varken några nya apotek eller försälj- ningsställen för receptfria läkemedel har tillkommit, är nätet av

Figur 11 återger komponenternas medelvärden för de fem senaste åren, och vi ser att Sveriges bidrag från TFP är lägre än både Tysklands och Schweiz men högre än i de

Det har inte varit möjligt att skapa en tydlig överblick över hur FoI-verksamheten på Energimyndigheten bidrar till målet, det vill säga hur målen påverkar resursprioriteringar

Detta projekt utvecklar policymixen för strategin Smart industri (Näringsdepartementet, 2016a). En av anledningarna till en stark avgränsning är att analysen bygger på djupa

 Påbörjad testverksamhet med externa användare/kunder Anmärkning: Ur utlysningstexterna 2015, 2016 och 2017. Tillväxtanalys noterar, baserat på de utlysningstexter och