a two-loop calculation in quantum field theory on orbifolds nobuhiro uekusa

A two-loop calculation in quantum field theory on orbifolds

Nobuhiro Uekusa

Description of physical quantities

Action principle

Virtual processes by quantum loop corrections

In 4D theory

2

LHC era higher energies

Description of physical quantities

Action principle

Virtual processes by quantum loop corrections

Invariance of theory

Conserved currents

In 4D theory

3

LHC era higher energies

Renormalizability

Finite number of interactionsNew counterterms not required

4

accurate prediction

Renormalizability

Finite number of interactionsNew counterterms not required

Invariance of theory does not forbid non-renormalizable interactions

A non-renormalizable interactionNew counterterms

5

accurate prediction

Renormalizability

Requirement in addition to invariance of theory?

6

Renormalizability

Requirement in addition to invariance of theory? Not compulsory

Irrelevant operatorsNegligible contributions to physical quantities

In 4D, usually non-renormalizable interactions are supposed to be suppressed by a UV cutoff of a theory.

7

Renormalizability

Requirement in addition to invariance of theory? Not compulsory

Irrelevant operatorsNegligible contributions to physical quantities

In 4D, usually non-renormalizable interactions are supposed to be suppressed by a UV cutoff of a theory.

Effective theory with a large cutoff can be predictable

without requiring renormalizabilityOnly ?

8in 4D

renormalizable and non-renormalizable

interactions coexist.

fields as 4D modes can have dim-4 operators

In 4D

In a theory with compactifed extra dim

Simliar to renormalizable

terms in 4D

If coefficients of other operators are small, such a theory might be predictable with a certain accuracy.

9

If coefficients of other operators are small, such a theory might be predictable with a certain accuracy.

10

The coefficients of higher-dimensional operators

UnknownShould be eventually determined

fields as 4D modes can have dim-4 operators

11

Some attitudes

Try to construct a consistent theory to specify all the non-renormalizable interactionsSearch for rules or orders for possible interactions at each given loop level

The coefficients of higher-dimensional operators

UnknownShould be eventually determined

12

Search for rules or orders for possible interactions at each given loop level

Quantum loop corrections

to 2-point functions in 5D

theory on orbifold S /Z21

13

The action for the real scalar field

The boundary conditions for

Possible Lagrangian counterterms

and

14

Mass termNo wave function

Mass termNo wave function

Mass termWave function

15

Mass termNo wave function

Mass termNo wave function

Mass termWave function

16

1-loop KK mode expansion

Sum of diagrams for KK modes

Momentum integralsDimensionless

17

1-loop KK mode expansion

Sum of diagrams for KK modes

Momentum integrals

0 0

0

0 2n

n

f f+2n

n

f f

n

Internal mode indep of external mode

18

1-loop KK mode expansion

Sum of diagrams for KK modes

Momentum integrals

Boundary terms

Bulk terms

19

Fractions Integral expression of Gamma function

2-loop

KK mode sum Poisson’s summation

Divergent part momentum integral with a

Calculation method

cutoff regularization

counterterm

20

Now (p ) divergence has been found

It needs to be taken into account in the starting action integral

2 2

Toward extraction of physical quantities without requiring

renormalizablility

21

An effect of higher terms

Take into account (p ) terms2 2

Equation of motion (Fourier transformed)

parameter

Propagator

22

An effect of higher terms

Propagator

Two poles

Unusual signDecaying mode

23

An effect of higher terms Two poles

Unusual signDecaying mode

Propagator

24

An effect of higher terms

as a loop effect

Unnatural degree with a mass larger than the cutoff

The correction is extracted with a tuning as in 4D

large

Propagator

25

Even higher loop

4-loop, (p ) corrections2 3

3 poles in propagator

p

p

k1

k2

k3

k4

K1+k2+k3

P-k1-k2-k3-k4

P-k1-k2

26

1Quantum loop corrections to 2-point functions in 5D theory on orbifold S /Z2

2-loop, (p ) div2 2 4-loop, (p ) div2 3

For extraction of corrections for 2-pt function, the UV cutoff needs to be orders of magnitude larger compared to the compactified scale

This behavior is in agreement with the conventional observation with that contributions of higher dim operators are small for a large cutoff.

Two or more poles in propagater

SUMMARY

27

Evaluation of bulk and boundary terms

Mode expansion

Boundary terms have off-diagonal components wrt n

On the other hand, bulk terms are diagonal wrt n

28

Evaluation of divergence

Fractions Integral expression of Gamma function

KK mode sum Poisson’s summation

Intuitive interpretation of bulk divergence

e.g.

29

Evaluation of divergence

30

Evaluation of divergence

3131

1Quantum loop corrections to 2-point functions in 5D theory on orbifold S /Z2

2-loop, (p ) div2 2 4-loop, (p ) div2 3

For extraction of corrections for 2-pt function, the UV cutoff needs to be orders of magnitude larger compared to the compactified scale

This behavior is in agreement with the conventional observation with that contributions of higher dim operators are small for a large cutoff.

Two or more poles in propagater

SUMMARY