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