AT FINITE T AND FOR STRONG COUPLING Mati Pter BME, ELTE, UD Jakovc - - PowerPoint PPT Presentation

INFRARED LIMIT OF QED AT FINITE T AND FOR STRONG COUPLING

Mati Péter BME, ELTE, UD Jakovác Antal ELTE

A semi-classical example

Classical conserved current + quantized EM field

distant future / past and We want to solve the EOM: The IR-catastrophe

A semi-classical example

Classical conserved current + quantized EM field

distant future / past and We want to solve the EOM:

By George Green:

The IR-catastrophe

From quantization: By using

A semi-classical example

The IR-catastrophe

We are interested in: (by using the decomposition: )

A semi-classical example

The IR-catastrophe

We are interested in: (by using the decomposition: ) Let’s define a random variable X : the number of emitted photons

A semi-classical example

The IR-catastrophe

The infrared-catastrophe

Semi-classical

The infrared-catastrophe

Semi-classical We can also calculate the average radiated energy: …after some calculations:

, where Going to the soft regime:

A semi-classical example

The IR-catastrophe

, where Going to the soft regime: Thus the probability of finding finite number of photons in the final state: 10 30 70

A semi-classical example

The IR-catastrophe

QED

p p p P+k p k In QED from quantum corrections: The loop integral diverges when the the photon momentum . An artifical mass can be introduced in order to avoid the infrared singularity.

Kinoshita-Lee-Neuenberg Theorem:

”The infrared divergences coming from loop integrals are canceled by IR divergences coming from phase space integrals of real photons.” (meaning :cross sections are IR safe) The IR-catastrophe

The spectral function

In general

The spectral function Consider a general QFT with the field . Using the completeness relation of states: For the time ordered two point function

Källén-Lehmann representation

It obeys the sum rule: for fermions for bosons

• ptional

Properties

The spectral function

The spectral function

PROPERTIES For a free theory at

• ne-particle state

• ne-particle state

bound states particle continuum For an interacting theory at

Properties

The spectral function

In general at

Properties

The spectral function

In general at We are interested in the IR region of the QED

Properties

The spectral function

The Bloch-Nordsieck model

Properties

The B-N model Bloch and Nordsieck (1937) constructed an effective theory for the low energy QED.

QED Bloch-Nordsieck

”Four velocity”

Low energy features:

• No antiparticles *
• No spin flips
• Fermionic scalar field (yes, I know… but this is the model)
• Full fermion propagator can be given in a closed form

*

Free theory

(retarded)

Properties

The B-N model

Expand by the coupling : Divergent loop-integral (in Feynman gauge, =1 ):

• Dimensional regularization
• Special frame u=(1,0,0,0)

(+ UV RENORM.)

( )

1-loop perturbation theory

The B-N model

Dyson-equation geometric series

1-loop propagator

1-loop perturbation theory

The B-N model

We need to handle the IR regime 2PI resummation

• Summing up the photon-loops (rainbow diagram)
• Treating G as full propagator
• ”Quasi particle picture”, dressing
• Details: A. Jakovac, Phys. Rev. D76, 125004 (2007). [hep-ph/0612268]

2PI resummation

The B-N model Numerical implementation…

1PI Effective Action 2PI Effective Action

The variational principle defines a self-consistant system of equations

2PI resummation

The B-N model

”Modified 2PI ” = 2PI + vertex corrections ( D-S eq.) We have a third equation!

1 2 3 (vertex function)

Dyson-Schwinger

The B-N model

”Modified 2PI ” = 2PI + vertex corrections ( D-S eq.) We have a third equation!

1 2 3 (vertex function) 3 2 1

WARD-IDENTITIES

Dyson-Schwinger

The B-N model

( A. Jakovac, P. Mati PHYSICAL REVIEW D85, 085006 (2012) )

Dyson-Schwinger

The B-N model (in 1937)

Benchmarking 1

The B-N model Here m can be set to 0 without the loss of generality. Note that there is no mass-gap. The spectral function for the various solutions

Benchmarking 2

The B-N model We can see the infrared sensitivity of the various solutions in more details on the l Log-Log plot of the propagators

Benchmarking 2

The B-N model However for strong coupling the PT completely breaks down in the IR.

Wait, there is still a problem…

The B-N model with the sum rule. Let’s check: But doesn’t exist.

Wait, there is still a problem…

The B-N model with the sum rule. Let’s check: But does not make sense. This problem can be solved by introducing a a wave function renormalization that . Anyway, this is a fingerprint of the IR singularity, which will be automatically solved on non-zero temperature.

?

THE FINITE TEMPERATURE CASE

The non-zero T

What is new?

• The hot medium assigns the frame of reference
• F-D and B-E distributions come into picture
• New kind of loop integrals (retarded Self-energy, propagators have

matrix structure; R/A & Keldysh formalism! )

The non-zero T

Some details

• At one-loop the loop integral would violate causality if the Fermi distribution wouldn’t

be set to zero. Blaizot et. al. Used the same assumptions which is the following: The fermion is a hard probe of the system, it is not part of the thermal bath.

• The calculation were performed in real time formalism which gives a 2x2 structure

to the propagators, hence it makes things more complicated

• An exact solution can be given for the case of . Otherwise numeric was used.
• The 2PI works well.

The non-zero T

The u=0 case

T raises Width spreads The excitations lifetime decreases

The non-zero T

The u=0 case

The bigger the coupling the more unstable the quasiparticle.

The non-zero T

The finite u case

Increasing u has the effect of shrink the width and hence increase the lifetime, which is quite intuitive if we think of u as a three velocity.

The non-zero T

2PI resummation

Again, only numerical solution can be achieved. They are not agreeing

• well. However let us

vary the coupling for the EXACT solution.

The non-zero T

2PI resummation

Easy now… They are not agreeing

• well. However let us

vary the coupling for the EXACT solution.

The non-zero T

2PI resummation

Almost perfect matching!

The non-zero T

2PI resummation

Actually by consistently rescaling the coupling we can map all exact graph on the 2PI.

The non-zero T

2PI resummation

Conclusion: the 2PI resummation works well in finite temperature: Apparently it seems it can reproduce the exact spectral function just need to apply a finite coupling rescaling, which can be understood as switching between renormalization scheme.

Actually there is more…

The non-zero T

2PI resummation

At extremely strong coupling ( ) a new peak occurs.

• Artifact?
• Plasmino?
• Is this present in

QED?

The non-zero T

2PI resummation

The ”gap” is growing by the temperature. There are evidences that in HTL calculations where a similar phenomenon occurs.

• Hisao Nakkagawa et. al
• Daisuke Satow et. al.

Literature

• [1] F. Bloch and A. Nordsieck, Phys. Rev. 52(1937) 54.
• N.N. Bogoliubov and D.V. Shirkov, Introduction to the theories to the

quantized elds (John Wiley & Sons, Inc., 1980)

• [3] H.M. Fried, Greens Functions and Ordered Exponentials(Cambridge

University Press, 2002)

• [4] H. A. Weldon, Phys. Rev. D 44, 3955 (1991).
• [6] A. Jakovac and P. Mati, Phys. Rev. D 85(2012) 085006

[arXiv:1112.3476 [hep-ph]].

• Catalin Catana Master Thesis
• [7] J. -P. Blaizot and E. Iancu, Phys. Rev. D 55(1997) 973 [hep-

ph/9607303].

• [8] J. -P. Blaizot and E. Iancu, Phys. Rev. D 56(1997) 7877 [hep-

ph/9706397]

• [11] N. P. Landsman and C. G. van Weert, Phys. Rept. 145, 141 (1987);
• M. Le Bellac, Thermal Field Theory, (Cambridge
• Univ. Press, 1996.)
• Edvard Munch – The scream

Thank You