INFRARED LIMIT OF QED AT FINITE T AND FOR STRONG COUPLING Mati Péter BME, ELTE, UD Jakovác Antal ELTE
The IR-catastrophe 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 A semi-classical example From quantization: By using
The IR-catastrophe A semi-classical example (by using the decomposition: ) We are interested in:
The IR-catastrophe A semi-classical example (by using the decomposition: ) We are interested in: Let ’s define a random variable X : the number of emitted photons
Semi-classical The infrared-catastrophe
Semi-classical The infrared-catastrophe We can also calculate the average radiated energy: … after some calculations:
The IR-catastrophe A semi-classical example , where Going to the soft regime:
The IR-catastrophe A semi-classical example , where Going to the soft regime: Thus the probability of finding finite number of photons in the final state: 10 30 70
The IR-catastrophe QED k In QED from quantum corrections: p p p p P+k 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 spectral function
The spectral function In general Consider a general QFT with the field . Using the completeness relation of states: For the time ordered two point function Källén -Lehmann representation
The spectral function Properties It obeys the sum rule: for fermions for bosons optional
PROPERTIES The spectral function For a free theory at one-particle state
The spectral function Properties For an interacting theory at one-particle state bound states particle continuum
The spectral function Properties In general at
The spectral function Properties In general at We are interested in the IR region of the QED
The Bloch-Nordsieck model
The B-N model Properties Bloch and Nordsieck (1937) constructed an effective theory for the low energy QED. QED Bloch-Nordsieck ” Four velocity ”
The B-N model Properties 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)
The B-N model 1-loop perturbation theory Expand by the coupling : Divergent loop-integral (in Feynman gauge, =1 ): • Dimensional regularization • Special frame u=(1,0,0,0) (+ UV RENORM.) ( )
The B-N model 1-loop perturbation theory Dyson-equation geometric series 1-loop propagator
The B-N model 2PI resummation 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] Numerical implementation …
The B-N model 2PI resummation 1PI Effective Action 2PI Effective Action The variational principle defines a self-consistant system of equations
The B-N model Dyson-Schwinger ” Modified 2PI ” = 2PI + vertex corrections ( D-S eq.) We have a third equation! 1 2 3 (vertex function)
The B-N model Dyson-Schwinger ” Modified 2PI ” = 2PI + vertex corrections ( D-S eq.) We have a third equation! 1 2 3 (vertex function) 1 2 WARD-IDENTITIES 3
The B-N model Dyson-Schwinger ( A. Jakovac, P. Mati PHYSICAL REVIEW D85, 085006 (2012) ) (in 1937)
The B-N model Benchmarking 1 The spectral function for the various solutions Here m can be set to 0 without the loss of generality. Note that there is no mass-gap.
The B-N model Benchmarking 2 We can see the infrared sensitivity of the various solutions in more details on the l Log-Log plot of the propagators
The B-N model Benchmarking 2 However for strong coupling the PT completely breaks down in the IR.
The B-N model Wait, there is still a problem … with the sum rule. Let ’s check: But doesn ’t exist.
The B-N model Wait, there is still a problem … 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
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