# Presentation: 840065

## Heavy quark diffusion at next-to-leading order

- Simon Caron-Huot
- McGill University
- Based on work with Guy D. Moore,
-  arXiv:0708.4232, arXiv:0801.2173

- Talk for “QGP Spectral Functions and Euclidean Correlators” at BNL, Apr. 2008.

## Outline of Talk

- Introduction
- Why calculate transport coefficients at NLO, & Why heavy quark diffusion?
- The O(*g*) corrections to **
- Our calculation
- Operator definition of **
- The HTL effective theory
- Some details
- Results

## 1. Introduction (Why calculate transport coefficients at NLO?)

- Perturbative series for the pressure: converges poorly unless
- Solely due to “soft scale” physics (*p*~*gT*)
- (Braaten&Nieto PRL76:1417, 1996)
- Expansion parameter isto be contrasted with

-  This expansion parameter can be   thought of as:

## 1.1. Introduction (II)

- Transport coefficients tend to be sensitive to *gT*-scale physics; this happens for: photon production rate, shear viscosity, jet quenching, heavy quark E loss...
- All of them could receive large O(*g*) corrections!
- - The tools needed to calculate them already exist (HTL effective theory)

## 1.2. Why calculate ‘nonrelativistic heavy quark diffusion’?

-  The simplest thing to calculate!
-  Nonrelativistic  momentum evolves continuously   Heavy (*M*<< *T*)  velocity evolves slowly
-  Langevin process:

- (Gaussian noise)

-  Only one parameter:

-  A force-force correlator:

## Corrections to . (First,  at LO)

-  QGP = gas of ~ free quarks   and gluons
-  Semiclassical scattering:
-  Momentum diffusion:

- (To get full leading order, one must go beyond this semiclassical approximation and use the full matrix elements)

- E.Braaten& M.H.Thoma,  PRD44,1298(1991)G.D.Moore& D.Teaney PRC71,hep-ph/0412346

##  at NLO: overlapping scatterings

-  Total scattering rate:

-  Scattering duration:  up to ~ *(**gT**)**-1*
-  Probability for two  scatterings event to  overlap: *g*

-  In QED, that wouldn’t do much: deflections are small
-  But in QCD: color can change!

##  at NLO: overlapping scatterings (II)

##  at NLO: virtual corrections

-  An O(*g*) fraction of *m**D**2* originates from soft gluons:

-  But soft gluon propagators & vertices suffer from O(1) HTL corrections!

##  at NLO: soft gluons

- The scattering target can be a soft gluon

-  The probability for this to happen is O(*g*):

-  For *k**~ **gT* the HTL corrections are O(1)!

## 2. Parenthesis: why  is nice

- In other transport coefficients, collinear splittings, including the LPM effect are important (photon production, jet E loss, shear viscosity...)

- Arnold, Moore & Yaffehep-ph/0109064

-  None of this is relevant for heavy quarks!

## 3. Our calculation

- Use the HTL effective theory: integrate out the scale *T* and focus on the scale *gT*.
- Operator definition of **

- E. Braaten and R. Pisarski,Nucl.Phys.B337:569,1990.

- Where:

- J. Casalderrey-Solana & D. Teaney,hep-ph/0605199

## 3.1. Operator definition of  

-  In regular gauges, the *A**i* terms are total derivatives ( 0),  and operator ordering does not matter at O(*g*):

## The real-time formalism

-  Better to work with:
-  Averaged field:
-  Differenced field:

-  The HTL theory takes a simple form in this basis.

-  Propagator:

- SCH arXiv:0710.5726

## 3.2. The HTL diagrams

-  Must include all two-loop HTL diagrams.  All propagators are soft (*p**~ **gT*) and resummed.

## 3.2. The HTL effective theory

## 3.3. Some details

- Integration would seem 4D: *p,|q|,q**0* and an angle;it can be beaten down to 3D.

## 3.3. Some details(too much?)

## 3.3. Some details...

## 3.3. Some details...

## 3.3. Some more details...

- Integrals are 4D: *p,q,q**0* and an angle.

## 4.1. The result

- Where:

- (LO constant)

- (NLO coefficient)

## 4.1. The result (II)

## 4.2. Discussion

- The NLO correction is very large.  Even for 3-flavors QCD at T=100GeV, I wouldn’t trust the perturbative series for **.
- What happened?
- Outlook:
- If expansions in (*g**2**N**c**T*/*m**D*) are no good for understanding the gluons, what else to do?

- Half of correction is “the full gluon self-energy screens *less* than its HTL approximation”.

## Backup: Classical (nonabelian) plasma physics

- Explain the relation with classical plasma physics