Final remarks

In order to incorporate the effect from momentum-dependent interactions in our computation of nonperturbative corrections, we go one step further in the derivative expansion for the effective action, including the κ−dependent wavefunction renormalization Zκ in our ansatz. This improved truncation scheme is the so-called LPAʼ [3,6].

Our LPAʼ ansatz for the effective action includes different wavefunction renormalizations in the spatial and temporal directions and may feature a running chemical potential:

The role of interactions in the phenomenon of Bose-Einstein Condensation (BEC) is a nontrivial fundamental question. In the nonrelativistic case, it was shown [1] that nonperturbative effects associated with interactions with nonzero momentum exchange affect significantly the critical parameters for condensation, with implications for cold atom physics. Here, we consider a relativistic context, with the application to pion BEC in isospin-dense QCD [2] media in mind. Besides its phenomenological motivation, the problem of relativistic BEC in isospin dense QCD represents a particularly interesting framework to develop efficient nonperturbative methods for dense systems: being Sign-Problem free, a robust reference from lattice simulations at finite isospin chemical potential is in principle available. As a second-order phase transition, dominated thus by zero modes, the phenomenon of BEC is particularly suited for the implementation of a Functional Renormalization Group (FRG) analysis [3]. To isolate the physical problem of BEC, we consider a toy model of a complex scalar field theory with U(1) symmetry at finite density. We concentrate then on analyzing in detail how the FRG and the different possible approximations that must complement it perform in this case.

SE =

d4x

−(∂0 + iµ)π∗(∂0 − iµ)π + (∇π∗) · (∇π) +m2π∗π + λ2(π∗π)2

,

π = (π1 + iπ2)/√2

m2 > 0

Z = Tr exp(−SE)

Vκ (α) = Vκ +(m2

κ − µ2κ)

2α+

λ2κ

4α2

Γκ =

d4x

1

2(∂νπ

i)(∂νπi) + µκ

π1∂0π

2 − π2∂0π1+ Vκ (α)

α = (π1)2 + (π2)2

Rκ(q2) = (κ2 − q2)θ(κ2 − q2)

∂κVκ =κ4

12π2

±

±µ2

κ

wκ+ 1

1E2

±

1 + 2nb

E2

±

wκ =

µ2κ(κ

2 + 2V κ) + (µ2

κ + αV κ )2

E2± = κ2 + 2αV

κ + 2V κ + 2µ2

κ ± 2wκ

nb

Γκ =

d4x

Z0κ

2∂0π

a∂0πa +

Zκ

2∂iπ

a∂iπa + µκ

π1∂0π

2 − π2∂0π1+ Vκ (α)

Introduction and motivation

Functional Renormalization Group analysis of relativistic Bose-Einstein Condensation

Letícia F. Palhares* and Jean-Paul Blaizot$*Univ. Fed. do Rio de Janeiro & Univ. Estadual do Rio de Janeiro (Brazil)

$Institut de Physique Théorique, CEA-Saclay (France)

Effective Theory

Results in LPA and discussion

References[1] J.-P. Blaizot, R. Mendez Galain and N. Wschebor, Europhys. Lett. 72, 705 (2005);[2] D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592 (2001);[3] J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363, 223 (2002); B. Delamotte, arXiv:cond-mat/0702365; J. P. Blaizot, arXiv:0801.0009;[4] J. I. Kapusta, Phys. Rev. D 24, 426 (1981);[5] E. E. Svanes and J. O. Andersen, Nucl.Phys. A857, 16 (2011);[6] N. Dupuis and K. Sengupta, Europhys. Lett. 80, 50007 (2007); C. Wetterich Phys. Rev. B 77, 064504 (2008);[7] J.-P. Blaizot and LFP, work in progress.

Towards LPA ʼ : momentum exchange effects

AcknowledgementsLFP acknowledges the hospitality at IPhT (CEA-Saclay), where great part of this work was developed. This work was partially supported by CAPES-Cofecub, CNPq and FAPERJ.

We have investigated the role of interactions on the BEC transition at finite temperature and densities in a complex scalar field theory with U(1) symmetry using the nonperturbative framework of the FRG.

Our results [7] within the Local Potential Approximation are consistent with what was obtained previously in a chiral model and we believe that the relativistic BEC physics is fully present (and clean) in our toy model. We show that the shifts of the critical parameters can be directly related to medium-modification of the physical mass, present also at zero temperature (and even in perturbative results).

It is clear how one may render, through sensible approximations, the FRG formalism a powerful tool to address in a nonperturbative fashion the phase structure of in-medium field theories. The adequacy of approximations and truncations is, however, a subtle system-dependent issue. Our formal aim in the on-going part of this work is to scrutinize the results provided by FRG and its approximations within a sufficiently simple theory containing a physically motivating phenomenon (BEC), gaining understanding of how this nonperturbative flow implements nontrivial contributions and what are the limitations of the approximations used. In the near future, results beyond the leading order in the derivative expansion [7] should eventually answer whether the puzzling features found in the description of the T − μ phase-diagram are physical or rather artefacts of the LPA ansatz.

The generalization of the results to a QCD chiral model for pion condensation at finite isospin density should in principle be straightforward.

We consider a complex scalar field with real mass ( ) and U(1) symmetry at finite temperature and density. The euclidean action SE, with the partition function given by , including the U(1) charge conservation constraint is:

Comparing the latter form with the problem of the linear sigma model at finite isospin density, one can see that this is a simplified version of it in which we consider only the charged pion directions and , , with being the minimum of the chiral potential.

−λ2v2 → m2

=

d4x

1

2∂νπ

1∂νπ1 +

1

2∂νπ

2∂νπ2 + µ

π1∂0π

2 − π2∂0π1+

+(m2 − µ2)

2[(π1)2 + (π2)2] +

λ2

4[(π1)2 + (π2)2]2

- QUASIPARTICLE SPECTRUM AROUND THE CONDENSATE:

0.0 0.5 1.0 1.50.00.51.01.52.02.53.03.54.0

ΜIm

Ωq0m

v2

Indeed, the toy model result for the spectrum of fluctuations a r o u n d t h e c o n d e n s a t e resembles that of an isospin dense chiral model, featuring a BEC of for . π+ µ > m

π+

π−

µ/m

The FRG formalism provides a systematic framework for the inclusion of nonperturbative effects in field theory calculations. This is implemented via the construction of a set of effective actions . which interpolate between the classical action and the full effective action , i.e. a flow in the effective action space parameterized by κ. One interesting and convenient definition of an exact flow is that of the original theory with a modified propagator which suppresses IR modes with , satisfying thus:

Functional RG [3]

normal phase BEC phase

CHAPTER 7. NONPERTURBATIVE ANALYSIS OF RELATIVISTIC... 144

Putting these features together, we conclude that one possible interesting definition of

an exact flow is the one defined as the original theory with a modified propagator which

suppresses IR modes with momentum q ! κ. Since the aim is that of interpolating between

the classical action S[φ] and the full effective action Γ[φ], the trajectory of the FRG flow in

effective action space should satisfy the following features at different scales κ:

κ = 0 all quantum fluctuations are included (Γκ=0 = Γ)

Intermediate κ IR modes (q2 ! κ2) suppressed; UV fluctuations (q2 > κ2) included

κ = Λ → ∞ all fluctuations are suppressed; physics is classical (Γκ=Λ = S)

Notice that the existence of such trajectory for a finite UV scale Λ relies on a nontrivial

hypothesis: the classical action must be the full effective action at some scale Λ, i.e., there

exists a scale Λ at (or above) which all fluctuations are indeed suppressed in the full the-

ory. Strictly speaking, this is in general not the case in quantum field theories. From the

experience with perturbative UV renormalization, we know that not even renormalizable

theories present this exact property. When one renormalizes the theory, integrating out the

UV modes with wavelengths κ ∈ [Λ,∞, it amounts to considering the same operators as in

the original theory, but with dressed parameters. Stated in a more precise way, the initial

effective action Γκ=Λ for the FRG flow is not the naıve classical theory, but rather an action

containing the same operators as the classical one with dressed coefficients. Some remaining

fluctuations with wavelengths κ > Λ are encoded in the dressing of the parameters of the

classical action.

All the subtleties and intricacies of the perturbative renormalization procedure are there-

fore implicitly included in the FRG formalism through its initial conditions. Once finite

initial conditions are given and a sufficiently large initial scale Λ is chosen, all results at the

end of the flow (κ = 0) are both finite and consistent.

7.2.1 The exact flow equation

Let us now illustrate the derivation of the exact flow equation that generates an interpolating

trajectory as described above by considering a scalar field theory. The general guideline,

however, is universal: starting from the classical deformation of the original theory through

- EXACT RG FLOW:

- TRUNCATIONS:

A standard procedure of deforming the theory yields the exact FRG flow equation for the κ-dependent effective action:

This flow equation actually encodes an infinite hierarchy of coupled exact RG equations involving the n−point functions.

∂κΓκ[πi] =

1

2Tr

∂κRκ

G−1 +Rκ

−1

G−1(Q)

ij=

δ2Γκ

δπi(Q)δπj(−Q)

x

full propagator=

where is the regulator and the full 2-point function appears:

An adequate truncation or approximation is needed to render the exact FRG flow a practical nonperturbative method. Common choices are the derivative expansion and the disconsideration of higher order n-point functions, which are often combined, as we shall do in what follows. Our aim here is to discuss the performance of different choices in the context of relativistic BEC in hot and dense systems.

The Local Potential Approximation (LPA) is the lowest order of a derivative expansion for the effective action. Our LPA ansatz has the form of the classical euclidean action, but with κ-dependent parameters:

An adequate truncation or approximation is needed to render the exact FRG flow a practical nonperturbative method. Common choices are the derivative expansion and the disconsideration of higher order n-point functions, which are often combined, as we shall do in what follows. Our aim here is to discuss the performance of different choices in the context of relativistic BEC in hot and dense systems.

which is also truncated beyond the 4-point function. Inserting this ansatz in the flow equation and solving integrals and Matsubara sums, we obtain (for Litim regulator: )

with being the Bose distribution and:

In this ansatz, we cannot access the flow of the chemical potential independently of that of the mass, so we set it constant and end up with 3 coupled differential equations.

- COLD AND DENSE FLOW:

- THERMAL FLOW:

- PHASE DIAGRAM FOR BEC: MEAN-FIELD VERSUS FRG-LPA

16

0.0 0.2 0.4 0.6 0.80

1. 109

2. 109

3. 109

4. 109

ΜM

n ΚT0,

ΜM3

0 1 2 3 4 50

1. 109

2. 109

3. 109

4. 109

ΚM

n ΚT0,

ΜM3

Figure 6: Density versus chemical potential µ [left; κ/Λ = 0(top), 0.1, 0.5, 1 (bottom)] and

flow variable κ [right; µ/M = 0.88(top), 0.51, 0.16, 0.0005 (bottom)].

0 1 2 3 4 5

0.000015

0.00001

5. 106

0

ΚM ΜmΚ2T0,

ΜM2

2.1.2 Numerical results at finite temperature

Using the same procedure and initial conditions, we analyze temperature effects. Fig. 7

shows the flow evolution of the parameters of the effective potential at finite temperature.

0 1 2 3 4 51.00.80.60.40.20.0

ΚM

mΚ2M2

0 1 2 3 4 50.81.01.21.41.61.82.0

ΚM

Λ Κ2

Figure 7: Flow at T/M ≈ 0.51 and µ2/M2 ≈ 1.04: m2κ versus κ (left) and λ2

κ versus κ (right).

The critical line obtained for Bose-Einstein condensation is displayed in Fig. 8.

16

Flow of the density:

µ/m = 0.0005

µ/m = 0.88

Flow of the physical running mass:

The second-order BEC transition occurs when the quadratic coefficient of the effective potential flows to zero at the physical theory (k=0).

13

0 200 400 600 800 10000

10000

20000

30000

Κ a.u.

mΚ2Μ0

a.u.

0 200 400 600 800 10001.51.61.71.81.92.0

Κ a.u.Λ Κ2 Μ0

Figure 2: Flow in the vacuum (T = µ = 0): m2κ versus κ (left) and λ2

κ versus κ (right).

0 1 2 3 4 5

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

ΚM

mΚ2M2

Figure 3: Flow for different µ’s (T = 0): m2κ versus κ for µ2/M2 =

0.026 (top), 0.260, 0.650, 0.996 (bottom, dashed, critical).

The flow for different values of chemical potential is shown in Figs. 3 and 4. The effective

mass m20(µ) decreases as expected when one increases the chemical potential µ and goes to

zero when µ → M . We find:

m20

µ2 = 0.996M2

M2= 0.0000693186 . (65)

13

Quadratic coefficient of :Vκ

µ2/m2 = 0.026

µ2/m2 = 0.996[CRITICAL]

The critical chem. potential is decreased by fluctuations wrt mean-field result. This is due to running mass e f f e c t s , t r i g g e r e d b y interactions (cf. below). Condensation occurs at the physical running mass.

• The second-order BEC transition point is shifted due to mass modifications in the medium (these are present even at 1-loop level).

• Possible LPA artefacts at low and high densities which we intend to investigate beyond LPA.

• Same features observed in chiral model [5].

18

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.250.0

0.2

0.4

0.6

0.8

1.0

ΜM

TM

Figure 9: Critical lines obtained within LPA (red, solid line) and MFA (black, dashed).

Verify T → 0 result: should be perpendicular to the µ−axis (theorem, I think...).

Similar numerical difficulties in this limit for quark-meson models.

Following Ref. [6] (section 5, in particular), one can obtain the flow equation for the

wavefunction renormalization by expanding the field content in the flow equation for the

effective action in the following way:

π(x) = ϕ+∆(x) , (73)

with ϕ being a constant background field (that corresponds to the full Q = 0 contribution

to the field) and ∆(x), a small inhomogeneous fluctuation, i.e. satisfying ∆(x) ϕ and

∆(Q = 0) = 0. One can then show that the wave function renormalization is given by:

Zκ =1

β2limQ2→0

∂

∂Q2

δ2Γκ

δ∆1(−Q)δ∆1(Q)

∆=0

. (74)

Indeed, if we make the substitution (73) in our LPA’ ansatz for the effective action, Eq. (72),

we obtain:

Γκ =

d4x

Zκ

2∂ν∆

1∂ν∆1+

Zκ

2∂ν∆

2∂ν∆2+ µκ

(ϕ1

+∆1)∂0∆

2 − (ϕ2+∆

2)∂0∆

1+

+Vκ

α = αϕ + δ(x)

, (75)

18

MFA [4] FRG-LPA

BEC phase

14

0 1 2 3 4 5

1.4

1.5

1.6

1.7

1.8

1.9

2.0

ΚM

Λ Κ2

Figure 4: Flow for different µ’s (T = 0): λ2κ versus κ for µ2/M2

=

0.026 (top), 0.260, 0.650, 0.996 (bottom, dashed, critical).

However, above µ2 ≈ 0.996M2, numerical instabilities appear, signaling the criticality before

the expected value µexpcrit = M .

Actually, this can be understood as the effect of the running of the physical mass (m2κ ≡

m2κ + µ2

) at zero temperature and finite µ, as illustrated by Fig. 5.

0.0 0.2 0.4 0.6 0.8 1.00.99650.99700.99750.99800.99850.99900.99951.0000

ΜM

mΚ02T0,

ΜM2

Figure 5: Physical mass at finite µ, obtained at the end of the flow (κ → 0), as a function

of µ.

14

16

0.0 0.2 0.4 0.6 0.80

1. 109

2. 109

3. 109

4. 109

ΜM

n ΚT0,

ΜM3

0 1 2 3 4 50

1. 109

2. 109

3. 109

4. 109

ΚM

n ΚT0,

ΜM3

Figure 6: Density versus chemical potential µ [left; κ/Λ = 0(top), 0.1, 0.5, 1 (bottom)] and

flow variable κ [right; µ/M = 0.88(top), 0.51, 0.16, 0.0005 (bottom)].

0 1 2 3 4 5

0.000015

0.00001

5. 106

0

ΚM ΜmΚ2T0,

ΜM2

2.1.2 Numerical results at finite temperature

Using the same procedure and initial conditions, we analyze temperature effects. Fig. 7

shows the flow evolution of the parameters of the effective potential at finite temperature.

0 1 2 3 4 51.00.80.60.40.20.0

ΚM

mΚ2M2

0 1 2 3 4 50.81.01.21.41.61.82.0

ΚM

Λ Κ2

Figure 7: Flow at T/M ≈ 0.51 and µ2/M2 ≈ 1.04: m2κ versus κ (left) and λ2

κ versus κ (right).

The critical line obtained for Bose-Einstein condensation is displayed in Fig. 8.

16

Quartic couplingQuadratic coupling

CHAPTER 7. NONPERTURBATIVE ANALYSIS OF RELATIVISTIC... 166

(MFA),

TMFcrit =

√

3

λ2(µ2 −m2) . (7.126)

As discussed for the results at zero-temperature, the shift of the critical line in LPA

as compared to the MFA is due to the effective running of the mass (i.e. the quadratic

coefficient of the potential) in a medium. The hotter the environment the larger is this

effect, as depicted in Fig. 7.11.

Figure 7.11: Mass spectrum as a function of the chemical potential µ for different temper-

atures: T/M = 0.30584(bottom), 0.611679, 1.01947, 1.42725, 2.03893 (top). The thick red

line is the “critical” curve mκ=0 = µ, while the dashed one is the reference mκ=0 = M .

The comparison between MFA (with λ2 = 2 andm = M) and our LPA results is shown in

Fig. 7.12. This plot explicitates the fact that the critical line within the LPA FRG analysis

is for most values of µI , but not all, below the one within MFA. Physically, however, one

expects that, at fixed µI , the treatment of fluctuations – as implemented by FRG – should

in principle always enlarge the disordered phase, i.e. the noncondensed one. This might be

yet another peculiar feature of the LPA FRG result.

The second-order critical line obtained within LPA for this toy model is qualitatively very

similar to the one in Fig. 7.2, computed within the pion chiral model, even though we have

used a simplified LPA ansatz (neglectingO([φ2]3) terms). Once again, this simple field-theory

appears to be a good toy model for the description of relativistic Bose-Einstein condensation.

T/M=0.3(bottom), ..., 2.04(top)

[‘CRITICAL’]

Zκ =1

β2lim

Q2→0

∂

∂q2δ2Γκ

δ∆1(−Q)δ∆1(Q)

∆=0

The flow of the wavefunction renormalizations are then obtained via the adequate projection of the momentum dependence of the full 2-point function, e.g.:

Here, Δ is a small inhomogeneous fluctuation around the π condensate.

The solution of the set of coupled flow equations obtained in this way is currently work in progress.

Rκ

Γκ[π]

S[π]

Γ[π]

q κ