Nonequilibrium chiral perturbation theory and pion decay functions

11 March 1999

Physics Letters B 449 Ž1999. 288–298

Nonequilibrium chiral perturbation theory and pion decay functions A. Gomez Nicola ´

a,1

, V. Galan-Gonzalez ´ ´

b,2

a

b

Departamento de Fısica Teorica, UniÕersidad Complutense, 28040 Madrid, Spain ´ ´ Theoretical Physics, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK Received 1 December 1998 Editor: R. Gatto

Abstract We extend chiral perturbation theory to study a meson gas out of thermal equilibrium. Assuming that the system is initially in equilibrium at Ti - Tc and working within the Schwinger-Keldysh contour technique, we define consistently the time-dependent temporal and spatial pion decay functions, the counterparts of the pion decay constants, and calculate them to next to leading order. The link with curved space-time QFT allows to establish nonequilibrium renormalisation. The short-time behaviour and the applicability of our model to a heavy-ion collision plasma are also discussed in this work. q 1999 Elsevier Science B.V. All rights reserved. PACS: 12.39.Fe; 11.10.Wx; 05.70.Ln; 12.38.Mh Keywords: Nonequilibrium thermal field theory; Chiral perturbation theory; Pion decay constants; Heavy-ion collisions; Quantum field theory in curved space-time

1. Introduction The chiral phase transition plays a fundamental role in the description of the plasma formed after a relativistic heavy-ion collision ŽRHIC., where it is imperative to use meson effective models to describe QCD. Two of the most successful approaches are the O Ž4. linear sigma model ŽLSM., valid only for Nf s 2 light flavours, and Chiral Perturbation Theory ŽChPT., based on derivative expansions compatible with the QCD symmetries, and whose lowest order action is the nonlinear sigma model ŽNLSM. w1x. In ChPT, the perturbative parameter is prLx , with p a meson energy Žlike masses, external momenta or temperature. and Lx , 1 GeV. Every meson loop is O Ž p 2rLx2 . and all the infinities coming from them can be absorbed in the coefficients of higher order lagrangians w1,2x. In thermal equilibrium at finite temperature T, the chiral symmetry is believed to be restored at Tc , 150–200 MeV w3x. In fact, near Tc , the mean-field LSM is well known to undergo a second-order phase 1 2

E-mail: [email protected] E-mail: [email protected]

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 6 7 - 2

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transition. The NLSM is equally valid for reproducing the phase transition, provided one works in the large-N limit w4x. Strictly within ChPT, the low-temperature meson gas has been studied w5,6x on expansion in T 2rLx2 , predicting the correct behaviour of the observables as T approaches Tc . The equilibrium assumption is not realistic if one is interested in the dynamics of the expanding plasma formed after a RHIC, where several nonequilibrium effects could be important. One of them is the formation of disoriented chiral condensates ŽDCC., regions in which the chiral field is correlated and has nonzero components in the pion direction w7x. As the plasma expands, long-wavelength pion modes —propagating as if they had an effective negative mass squared— can develop instabilities growing fast as the field relaxes to the ground state, an observable consequence being coherent pion emission w8x. This issue has been extensively studied in the literature, mostly within the LSM assuming initial thermal equilibrium at Ti ) Tc , either encoding the cooling mechanism in the time dependence of the lagrangian parameters w8–11x or describing the plasma expansion in proper time and rapidity w12x. This phenomenon has also been studied using Gross-Neveu models w13x. Another important nonequilibrium observable is the photon and dilepton production w14x, to which the anomalous meson sector could significantly contribute w10x. In this work we will construct an effective ChPT-based model to describe a meson gas out of thermal equilibrium, as an alternative to the LSM approach. Our only degrees of freedom will be then the Nambu-Goldstone bosons ŽNGB. and we will consider the most general low-energy lagrangian compatible with the QCD symmetries. We will restrict here to Nf s 2 Žwhere the NGB are just the pions. and to the chiral limit Žmassless quarks., which is the simplest approximation allowing to build the model in terms of exact chiral symmetry. One of the novelties of our approach is to exploit the analogy between ChPT and the physical regime where the system is not far from equilibrium and then a derivative expansion is consistent.

2. The NLSM and ChPT out of equilibrium We will take the system in thermal equilibrium for t F 0 at a temperature Ti - Tc and for t ) 0 we let the lagrangian parameters be time-dependent. We are also assuming that the system is homogeneous and isotropic. The generating functional of the theory can then be formulated in the path integral formalism, by letting the time integrals run over the Schwinger-Keldysh contour C displayed in Fig. 1 w15–18x. We will eventually let t i ™ y` and t f ™ q`, although we will show that our results are independent of t i and t f . We remark that, even in that limit, the imaginary-time leg of C has to be kept, since it encodes the KMS equilibrium boundary conditions w17–19x. With these assumptions, our low-energy model will be the following nonequilibrium NLSM S wU x s

HC

d4 x

f 2Ž t. 4

tr EmU † Ž x ,t . E m U Ž x ,t .

Ž 1.

where HC d 4 x ' HC dtHd 3 x, UŽ x,t . g SUŽ2. is the NGB field, satisfying UŽ x,t i q i bi . s UŽ x,t i . with bi s Tiy1 , and f Ž t . is a real function which in equilibrium and to the lowest order Žsee Section 4. would be f s fp , 93 MeV Žthe pion decay constant. i.e, f Ž t F 0. s f. Note that f Ž t . cannot be analytic at t s 0 and, in particular, it

Fig. 1. The contour C in complex time t. The lines C1 and C2 run between t i q i e and t f q i e and t f y i e and t i y i e respectively, with e ™0q.

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could be discontinuous, like the meson mass in quenched LSM approaches w8–11x. This is a consequence of the nature of our approach, since the system is driven off equilibrium instantaneously. An alternative, which we will not attempt here, is to choose f Ž t . analytic ; t, having equilibrium only at t s t i w20x. Thus, the temporal evolution of our results will start at t s 0q, an infinitesimally small response time. As we will see below, our approach is consistent because the discontinuities at t s 0 appear to NLO. We will parametrise the field U as U Ž x ,t . s

1 f Ž t.

f 2 Ž t . y p 2 Ž x ,t .

½

1r2

I q itap a Ž x ,t . ;

5

a s 1,2,3,

Ž 2.

where p 2 s p apa , p a the pion fields satisfying p a Ž t i q i bi . s p a Ž t i . and I and ta are the identity and Pauli matrices. Note that with the choice Ž2. we recover the canonical kinetic term in the action after expanding U in powers of p . Other choices amount to a time-dependent normalisation of the pion fields and should not have any effect on the physics Žsee Section 4.. For instance, if we redefine p˜ a s p a f Ž0.rf Ž t ., the action for the p˜ a Ž x,t . fields is just the equilibrium NLSM multiplied by the time-dependent scale factor f 2 Ž t .rf 2 Ž0. Žsee below.. Our action Ž1. is manifestly chiral invariant ŽUŽ x . ™ LUŽ x . R † .. Notice that we work in the chiral limit and hence there are no explicit symmetry-breaking pion mass terms in the action. The conserved axial and vector currents for the chiral symmetry can be derived by applying the standard procedure w1,2x, so that the axial current reads Ama Ž x ,t . s i

f 2Ž t. 4

tr t a Ž U †Em U y UEm U † .

Ž 3.

Let us now discuss how to establish a consistent nonequilibrium ChPT. The new ingredient we need is the temporal variation of f Ž t .. We will then consider f˙Ž t . f 2Ž t.

p

,O

ž / Lx

,

f¨Ž t . f 3Ž t.

,

f˙Ž t . f 4Ž t.

2

p2

,O

ž / Lx2

,

Ž 4.

and so on, the rest of the chiral power counting being the same as in equilibrium. Therefore, in our approach we treat the deviations of the system from equilibrium perturbatively, following the ChPT guidelines. Thus, we will expand our action Ž1. to the relevant order in pion fields and take into account all the contributing Feynman diagrams. The loop divergences should be such that they can be absorbed in the coefficients of higher order lagrangians, which in general will require the introduction of new time-dependent counterterms Žsee below.. Notice also that according to Ž4., we can always describe the short-time nonequilibrium regime, just by expanding f Ž t . around t s 0q. In fact, for times t F fpy1, that is equivalent to a chiral expansion, since then f˙Ž0q . trf Ž0q . s O Ž prLx . and so on. Nonetheless, we stress that the conditions Ž4. do not imply working at short-times, but just to remain close enough to equilibrium. To leading order in p fields, the action Ž1., after using Ž2., reads S0 w p x s y 12

HCd

4

xp a Ž x ,t . I q m2 Ž t . p a Ž x ,t .

with m2 Ž t . s y

f¨Ž t . f Ž t.

,

Ž 5.

where we have partial integrated in C. Thus, the leading order nonequilibrium effect of our model can be written as a time-dependent pion mass term, which, as commented before, is a common feature of nonequilibrium models w9–12x. Notice that m2 Ž t . can be negative, so that our model accommodates unstable pion modes,

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whose importance we have discussed before. Note also that this mass term does not break the chiral symmetry, i.e, the axial current is classically conserved. Indeed, to leading order we have, from Ž3., Ama Ž x ,t .

LO

s yf Ž t . Em p a Ž x ,t . q dm 0 f˙Ž t . p a Ž x ,t . ,

m

2

Ž 6.

a

which satisfies E Am s 0 using I q m Ž t . p s 0, the equations of motion to the same order. Had we included the pion mass term mp —explicitly breaking the symmetry— the instabilities threshold, to leading order, would have been m 2 Ž t . - ymp2 instead. It is very interesting to rephrase our model as a NLSM in a curved space-time background gmn , which reads w21,22x, f 2 Ž 0.

d 4 x Ž y g . g mn tr Em U † Ž x . En U Ž x . q j SR w U, R x Ž 7. 4 C plus U independent terms, where g s det g and the last term accounts for possible couplings between the pion fields and the scalar curvature RŽ x . Žlike RŽ x . f 2 for a free scalar field f w21x.. Now, notice that our nonequilibrium model Ž1. is obtained by writing UŽ x . in the p˜ parametrisation discussed before Ži.e, with f Ž t . replaced by f Ž0. in Ž2.., choosing j s 0 Žminimal coupling. and a spatially flat Robertson-Walker ŽRW. space-time in conformal time, whose line element is ds 2 s a2 Žh .w dh 2 y dx 2 x, with the scale factor aŽh . s f Žh .rf Ž0.. Our effective theory is then not only suitable for a RHIC environment, but also in a cosmological framework. Notice also that if we take j / 0, the lowest order SR term we can construct has the form of an effective mass term breaking explicitly the chiral symmetry. In fact, it is not difficult to see that we could cancel the m 2 Ž t . term in Ž5. by choosing j s 1r6, which is the value rendering the theory scale invariant w21x. This is just a consequence of the lagrangian chiral and conformal symmetries being incompatible in a curved background w22x or, equivalently, at nonequilibrium. In other words, for j s 0 —which is our choice, since we want to preserve chiral symmetry, as in Ref. w22x — we may interpret the m2 Ž t . term, in the chiral limit, as the minimal coupling with the background yielding chiral invariance. The above equivalence turns out to be very useful to renormalise our model, consistently with ChPT. In fact, all the one-loop divergences arising from Ž7. can be absorbed in the coefficients of the O Ž p 4 . action S4 , which consists of the Minkowski terms with indices raised and lowered with gmn plus new chiral-invariant couplings of pion fields with the curvature w22x. In the chiral limit, those new terms read Sg wU x s

S4R w p x s

'

H

L11 R Ž x . g mn q L12 R mn trEm U † Ž x . En U Ž x .

4

HCd x Ž'y g .

s y 12

HCd

4

xp a f 1 Ž t . E t2 y f 2 Ž t . = 2 q m12 Ž t . p a q O Ž p 4 .

Ž 8.

where Rmn is the Ricci tensor, L11 and L12 are two new low-energy constants w22x and we have given the two-pion contributions in the parametrisation Ž2., after partial integration, with our RW metric, where f 1 Ž t . s 12 Ž 2 L11 q L12 .

f 2 Ž t . s 4 Ž 6 L11 q L12 .

m12 Ž t . s y

f¨Ž t . f 3Ž t. f¨Ž t .

f 3Ž t.

y L12

q L12

f 1 Ž t . f¨Ž t . q f˙1 Ž t . f˙Ž t . f Ž t.

2

f˙Ž t .

,

f 4Ž t. f˙Ž t . f 4Ž t.

2

,

q 12 f¨1 Ž t . .

Ž 9.

for t ) 0 and f i Ž t F 0. s 0. The above terms are the only ones in S4 containing two pions and they will renormalise purely nonequilibrium infinities —which are time-dependent and vanish for t F 0—. It is important

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to bear in mind that to cancel the one-loop new divergences only L11 needs to be renormalised, whereas L12 s Lr12 w22x. We will come back to this point below. Next, we will concentrate on the Green functions time-ordered along C w17,18x. Unless otherwise stated, we will be using the parametrisation Ž2. in the remaining of this work. The two-point function defines the pion propagator G a b Ž x, y . s yi ²TC p a Ž x .p a Ž y .:, which to leading order G 0ab Ž x, y . s d a b G 0 Ž x, y ., by isospin invariance, and

 I x q m2 Ž x 0 . 4 G 0 Ž x , y . s ydC Ž x 0 y y 0 . d Ž3. Ž x y y .

Ž 10 .

with KMS equilibrium conditions G 0) Ž x,t i y i bi ; y . s G 0- Ž x,t i ; y ., the advanced and retarded propagators being defined as customarily along C. Notice that GŽ x, xX . s GŽ t,tX , x y xX . due to the nonequilibrium lack of time translation invariance. Therefore, we will define, as customarily, the ‘‘fast’’ temporal variable t y tX and the ‘‘slow’’ one t ' Ž t q tX .r2, so that F Ž q0 , v q ,t . and F Ž v q ,t,tX ., with v q2 s < q < 2 , will denote, respectively, the fast and mixed Žin which only the spatial coordinates are transformed. Fourier transforms of F Ž x, xX .. Note that F Ž q0 , v q ,t . depends separately on q0 and v q because of the thermal loss of Lorentz covariance and has the extra nonequilibrium t-dependence. Then, in the mixed representation, Ž10. becomes d2 dt 2

q v q2 q m 2 Ž t . G 0 Ž v q ,t ,tX . s ydC Ž t y tX .

Ž 11 .

The general solution of Ž11. is only known explicitly for some particular choices of m2 Ž t . w21,17,18x. Formally, we can write it as a Schwinger-Dyson equation as G 0 Ž v q ,t ,tX . s G 0e q Ž v q ,t y tX . q

2

HCdzm Ž z . G

eq 0

Ž v q ,t y z . G 0 Ž v q , z ,tX .

Ž 12 .

with G 0e q Ž v q ,t y tX . the equilibrium solution of Ž11., i.e, with m2 Ž t . s 0. Another object of interest for our purposes is the Lehman spectral function r Ž x, y . s G ) Ž x, y . y G - Ž x, y . w16x, which in equilibrium to leading order is r 0e q Ž q . s y2p isgnŽ q0 . d Ž q 2 . w19x. Note that, by construction, G ) Ž x, y . s G - Ž y, x ., so that r Ž x, y . s yr Ž y, x . and r Ž q0 , v q ,t . s yr Žyq0 , v q ,t .. The normalisation of r 0 is 1 2p i

q`

Hy` q

0

r 0 Ž q0 , v q ,t . s

d r 0 Ž v q ,t ,tX . dt

Ž 13 .

s y1, tst

X

which can be readily checked by using Ž12. and r 0 Ž v q ,t,t . s 0. 3. Next to leading order propagator We will now obtain the NLO correction to the propagator. For that purpose, we need the action in Ž1. up to four-pion terms: S w p x s S0 w p x q

1 2

HCd

4

x

½

1 f 2Ž t.

a

m

b

1 2

2 2

Em p E p pap b q Ž p .

ž

f¨Ž t . f Ž t.

y

f˙2 Ž t . f 2Ž t.

/

qO Žp 6 .

5

Ž 14 .

plus the two-pion terms in Ž8.. The two diagrams contributing are, respectively, Ža. and Žb. in Fig. 2.

Fig. 2. Diagrams contributing to the NLO pion propagator Ža,b. and axial-axial correlator Žc.. The black dot in b. represents the interaction coming from S4R in Ž8..

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Let us concentrate on G 11 Ži.e, t,tX g C1 in Fig. 1. and t and tX positive, which is the relevant case for our purposes, as commented above. We will use dimensional regularisation ŽDR., so that evaluating the above diagrams, using Ž10. with d Ž d . Ž0. s 0, and after some algebra, we obtain in the mixed representation to NLO ) G 11

X

Ž t ,t .

) s G 0,11

qi

X

Ž t ,t . Ž 1 y t

½H

0

t

X

1 2

f 1Ž t . q f 1Ž t .

coth q

2

Ti2 12 f

2

cos v q Ž t q tX .

X

1

q

t

Ht d t˜ D Ž t˜, v . G

y

. q 2v

bi v q

d t˜ D1 Ž t˜, v q . G 0) Ž t˜,t . G 0) Ž t˜,tX . q D2 Ž t˜. G˙ 0) Ž t˜,t . G˙ 0) Ž t˜,tX .

H0 d t˜ D Ž t˜, v . G

y

i

1

X

q

0

0

Ž t˜,t . G 0- Ž t˜,tX . q D2 Ž t˜. G˙ 0- Ž t˜,t . G˙ 0- Ž t˜,tX .

Ž t˜,t . G 0) Ž t˜,tX . q D2 Ž t˜. G˙ 0- Ž t˜,t . G˙ 0) Ž t˜,tX .

5

Ž 15 .

-Ž )Ž X . and G11 t,tX . s G11 t ,t , where we have suppressed for simplicity the v q dependence of the propagators, the ˜ dot denotes drdt,

D1 Ž t˜, v q . s

1 2

f Ž t˜.

ž

6

f¨Ž t˜. f Ž t˜.

y5

ž / /

G 0 Ž t˜. f 2 Ž t˜.

y v q2 G 0 Ž t˜. y 2G¨ 0 Ž t˜. q 4

f Ž t˜.

q i v q2 f 2 Ž t˜. y f 1 Ž t˜. y i

D2 Ž t˜. s

2

f˙Ž t˜.

f˙1 Ž t˜. f˙Ž t˜. f Ž t˜.

,

q 12 f¨1 Ž t˜. ,

f˙Ž t˜. f Ž t˜.

G˙ 0 Ž t˜.

Ž 16 .

Ž 17 .

and G 0 Ž z 0 . ' G 0 Ž z, z . is the equal-time correlation function. We observe that Ž15. is t i and t f independent, which is a good consistency check. Notice also that by replacing the equilibrium propagators in Ž15., we recover eq) G 11

X

Ž tyt .

eq) s G 0,11

X

ž

Ž tyt . 1y

T2 12 f 2

/

Ž 18 .

which agrees with w4x Žnote that we have derived it for the contour C, including both imaginary-time and real-time thermal field theory. and is finite in the chiral limit, where there is no tadpole renormalisation in DR w2x. However, out of equilibrium, the NLO propagator is in general divergent, even in the chiral limit, and the infinities have to be absorbed in the two-pion counterterms in Ž8..

4. The nonequilibrium pion decay functions In a thermal bath, the concepts of LSZ and asymptotic states are subtle, and so is then the extension of low-energy theorems like PCAC. Thus, pion decay constants are more conveniently defined through the thermal ab Ž axial-axial correlator Amn x, y . s ²TC Ama Ž x . Amb Ž y .:. At T / 0 the loss of Lorentz covariance in the tensorial structure of Amn implies that one can define two independent and complex fps Žspatial. and fpt Žtemporal., their

A. Gomez Nicola, V. Galan-Gonzalezr Physics Letters B 449 (1999) 288–298 ´ ´ ´

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real and imaginary parts being related respectively with the pion velocity and damping rate in the thermal bath w23x. Nevertheless, to one-loop in the chiral limit one has w5,4,23x fps Ž T .

2

2

s fpt Ž T .

ž

sf 2 1y

T2 Tc2

/

,

Ž 19 .

with Tc s '6 fp , 228 MeV. Despite it being just the lowest order in the low temperature expansion, Ž19. predicts the right behaviour and a reasonable estimate for the critical temperature, although, strictly speaking, fp ŽT . is not the order parameter w4x. To higher orders, fps / fpt and Im fps,t / 0 w23x. ab Let us then analyze Amn in our nonequilibrium model. The relevant quantity, as far as fp is concerned, is the ) ab spectral function rmn s Amn y Amn , with Amn s d a bAmn . We readily realise that rmn Ž q0 ,q,t . s yrnmŽyq0 ,y q,t .. Then, from rotational symmetry,

r i j Ž q0 ,q,t . s qi q j r LŽ q0 , v q ,t . q d i j r d Ž q0 , v q ,t .

Ž 20 .

with r L, d Ž q0 . s yr L, d Žyq0 . 3 and r j0 Ž q0 ,q,t . s q j rS Ž q0 , v q ,t .. Therefore, rmn is characterized, in principle by the four functions r L , r d , rS and r 00 . However, they are related through the Am conservation Ward Identity ŽWI. Emx r mn Ž x, y . s Enyr mn Ž x, y . s 0, which also holds in our model. Thus, we get i

q 0r 00 Ž q0 , v q ,t . y v q2 rS Ž q0 , v q ,t . y q 0rS Ž q0 , v q ,t . y v q2 r LŽ q0 , v q ,t . q

2 i

2

r˙ 00 Ž q0 , v q ,t . s 0,

r˙S Ž q0 , v q ,t . q r d Ž q0 , v q ,t . s 0,

Ž 21.

where the dot denotes ErEt . Thus, only two components of rmn are independent, as in equilibrium w4x, where there are no time derivatives in the above equation. At T s 0 one has r L s 2p fp2 sgnŽ q 0 . d Ž q 2 ., since there exist NGB states. That is not the case at T / 0, where the pion dispersion relation is not in general a d-function w4x. In fact, to define properly fp ŽT . requires taking the v q ™ 0q limit, in which a zero-energy excitation still exists w4x, although to NLO there is no need to take that limit. Extending these ideas to nonequilibrium, we will define the time-dependent pion decay functions ŽPDF. as fps Ž t .

2

1

`

lim q

s 2p

fps Ž t . fpt Ž t . s

v q™0

1 2p

fps Ž t . gp Ž t . s y

Hy`dq q 0

0

r L Ž q0 , v q ,t . s lim q i v q™0

d dt

r L Ž v q ,t ,tX . tst

Ž 22 .

X

`

lim q

v q™0

i 2p

Hy`dq

0

rS Ž q0 , v q ,t . s lim q rS Ž v q ,t ,t .

`

lim q

v q™0

Ž 23 .

v q™0

Hy`dq q 0

0

rS Ž q0 , v q ,t . s 12 lim q v q™0

d dt

rS Ž v q ,t ,tX . tst

X

Ž 24 .

The functions fps Ž t . and fpt Ž t . are the nonequilibrium counterparts of the spatial and temporal pion decay constants respectively, whereas gp Ž t . vanishes in equilibrium. However, the above PDF are related through the WI. Integrating in q0 in Ž21., we get fps Ž t . gp Ž t . s

1 d 2 dt

fpt Ž t . fps Ž t . ,

Ž 25 .

so that only two PDF are independent, as in equilibrium w23x. Let us now check the consistency of our definitions to leading order. From Ž6., r LLO Ž v q ,t,tX . s if Ž t . f Ž tX . r 0 Ž v q ,t,tX . and r d s 0, so that, using Ž13. yields 3

Our r L and r d correspond in in the notation of w4x, to sgnŽ q 0 . rAL q02 r v q2 q 2 and sgnŽ q 0 . rAT respectively.

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2 fps Ž t .LO s f 2 Ž t ., i.e, the PDF coincides with f Ž t . to leading order, as it should be. Similarly, we find tŽ . fp t LO s fps Ž t .LO s f Ž t . and gp Ž t .LO s f Ž t˙., so that our definitions are consistent to leading order. To NLO, we need the axial current up to O Žp 3 .. From Ž3.,

Ama Ž x ,t . s Ama Ž x ,t .

LO

1 y 2 f Ž t.

ž

p a Em p 2 y p 2Em p a y dm 0

f˙Ž t . f Ž t.

/

p 2p a q O Ž p 5 .

Ž 26 .

with AmLO in Ž6.. Thus, according to our chiral power counting, we have three types of NLO corrections to Amn . The first is the NLO correction to the pion propagator we have evaluated in Section 3, coming from the product of the O Žp . terms above. The second is the product of the O Žp . with the O Žp 3 ., represented by diagram c. in Fig. 2, and the third comes from the modification in Am due to the action Ž8., which amounts to prefactors w1 q f 1Ž t .x and w1 q f 2 Ž t .x in A 0 and A j respectively. Then, evaluating rmn to NLO, after using Ž15. Žwe take, without loss of generality, both t,tX g C1 and positive. and Ž22. – Ž23., we finally arrive to fps Ž t .

2

fpt

2

s f 2 Ž t . 1 q 2 f 2 Ž t . y f 1 Ž t . y 2 iG 0 Ž t . 2

Ž t . s f Ž t . 1 q f 2 Ž t . y 2 iG 0 Ž t .

Ž 27 . Ž 28 .

for t ) 0. This is the main result of this work. It provides the NLO relationship between the PDF and f Ž t . 4 . Notice that fps Ž t . / fpt Ž t . to NLO, unlike the equilibrium case, due to the effect of nonequilibrium renormalisation. However, note that w fps Ž t .x 2 y w fpt Ž t .x2 s f 2 Ž t .w f 2 Ž t . y f 1Ž t .x, which is finite, since it depends only on L12 , which does not renormalise. This is indeed an interesting consistency check, because the one-loop infinities appearing in G 0 Ž t . can then be absorbed in L11 , rendering both fps Ž t . and fpt Ž t . finite. We also remark that we did not need to take v q ™ 0q in Ž22. – Ž23. to arrive to Ž27. – Ž28. Žthere are still NGB to NLO.. Note also that both fps,t are real to this order. We have performed the following consistency checks on Ž27. – Ž28.: first, the equilibrium result Ž19. is recovered Žfor the contour C . simply by replacing G 0e q s yiT 2r12 and f 1 s f 2 s 0. Second, by calculating gp Ž t . from rS , through Ž24., we check explicitly that the WI Ž25. holds and, third, we have calculated Amn in the p˜ parametrisation, arriving to the same result. Therefore, Ž27. – Ž28. allow to express nonequilibrium observables Žlike decay rates, masses, etc. to one loop in ChPT, in terms of the physical fp Ž t ., which could be measured, for instance, in nonequilibrium lepton decays p ™ ln l . At this stage one can follow different approaches. Exact knowledge of f Ž t . would require to solve self-consistently the plasma hydrodynamic equations or, equivalently, Einstein equations for the metric. Alternatively, one can treat f Ž t . as external —so that Ž27. – Ž28. provide the system response— and study simple choices consistently with Ž4. w20x. In what follows, we shall take f Ž t . arbitrary and expand it near t s 0q, analysing thus the short-time evolution. For short times, the particular form of f Ž t . is not important and we can parametrise the nonequilibrium dynamics in terms of the values of f Ž t . and its derivatives at t s 0q. As we discussed in Section 2, this approach is justified for times t - t max with t max , 1rfp Ž0. , 2 fmrc Žcompare to the typical plasma time scales 5–10 fmrc w12x.. The general solution of Ž11. with KMS conditions at t i can be constructed in terms of two independent solutions to the homogeneous equation, which have to be continuous and differentiable ; t g C so that the solution is uniquely defined w17x. Therefore, they have to match the equilibrium solution and its first time derivative at t s 0. With these conditions and expanding both f Ž t . and the solutions near t s 0q we find to the lowest order i bi v q G 11 coth 1 y m2 t 2 q O Ž m4 t 4 . Ž 29 . 0 Ž v q ,t ,t . s y 2 vq 2 for t ) 0, with m2 s yf¨Ž0q .rf Ž0q .. For m 2 - 0 we see the unstable modes threshold, making the pion correlation function grow with time. The effect of those modes is not important for short times though, where 4

G 0 Ž t ., f 1Ž t . and f 2 Ž t . depend implicitly on f Ž t ., through Ž10. and Ž9..

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the exponential growth of the correlator is not appreciable. Observe that in Ž29. the time dependence factorises, so that the momentum dependence is the same as in equilibrium and then we can integrate it in DR, yielding the finite answer Ž18.. Then, from Ž27. – Ž28. we get fps,t Ž t .

2

s f Rs,t

2

½

1y

Ti2 Tc2

ž

q 2 Ht y m 2 1 y

Ti2 Tc2

/

y H 2 t 2 q O Ž p 3rLx2 .

5

Ž 30 .

for t ) 0, where H s f˙Ž0q .rf Ž0q ., with the renormalised constants f Rs

2

s f 2 Ž 0q . q 4 Ž L12 y 6 L11 . m 2 y L12 H 2

f Rt

2

s f 2 Ž 0q . q 4 y Ž L12 q 6 L11 . m 2 q L12 H 2

Ž 31 .

and where the H and m parameters Žwhich are O Ž p . and play the role of the Hubble constant and the deceleration parameter in the Universe expansion. also get renormalised, in terms of f˙1,2 Ž0q ., f¨1,2 Ž0q . and so on, but those are subleading contributions. Thus, for short times, all the effect of the S4 terms, which is Ti independent, is to redefine fp Ž0q. , since there are no infinities coming from G 0 in DR. We insist that this is just the effect of truncating the series in t and it is not true in general. Notice that Ž31. implies necessarily a nonzero jump D f s f Ž0q. y f Žsee our comments in Section 2. so that the divergent part of L11 can be absorbed in f 2 Ž0q. rendering a finite D f Rs,t s D fps,t. In fact, that effect is very small compared to the other contributions in Ž30., and so it is the difference fps Ž t . y fpt Ž t ., since Lr11 , Lr12 , 10y3 w22x. Notice also that for the particular case H 2 s m 2 ) 0, renormalising as f 2 Ž0q. s f 2 q 24 m2 L11 we get fps s fpt and D fp s 0. Finally, we will estimate some physical effects related to fp Ž t .. For that purpose, we will ignore, for simplicity, the effect of L11 and L12 and, based upon Ž19., define the plasma effective temperature as fp2 Ž t . s f 2 w1 y T 2 Ž t .rTc2 x. Therefore, we can also define a critical time as T Ž t c . s Tc and a freezing time T Ž t f . s 0. Thus, we will impose 0 - T Ž t . - Tc and then, through our short-time results for fp Ž t ., determine either t c or t f , depending on the initial conditions ŽT Ž t . is just quadratic in time to this order.. Notice that we are following a similar approach as in equilibrium when one extrapolates Ž19. until T s Tc . Let us then take typical values Ti , < H <, < m < , 100 MeV and retain only the leading order in x ' Ti2rTc2 , consistently with the chiral expansion. Then, if H s 0, the system cools down until t f2 m 2 , yx Ž1 q x .Ž t f , 0.2 fmrc . if m2 - 0, whereas for m2 ) 0 it is heated until t c2 m2 s 1Ž t c , 2 fmrc ., independent of Ti . For H ) 0, there is cooling until t f < H < , xr2Ž t f , 0.2 fmrc .. Finally, for H - 0 and m2 ) 0 there is heating until t c < H < , Ž1 y 3 xr4.r2Ž t c , 2 fmrc ., whereas if m 2 - 0, there is heating until a maximum t m < H < , Ž1 q xr2.r2 and then cooling down until t f < H < , 1 q x Ž t f , 2.3 fmrc .. We observe that the effect of the unstable modes Ž m2 - 0. is always to cool down the system and that the freezing time for H - 0 is much longer than that for H ) 0. Some of these time scales are indeed longer than those to which our short-time approximation remains valid, but they have to be understood as estimates, similarly to estimating Tc at equilibrium through Ž19., even though the low T approach is less reliable near T , Tc . Comparing with w12x, naively identifying the LSM order parameter Õ Ž t . , fp Ž t . Žin proper time., we see a similar short-time evolution, although our estimates for the time evolution duration are somewhat lower. This was expected, since the initial values in Ref. w12x correspond to Ti , 200 MeV and < H < , 400 MeV, which are too high for our low-energy approach. An important remark is that in typical simulations like w12x, Õ Ž t . reaches a stationary value, about which it oscillates Žthermalisation.. It is clear that we cannot predict that type of behaviour only within our short-time approach, quadratic in time, but only estimate the time scales involved —similarly as to why ChPT cannot see the phase transition—. Therefore, in view of the above estimates, we believe that our ChPT model may be useful for studying the different nonequilibrium observables evolution, from a stage where some cooling has already taken place onwards. In principle, we could approach closer to Tc by considering enough orders in our ChPT, although in practice, beyond one-loop, some resummation method, like large N, will need to be implemented.

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5. Conclusions and Outlook We have extended chiral lagrangians and ChPT out of thermal equilibrium. The chiral power counting requires all time derivatives to be O Ž p . and to lowest order our model is a NLSM with f ™ f Ž t .. This model accommodates unstable pion modes and corresponds to a spatially flat RW metric in conformal time with scale factor aŽ t . s f Ž t .rf Ž0. and minimal coupling. We have exploited this analogy to establish the renormalisation procedure, which allows to construct the fourth order lagrangian absorbing all the loop divergences, which in general will be time-dependent. We have applied our model to study the time-dependent pion decay functions, extending the equilibrium pion decay constants. In general there are two independent PDF, as in equilibrium, and to NLO in ChPT they already differ, unlike equilibrium, due to renormalisation. We have obtained them to NLO in terms of the equal-time correlation function, analysing their lowest order short-time coefficients and their dependence with Ti , and discussing the relevant time scales involved within the context of a RHIC plasma. Among the aspects of our model which are worth to be studied further are the long-time evolution, by choosing suitable parametrisations for f Ž t ., including the analytic approach, and the behaviour of the two-point correlation function at different space points, which would allow us to investigate the formation of regions of unstable vacua ŽDCC. w20x. Other applications and extensions, to be explored in the future include photon production in the pion sector Žby gauging the theory and including p 0 anomalous decay., the quark condensate time dependence Žby including the mass explicit symmetry-breaking terms., the Nf s 3 case, large N resummation and proper time evolution.

Acknowledgements We are grateful to T. Evans and R. Rivers for countless and fruitful discussions, as well as to R.F. Alvarez-Estrada, A. Dobado and A.L. Maroto for providing useful references and comments. A.G.N wishes to thank the Imperial College group for their kind hospitality during his stay there and has received financial support through a postdoctoral fellowship of the Spanish Ministry of Education and CICYT, Spain, project AEN97-1693.

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