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Quasiparticles in ultrarelativistic plasmas
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 606 (1996) 347-356
Quasiparticles in ultrarelativistic plasmas Jean-Paul Blaizot 1 Service de Physique Thdorique, CEA Saclay, 91191 Gif-sur-Yvette, France 2
Received 27 May 1996
Abstract
In ultrarelativistic plasmas, the existence of unscreened magnetic interactions and the presence of massless particles strongly affect the properties of the quasiparticles and of the collective modes. In this note, I discuss the peculiar splitting of the fermion dispersion relation at low momentum, which exhibits features of a collective phenomenon. I then show how the unscreened soft magnetic interactions affect the lifetime of quasiparticles.
1. Introduction
The general theme of the discussion will be how, in ultrarelativistic plasmas, quasiparticle properties are affected by unscreened long range interactions, or by the presence of massless particles to which they can couple. In analysing the general features of the phenomena uncovered in studying ultrarelativistic plasmas, such as the quark-gluon plasma, we may be able to extract general properties which could be of relevance to other physical systems, perhaps more amenable to direct observation. Thus, for example, I shall point out possible analogies with similar situations encountered in low dimensional condensed matter systems. In the weak coupling regime, the basic degrees of freedom of an ultrarelativistic plasma can be classified according to a hierarchy of scales controlled by the temperature T (T >> m, where m is the mass of the plasma constituents), and the coupling strength g (for a recent review, see e.g. Ref. [ 1] ). Two types of degrees of freedom emerge: l Affiliated with CNRS. 2 Laboratoire de la Direction des Sciences de la Mati6re du Commissariat ~ l'Energie Atomique. 0375-9474/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved Pll S0375-9474(96)00209-6
348
J.-t~ Blaizot/Nuclear Physics A 606 (1996) 347-356
i) The plasma particles, which have typical momenta of order T and a thermal wavelength of order l/T, comparable to their average relative distance (r0 "~ n -1/3 "-"
1/T). ii) The collective excitations which develop at a particular wavelength ~ 1/gT. For example the inverse screening length is ,t~ 1 ,~ gX/-~/T ~ gT. Since g << 1, 1/gT >> I/T, and excitations at wavelength 1/gT necessarily involve many particles (nh~ ~ 1/g 3 >> 1), i.e. they are collective. Similar considerations apply to cold and dense plasmas, with a chemical potential/1, >) m. I shall concentrate on two particular properties of the excitations of an ultrarelativistic plasma. The first relates to the fermion spectrum at small momenta p < gT, which is split [2]. I shall show that this phenomenon has many of the characteristic features of a collective excitation in many body systems. In particular, it can be simply understood in terms of a schematic model analogous to that introduced by Brown to describe the giant resonances in nuclei [3]. I shall suggest that the physics responsible for the splitting of the spectrum is similar to that giving rise to the pseudo-gap observed in studies of the two dimensional Hubbard model at low temperature [4]. Then I shall consider the lifetime of the quasiparticle excitations. When long range interactions are present, the perturbative estimate of this quantity is plagued with infrared divergences. A non-perturbative approach allows to resume all the dominant divergences, leading to quasiparticles with an unusual decay behaviour.
2. The spectrum of soft fermionic excitations
For simplicity, I consider here the interaction of a Dirac fermion ~ and a massless scalar field ~b with a Yukawa coupling g ~ b ~ . However, since most of the discussion extends to QED (and, in fact, to QCD as well) I refer to the fermion as an "electron" and to the scalar particle as a "photon". In presence of the interaction, the inverse propagator is G -1 = Go I + 27, where Go I is the inverse of the free Dirac propagator. At order g2, the dominant contribution to the mass operator at high temperature and/or high chemical potential can be written as follows [5]:
27(P'w)~gZ
/
p+k + nk
( 2~-)d3k3 2wkl
+ ev+k _
c~
,-~ ~
f p+k 'k nk
/
+ A+v+kw_ep+k . . . . + w k j y°
1
k dk ( f ~ + f ~ + 2nk) 0
w -dp cos c o s 00
y0_Y
Pcos0
. (1)
-I
In these expressions, 1
a ~ = ~Ep ( e p ± ( o l . p + myo))
(2)
are the projectors on positive and negative energy solutions, respectively ( ( A + ) 2 = A ±, A + + A - = 1), ep = V/p 2 + m 2. Furthermore,
J.-l~ Blaizot/Nuclear Physics A 606 (1996) 347-356
j~_
1
349
1
e~,,,±u~ + I '
nk - e~O~k _ 1
(3)
are the occupation numbers for the electrons ( f p ) , the positrons ( f + ) , and the photons (nk). The photon energy is ogk = k. In order to get the second line in Eq. (1), we have used the fact that o9, p ~ gT while the dominant contribution to the integral comes from the region k ~ T or #. Thus, for example, we have replaced ep+k by k + p cos 0, with 0 the angle between p and k. Note that, in the regime considered, the angular integral factorizes in Eq. (1). This simplification is quite analogous to that in Fermi liquid theory where all momenta lie on the Fermi surface. The energy denominators are approximately o9± p cos 0 ~ gT, glz. They are associated to virtual transitions where hard particles scatter on the soft one, with little deflection, and a possible change in their quantum numbers. For example, a hard electron can annihilate on a soft positron and "turn into" a hard photon. Aside from the changes in quantum numbers, such processes are reminiscent of the familiar Landau damping of electromagnetic waves in ordinary plasmas. They contribute an imaginary part in the region Io91 < P [6]. Let us consider in more detail what happens at very small p. Then, one can ignore the term in "y • p in v, and a simple calculation gives M2 X(og) = - - 3 " 0 , o9
(4)
where M = (g/47r) V/'/z2 + 7r2T2. At small p, the spectral weight of the Landau damping processes is concentrated at o9 = 0, which results in the imaginary part of 2(o9) being a delta function at w = 0. The spectrum is obtained from the poles of G(og) =
3"0 --o9 + M2/O9"
(5)
For 3'o = 1, there are two poles, ¢o± = ± M . There exists symmetrical poles, at -o9±, corresponding to 3"o = - l . The states at ± M are therefore doubly degenerate. This degeneracy is removed by a small fermion mass, or at finite momentum [7]. This is why, at finite momentum, the fermion dispersion relation at positive frequency appears to be split. However, this apparent splitting, which involves two states with different eigenvalues of 3"0, partly results from the particle-antiparticle symmetry. In particular, the two states at positive frequency do not merge into the same state as the interaction is turned off. The real dynamical effect is the splitting of, for example, the 3/o = 1 single particle strength into the two states at o9~: = ± M . The origin of this splitting of the 3'o = 1 state is the coupling of the electron with a large number of quasi degenerate states. This mechanism is quite general, and can be understood with the help of a "schematic model" in which a quantum state 10), with energy E0, couples with equal strength V to a large number N of states [i), uniformly distributed throughout the energy interval [ - - A E / 2 , A E / 2 ] ( A E / 2 ~< E0). In the calculation performed above, AE = 2p, corresponding to the phase space of the
350
J.-P. Blaizot/Nuclear Physics A 606 (1996) 347-356
Landau damping processes, and V v / N ~ gT. The equation determining the modes is then [ 5 ] V2
co - E 0 = ~ i
09 ~ E i
,~
NV 2 AE
w + AE/2
..In - - 09
I
AE/2
(6)
Assuming oJ large compared to AE, one can expand the logarithm and get NV 2
co - E0 ~ - co
(7)
One can now let E0 --, 0 (in this schematic model, E0 plays the role of a small mass term), and recover the two solutions at co = zLNv/NV-~ which correspond to the two poles at co+ = ± M . In the limit considered, these two solutions share the same single particle strength, i.e. 1/2. A structure somewhat similar to that studied here has been found in the study of the electron gas at intermediate densities [8]. There, the coupling of an electron hole to collective plasmons gives rise to a peak in the imaginary part of the hole self-energy which is sufficiently strong to produce a new quasiparticle, dubbed the "plasmaron". However, in the case of the non-relativistic electron gas, the validity of the perturbative approach is less clear, and it was indeed found that vertex corrections tend to suppress the structure [ 8 ]. More interestingly, it has recently been observed, in the study of the Hubbard model in two dimensions, that a pseudo-gap develops in the electron spectrum at very low temperature [4]. In this case, the nearly massless modes to which the electron couples are the long wavelength spin waves. Thus, at very low temperature, the self-energy of the electron is [4] 2(co) = C
d2q 1 l (277-)2 q2 + ( - 2 co + ie - / / ' F " q '
(8)
where C is a constant proportional to the temperature T and to the Hubbard coupling U, and ( is the antiferromagnetic correlation length. It is argued in Ref. [4] that below some cross over temperature, ( grows exponentially. A simple calculation gives Im 2 7 ( w ) =
C
1
8 x/co + 4 ( - 2
(9)
so that, for small ( - 2 , Im 27(
J.-P. Blaizot/Nuclear Physics A 606 (1996) 347-356
351
analyzed by an equation similar to Eq. (6). One finds in particular that the splitting is proportional to gx/N, reflecting the cooperative contribution of the N atoms. The analogy with the splitting of the fermion spectrum is striking, even though the mode which is split is here bosonic, i.e. it is the photon mode.
3. The lifetime of quasiparticles The validity of the quasiparticle description assumes that the damping rate y of quasiparticle excitations, obtained from the exponential decay of the retarded propagator, S R ( t , p ) ~ e x p [ - i w ( p ) t ] e x p [ - y ( p ) t ] , is small compared to the quasiparticle energy w ( p ) . In hot gauge theories, the typical quasiparticle energy is the temperature T, while one expects y ~ g2T [10-12]. The same damping rate is expected for the collective excitations, whose typical energies are ,~ gT. This suggests that, indeed, in the weak coupling regime, quasiparticles are well defined, and collective modes are weakly damped. However, the computation of y in perturbation theory is plagued with infrared divergences. Such divergences occur in all orders and can be eliminated only by a nonperturbative calculation of the quasiparticle propagator. Such a calculation has been done recently [13], and leads to the conclusion that quasiparticles do indeed exist, although they do not correspond to the usual exponential decay indicated above, but to a more complicated behaviour, SR (t, p) ~ exp [ - i w (p) t] exp [ -ceTt ln(tOplt) ], where Wpl ~ gT is the plasma frequency. Before I explain how this result was obtained, let me recall how, and why, infrared divergences occur in the calculation of the lifetime of a quasiparticle in a hot QED plasma. Physically, what limits the lifetime of a quasiparticle excitation is the collisions of the quasiparticle with the other particles in the plasma [ 14]. The collision rate can be estimated directly in the form y = no-, where n ,'-, T 3 is the density of plasma particles, and o- the collision cross section. Restricting ourselves first to the Coulomb interaction, we can write o- = f dq2(do-/dq2), with do-/dq 2 ,-~ g4/q4. Thus, 3/
g4 T 3 /
1 dq 2 c~'
(10)
which is badly infrared divergent. One knows, however, that in the plasma the Coulomb interaction is screened, so that the effective electric photon propagator is not 1/q 2 but 1/(q2 + m 2 ) , where mo ~ gT is the Debye screening mass. With this correction taken into account, the collision rate becomes Y ~ g4T 3 l m--~D ~ g2 T,
(11)
which is now finite, and of order g2T, as announced. However, screening corrections are not sufficient to eliminate all the divergences due to the magnetic interactions. To see that, consider the transverse part of the polarization tensor. At small frequency and momentum, it is imaginary:
,/.-/~ Blaizot/Nuclear Physics A 606 (1996) 347-356
352
II(q0, q) ~ .37"r 2 q0
(12)
When its contribution is included in the magnetic photon propagator, one obtains the following contribution to y:
y ,-~ g4T3
//
q
dq
.+ dqoa4
, (3~w~lqo/4q) 2.
(13)
--q
The integral over q0 can be calculated, with the result '~pl /'
y ~ gZT j
dq
(14)
which remains logarithmically divergent. This infrared problem occurs both in abelian and non-abelian gauge theories. In QCD, it is commonly bypassed by advocating the infrared cut-off provided by a possible magnetic mass ~ g2T, so that y ~ g2Tln ( l / g ) . But such a solution cannot apply for QED where one does not expect any magnetic screening [ 15]. The physical origin of the remaining infrared divergences can be traced back to the collisions involving the exchange of very soft, unscreened, magnetic photons. This is reflected in the fact that the dominant contribution to the integral (13) is concentrated at very small q0. In fact,
1 4 6(qo) q4 + (37rw~lqo/4q)2 '~'q-~O 3W2p] q
(15)
A similar observation can be made when calculating y from the imaginary part of the self-energy, on the unperturbed mass-shell w = p: y = - ~ p t1r ( p l m
S ( w + i r l , p) ) o~=l, .
(16)
In the Matsubara formalism, we have:
X(p) = - g 2 T E.
~
d3q
y~S°(p+q)y'D/~'(q)'
(17)
q0=/wm
where k = p + q , P0 = iw~, = i(2n + I)~-T and ton, = 27rmT, with integers n and m. It may he verified that the infrared logarithmic divergence in Eq. (14) arises entirely from the magnetic contribution of the term q0 = itom = 0 in the Matsubara sum of Eq. (17). One finds similar divergences in all the diagrams contributing to X, which have an arbitrary number of magnetic photon lines carrying zero Matsubara frequency. The fact that the dominant divergences are, in the Matsubara formalism, concentrated in the sector with zero Matsubara frequency, is an important simplification which will allow us to resume them in close form. Thus, one can ignore fermion loop insertions on static photon propagators (the transverse polarisation tensor at zero frequency is
J.-P, Blaizot/Nuclear Physics A 606 (1996) 347-356
353
proportional to q2, and represents a minor modification of the photon propagator). Therefore, in order to isolate the dominant divergences, we may use the "quenched approximation" [ 16], in which the retarded fermion propagator can be written as the following functional integral
SR(x,y) = f [ d a ] G R ( x , y l a )
exp{
l(a, Do,a)0} '
(18)
where G(x, ylA) is the fermion propagator in the presence of a static background gauge field, and
,/
(a, Do~A)o = ~
d3x d 3 y a ' ( x ) Dgi~j(x - y ) A . i ( y ) .
(19)
In this equation, the factor I/T has its origin in the restriction to the zero Matsubara frequency. The propagator Doij is that of a free static photon, d3q D~i(x) =_ J ~ e i q ' x
(20)
D~J(q)
with, in the Coulomb gauge, Dioi(q) = 8ij/q 2. (The final result can be shown to be gauge independent.) In the kinematical regime of interest, an approximate expression for GR(x,y[A) is obtained by neglecting the recoil of the fermion in the successive emissions or absorptions of very soft photons. More precisely, we can approximate a typical fermion propagator entering the perturbative expansion of GR(x, y[A) by
So(w,p+q)=
-wyo+(p+q) .y (w + it/)2 -- Ep+q 2
-1 w -- ¢~p - v ' q + i ~ 7
y0 - ,b. T 2 '
(21)
where q is a linear combination of the internal photons momenta and v =-- cgep/Op (v = p for the ultrarelativistic fermion). This is the familiar structure encountered in most treatments of infrared divergences in QED and which is economically exploited within the Bloch-Nordsieck model (see, e.g. Ref. [17]). In this model, G(x,y[A), satisfies the following equation (Da = c9~ + igA,~)
- i ( v . Dx) G(x,y[A) = 6 ( x - y),
(22)
which can be solved exactly. For retarded boundary conditions, and for static fields:
G~(x, ytA) = i O(xo - Yo) 8 (3) (x - y - V(Xo - Yo) ) U(x, y),
(23)
where U(x, y) is the parallel transporter
which involves the integral of the gauge potential along the straight line trajectory of the particle.
J.-P Blaizot/NuclearPhysicsA 606 (1996)347-356
354
The retarded propagator SR(x -- y) is calculated by inserting the expressions (23), (24) of GR (x, y lA) in the functional integral (18). Its Fourier transform with respect to x - y can be written as
SR(t, p) = iO(t)e -it~v'p) A(t),
(25)
where
A(t) -----/[ d a ] U(x,x - vt) exp {-½ (a, Dol a)o)
(26)
contains all the non-trivial time dependence. The functional integral is easily done:
{ i/ g2
A(t) = exp
--~T
dsl
0
ds2viDidJ(v(s2 - s2)) vj 0
)
. (27)
In this equation, Do/(X) is the coordinate space representation of the magnetostatic photon propagator (see Eq. (20)). The Sl and s2 integrations in Eq. (27) can be done by going to the Fourier representation. One obtains thus:
d 3 q b((~q- )~ 2 ( 1 - c o s t ( t , . q ) ) ) , A(t) =exp { -gZT f (27r)3
(28)
where viDoi(q)L,.j =_f)(q). The integral in Eq. (28) has no infrared divergence, but one can verify that the expansion of A(t) in powers of g2 generates the most singular pieces of the usual perturbative expansion for the self-energy. (The integral in Eq. (28) presents an ultraviolet logarithmic divergence. However, one should recall that the restriction to the static photon mode implies that such an integral is to be cut off at momenta q O)pl.) Calculating the integral in Eq. (28), one finds that at times t >> 1/tOpl the function A(t) is of the form ( a = g2/47r) ,-~
A(wplt >> 1) _~ exp (-o~Tt In OJpit) .
(29)
A measure of the decay time ~- is given by 1
o~Tlnwp,7
o~T ( l n - ~
lnln~
+
)
(30)
'7"
Since aT ~ gmpl, 7" l/g2Tln(1/g). This corresponds to a damping rate y ~ l/~g 2 T l n ( l / g ) , similar to that obtained in a one loop calculation with an infrared cut-off ~
g2T. However, contrary to what perturbation theory predicts, A(t) is decreasing faster than any exponential. It follows that the Fourier transform
S8( oo,p)
/ I --oo
dte-i~°tSR(t,p) = i l dte it(~'-v'p+iO) A(t), ,I 0
(31)
.I.-P Blaizot/NuclearPhysicsA 606 (1996)347-356
355
exists for any c o m p l e x ( a n d finite) co. Thus, the retarded propagator SR(w) is an entire function, w i t h sole singularity at Im oJ --~ - ~ . T h e p r e v i o u s analysis can be extended to the d a m p i n g o f collective excitations for which the s a m e infrared difficulty arises. By the same steps as before, we obtain the retarded p r o p a g a t o r for the two f e r m i o n i c m o d e s -4- in the f o r m O~ g.
Sa:(w,p) =iz+(p) /
d t e ''~ . . . . ±cp)+i~)A~(t),
tl
o
A± ( t ) = A ( I v ± l t ) ,
(32)
with the f u n c t i o n A ( t ) given by Eq. ( 2 8 ) . In these equations, the subscripts :1: refer to the t w o m o d e s o f the soft e l e c t r o n [ 2 ] , with energies w+(p) and residues z+(p), and v+(p)
~ Ow±(p)lop as discussed in the previous section.
Acknowledgements I wish to express m y gratitude to E d m o n d Iancu and J e a n - Y v e s Ollitrault with w h o m the results presented here were obtained. I am also indebted to Gerry B r o w n w h o first m a d e m e r e c o g n i z e the p o w e r and beauty o f Fermi L i q u i d theory.
References I 1 ] J.-P. Blaizot, E. lancu and J.-Y. Ollitrault, in: Quark Gluon Plasma 2, ed. R.C. Hwa (World Scientific, 12] [3 ] 141 15] [61 17} I8 ] [91 [ 101 111] [121
[ 13]
Singapore, 1995) p. 135. V.V. Klimov, Sov. J. Nucl. Phys. 33 (1981) 934; Soy. Phys. JETP 55 (1982) 199. G.E. Brown, Many Body Problems (North-Holland, Amsterdam, 1972). Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33 (1996) 159, and references therein. J.-P. Blaizot and J.-Y. Ollitrault, Phys. Rev. D 48 (1993) 1390. H.A. Weldon, Phys. Rev. D 26 (1982) 1394. G. Baym, J.-E Blaizot and B. Svetitsky, Phys. Rev. D 46 (1992) 4043. L. Hedin, B.I. Lundqvist and L. Lundqvist, Solid State Comm. 5 (1967) 237; P. Minnhagen, J. Phys. C 7 (1974) 3013. G.S. Agarwal, Phys. Rev. l~tt. 53 (1984) 1732; E Bernardot et al., Europhys. Lett. 17 (1992) 33. R.D. Pisarski, Phys. Rev. Lett. 63 (1989) 1129. E. Braaten and R.D. Pisarski, Phys. Rev. Lett. 64 (1990) 1338; Phys. Rev. D 42 (1990) 2156; Nucl. Phys. B 337 (1990) 569. V.V. Lebedev and A.V. Smilga, Phys. Lett. B 253 (1991) 231; Ann. Phys. 202 (1990) 229; Physica A 181 (1992) 187; R.D. Pisarski, Phys. Rev. D 47 (1993) 5589; T. Altherr, E. Petitgirard and T. del Rio Gaztelurrutia, Phys. Rev. D 47 (1993) 703; S. Peign6, E. Pilon and D. Schiff, Z. Phys. C 60 (1993) 455; A.V. Smilga, Phys. At. Nuclei 57 (1994) 519; R. Baier and R. Kobes, Phys. Rev. D 50 (1994) 5944; A. Ni6gawa, Phys. Rev. Lett. 73 (1994) 2023; E Flechsig, H. Schulz and A.K. Rebhan, Phys. Rev. D 52 (1995) 2994. J.P. Blaizot and E. lancu, Phys. Rev. Lett. 76 (1996) 3080.
356
J.-P. Blaizot/Nuclear Physics A 606 (1996) 347-356
114] G. Baym, H. Monien, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett. 64 (1990) 1867. [ 151 E. Fradkin, Proc. Lebedev Phys. Inst. 29 (1965) 7; J.E Blaizot, E. lancu and R. Parwani, Phys. Rev. D 52 (1995) 2543. [ 16] J.P. Blaizot and E. lancu, Nucl. Phys. B 459 (1995) 559. [ 17] N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, NewYork, 1959).
Nuclear Physics A 606 (1996) 347-356
Quasiparticles in ultrarelativistic plasmas Jean-Paul Blaizot 1 Service de Physique Thdorique, CEA Saclay, 91191 Gif-sur-Yvette, France 2
Received 27 May 1996
Abstract
In ultrarelativistic plasmas, the existence of unscreened magnetic interactions and the presence of massless particles strongly affect the properties of the quasiparticles and of the collective modes. In this note, I discuss the peculiar splitting of the fermion dispersion relation at low momentum, which exhibits features of a collective phenomenon. I then show how the unscreened soft magnetic interactions affect the lifetime of quasiparticles.
1. Introduction
The general theme of the discussion will be how, in ultrarelativistic plasmas, quasiparticle properties are affected by unscreened long range interactions, or by the presence of massless particles to which they can couple. In analysing the general features of the phenomena uncovered in studying ultrarelativistic plasmas, such as the quark-gluon plasma, we may be able to extract general properties which could be of relevance to other physical systems, perhaps more amenable to direct observation. Thus, for example, I shall point out possible analogies with similar situations encountered in low dimensional condensed matter systems. In the weak coupling regime, the basic degrees of freedom of an ultrarelativistic plasma can be classified according to a hierarchy of scales controlled by the temperature T (T >> m, where m is the mass of the plasma constituents), and the coupling strength g (for a recent review, see e.g. Ref. [ 1] ). Two types of degrees of freedom emerge: l Affiliated with CNRS. 2 Laboratoire de la Direction des Sciences de la Mati6re du Commissariat ~ l'Energie Atomique. 0375-9474/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved Pll S0375-9474(96)00209-6
348
J.-t~ Blaizot/Nuclear Physics A 606 (1996) 347-356
i) The plasma particles, which have typical momenta of order T and a thermal wavelength of order l/T, comparable to their average relative distance (r0 "~ n -1/3 "-"
1/T). ii) The collective excitations which develop at a particular wavelength ~ 1/gT. For example the inverse screening length is ,t~ 1 ,~ gX/-~/T ~ gT. Since g << 1, 1/gT >> I/T, and excitations at wavelength 1/gT necessarily involve many particles (nh~ ~ 1/g 3 >> 1), i.e. they are collective. Similar considerations apply to cold and dense plasmas, with a chemical potential/1, >) m. I shall concentrate on two particular properties of the excitations of an ultrarelativistic plasma. The first relates to the fermion spectrum at small momenta p < gT, which is split [2]. I shall show that this phenomenon has many of the characteristic features of a collective excitation in many body systems. In particular, it can be simply understood in terms of a schematic model analogous to that introduced by Brown to describe the giant resonances in nuclei [3]. I shall suggest that the physics responsible for the splitting of the spectrum is similar to that giving rise to the pseudo-gap observed in studies of the two dimensional Hubbard model at low temperature [4]. Then I shall consider the lifetime of the quasiparticle excitations. When long range interactions are present, the perturbative estimate of this quantity is plagued with infrared divergences. A non-perturbative approach allows to resume all the dominant divergences, leading to quasiparticles with an unusual decay behaviour.
2. The spectrum of soft fermionic excitations
For simplicity, I consider here the interaction of a Dirac fermion ~ and a massless scalar field ~b with a Yukawa coupling g ~ b ~ . However, since most of the discussion extends to QED (and, in fact, to QCD as well) I refer to the fermion as an "electron" and to the scalar particle as a "photon". In presence of the interaction, the inverse propagator is G -1 = Go I + 27, where Go I is the inverse of the free Dirac propagator. At order g2, the dominant contribution to the mass operator at high temperature and/or high chemical potential can be written as follows [5]:
27(P'w)~gZ
/
p+k + nk
( 2~-)d3k3 2wkl
+ ev+k _
c~
,-~ ~
f p+k 'k nk
/
+ A+v+kw_ep+k . . . . + w k j y°
1
k dk ( f ~ + f ~ + 2nk) 0
w -dp cos c o s 00
y0_Y
Pcos0
. (1)
-I
In these expressions, 1
a ~ = ~Ep ( e p ± ( o l . p + myo))
(2)
are the projectors on positive and negative energy solutions, respectively ( ( A + ) 2 = A ±, A + + A - = 1), ep = V/p 2 + m 2. Furthermore,
J.-l~ Blaizot/Nuclear Physics A 606 (1996) 347-356
j~_
1
349
1
e~,,,±u~ + I '
nk - e~O~k _ 1
(3)
are the occupation numbers for the electrons ( f p ) , the positrons ( f + ) , and the photons (nk). The photon energy is ogk = k. In order to get the second line in Eq. (1), we have used the fact that o9, p ~ gT while the dominant contribution to the integral comes from the region k ~ T or #. Thus, for example, we have replaced ep+k by k + p cos 0, with 0 the angle between p and k. Note that, in the regime considered, the angular integral factorizes in Eq. (1). This simplification is quite analogous to that in Fermi liquid theory where all momenta lie on the Fermi surface. The energy denominators are approximately o9± p cos 0 ~ gT, glz. They are associated to virtual transitions where hard particles scatter on the soft one, with little deflection, and a possible change in their quantum numbers. For example, a hard electron can annihilate on a soft positron and "turn into" a hard photon. Aside from the changes in quantum numbers, such processes are reminiscent of the familiar Landau damping of electromagnetic waves in ordinary plasmas. They contribute an imaginary part in the region Io91 < P [6]. Let us consider in more detail what happens at very small p. Then, one can ignore the term in "y • p in v, and a simple calculation gives M2 X(og) = - - 3 " 0 , o9
(4)
where M = (g/47r) V/'/z2 + 7r2T2. At small p, the spectral weight of the Landau damping processes is concentrated at o9 = 0, which results in the imaginary part of 2(o9) being a delta function at w = 0. The spectrum is obtained from the poles of G(og) =
3"0 --o9 + M2/O9"
(5)
For 3'o = 1, there are two poles, ¢o± = ± M . There exists symmetrical poles, at -o9±, corresponding to 3"o = - l . The states at ± M are therefore doubly degenerate. This degeneracy is removed by a small fermion mass, or at finite momentum [7]. This is why, at finite momentum, the fermion dispersion relation at positive frequency appears to be split. However, this apparent splitting, which involves two states with different eigenvalues of 3"0, partly results from the particle-antiparticle symmetry. In particular, the two states at positive frequency do not merge into the same state as the interaction is turned off. The real dynamical effect is the splitting of, for example, the 3/o = 1 single particle strength into the two states at o9~: = ± M . The origin of this splitting of the 3'o = 1 state is the coupling of the electron with a large number of quasi degenerate states. This mechanism is quite general, and can be understood with the help of a "schematic model" in which a quantum state 10), with energy E0, couples with equal strength V to a large number N of states [i), uniformly distributed throughout the energy interval [ - - A E / 2 , A E / 2 ] ( A E / 2 ~< E0). In the calculation performed above, AE = 2p, corresponding to the phase space of the
350
J.-P. Blaizot/Nuclear Physics A 606 (1996) 347-356
Landau damping processes, and V v / N ~ gT. The equation determining the modes is then [ 5 ] V2
co - E 0 = ~ i
09 ~ E i
,~
NV 2 AE
w + AE/2
..In - - 09
I
AE/2
(6)
Assuming oJ large compared to AE, one can expand the logarithm and get NV 2
co - E0 ~ - co
(7)
One can now let E0 --, 0 (in this schematic model, E0 plays the role of a small mass term), and recover the two solutions at co = zLNv/NV-~ which correspond to the two poles at co+ = ± M . In the limit considered, these two solutions share the same single particle strength, i.e. 1/2. A structure somewhat similar to that studied here has been found in the study of the electron gas at intermediate densities [8]. There, the coupling of an electron hole to collective plasmons gives rise to a peak in the imaginary part of the hole self-energy which is sufficiently strong to produce a new quasiparticle, dubbed the "plasmaron". However, in the case of the non-relativistic electron gas, the validity of the perturbative approach is less clear, and it was indeed found that vertex corrections tend to suppress the structure [ 8 ]. More interestingly, it has recently been observed, in the study of the Hubbard model in two dimensions, that a pseudo-gap develops in the electron spectrum at very low temperature [4]. In this case, the nearly massless modes to which the electron couples are the long wavelength spin waves. Thus, at very low temperature, the self-energy of the electron is [4] 2(co) = C
d2q 1 l (277-)2 q2 + ( - 2 co + ie - / / ' F " q '
(8)
where C is a constant proportional to the temperature T and to the Hubbard coupling U, and ( is the antiferromagnetic correlation length. It is argued in Ref. [4] that below some cross over temperature, ( grows exponentially. A simple calculation gives Im 2 7 ( w ) =
C
1
8 x/co + 4 ( - 2
(9)
so that, for small ( - 2 , Im 27(
J.-P. Blaizot/Nuclear Physics A 606 (1996) 347-356
351
analyzed by an equation similar to Eq. (6). One finds in particular that the splitting is proportional to gx/N, reflecting the cooperative contribution of the N atoms. The analogy with the splitting of the fermion spectrum is striking, even though the mode which is split is here bosonic, i.e. it is the photon mode.
3. The lifetime of quasiparticles The validity of the quasiparticle description assumes that the damping rate y of quasiparticle excitations, obtained from the exponential decay of the retarded propagator, S R ( t , p ) ~ e x p [ - i w ( p ) t ] e x p [ - y ( p ) t ] , is small compared to the quasiparticle energy w ( p ) . In hot gauge theories, the typical quasiparticle energy is the temperature T, while one expects y ~ g2T [10-12]. The same damping rate is expected for the collective excitations, whose typical energies are ,~ gT. This suggests that, indeed, in the weak coupling regime, quasiparticles are well defined, and collective modes are weakly damped. However, the computation of y in perturbation theory is plagued with infrared divergences. Such divergences occur in all orders and can be eliminated only by a nonperturbative calculation of the quasiparticle propagator. Such a calculation has been done recently [13], and leads to the conclusion that quasiparticles do indeed exist, although they do not correspond to the usual exponential decay indicated above, but to a more complicated behaviour, SR (t, p) ~ exp [ - i w (p) t] exp [ -ceTt ln(tOplt) ], where Wpl ~ gT is the plasma frequency. Before I explain how this result was obtained, let me recall how, and why, infrared divergences occur in the calculation of the lifetime of a quasiparticle in a hot QED plasma. Physically, what limits the lifetime of a quasiparticle excitation is the collisions of the quasiparticle with the other particles in the plasma [ 14]. The collision rate can be estimated directly in the form y = no-, where n ,'-, T 3 is the density of plasma particles, and o- the collision cross section. Restricting ourselves first to the Coulomb interaction, we can write o- = f dq2(do-/dq2), with do-/dq 2 ,-~ g4/q4. Thus, 3/
g4 T 3 /
1 dq 2 c~'
(10)
which is badly infrared divergent. One knows, however, that in the plasma the Coulomb interaction is screened, so that the effective electric photon propagator is not 1/q 2 but 1/(q2 + m 2 ) , where mo ~ gT is the Debye screening mass. With this correction taken into account, the collision rate becomes Y ~ g4T 3 l m--~D ~ g2 T,
(11)
which is now finite, and of order g2T, as announced. However, screening corrections are not sufficient to eliminate all the divergences due to the magnetic interactions. To see that, consider the transverse part of the polarization tensor. At small frequency and momentum, it is imaginary:
,/.-/~ Blaizot/Nuclear Physics A 606 (1996) 347-356
352
II(q0, q) ~ .37"r 2 q0
(12)
When its contribution is included in the magnetic photon propagator, one obtains the following contribution to y:
y ,-~ g4T3
//
q
dq
.+ dqoa4
, (3~w~lqo/4q) 2.
(13)
--q
The integral over q0 can be calculated, with the result '~pl /'
y ~ gZT j
dq
(14)
which remains logarithmically divergent. This infrared problem occurs both in abelian and non-abelian gauge theories. In QCD, it is commonly bypassed by advocating the infrared cut-off provided by a possible magnetic mass ~ g2T, so that y ~ g2Tln ( l / g ) . But such a solution cannot apply for QED where one does not expect any magnetic screening [ 15]. The physical origin of the remaining infrared divergences can be traced back to the collisions involving the exchange of very soft, unscreened, magnetic photons. This is reflected in the fact that the dominant contribution to the integral (13) is concentrated at very small q0. In fact,
1 4 6(qo) q4 + (37rw~lqo/4q)2 '~'q-~O 3W2p] q
(15)
A similar observation can be made when calculating y from the imaginary part of the self-energy, on the unperturbed mass-shell w = p: y = - ~ p t1r ( p l m
S ( w + i r l , p) ) o~=l, .
(16)
In the Matsubara formalism, we have:
X(p) = - g 2 T E.
~
d3q
y~S°(p+q)y'D/~'(q)'
(17)
q0=/wm
where k = p + q , P0 = iw~, = i(2n + I)~-T and ton, = 27rmT, with integers n and m. It may he verified that the infrared logarithmic divergence in Eq. (14) arises entirely from the magnetic contribution of the term q0 = itom = 0 in the Matsubara sum of Eq. (17). One finds similar divergences in all the diagrams contributing to X, which have an arbitrary number of magnetic photon lines carrying zero Matsubara frequency. The fact that the dominant divergences are, in the Matsubara formalism, concentrated in the sector with zero Matsubara frequency, is an important simplification which will allow us to resume them in close form. Thus, one can ignore fermion loop insertions on static photon propagators (the transverse polarisation tensor at zero frequency is
J.-P, Blaizot/Nuclear Physics A 606 (1996) 347-356
353
proportional to q2, and represents a minor modification of the photon propagator). Therefore, in order to isolate the dominant divergences, we may use the "quenched approximation" [ 16], in which the retarded fermion propagator can be written as the following functional integral
SR(x,y) = f [ d a ] G R ( x , y l a )
exp{
l(a, Do,a)0} '
(18)
where G(x, ylA) is the fermion propagator in the presence of a static background gauge field, and
,/
(a, Do~A)o = ~
d3x d 3 y a ' ( x ) Dgi~j(x - y ) A . i ( y ) .
(19)
In this equation, the factor I/T has its origin in the restriction to the zero Matsubara frequency. The propagator Doij is that of a free static photon, d3q D~i(x) =_ J ~ e i q ' x
(20)
D~J(q)
with, in the Coulomb gauge, Dioi(q) = 8ij/q 2. (The final result can be shown to be gauge independent.) In the kinematical regime of interest, an approximate expression for GR(x,y[A) is obtained by neglecting the recoil of the fermion in the successive emissions or absorptions of very soft photons. More precisely, we can approximate a typical fermion propagator entering the perturbative expansion of GR(x, y[A) by
So(w,p+q)=
-wyo+(p+q) .y (w + it/)2 -- Ep+q 2
-1 w -- ¢~p - v ' q + i ~ 7
y0 - ,b. T 2 '
(21)
where q is a linear combination of the internal photons momenta and v =-- cgep/Op (v = p for the ultrarelativistic fermion). This is the familiar structure encountered in most treatments of infrared divergences in QED and which is economically exploited within the Bloch-Nordsieck model (see, e.g. Ref. [17]). In this model, G(x,y[A), satisfies the following equation (Da = c9~ + igA,~)
- i ( v . Dx) G(x,y[A) = 6 ( x - y),
(22)
which can be solved exactly. For retarded boundary conditions, and for static fields:
G~(x, ytA) = i O(xo - Yo) 8 (3) (x - y - V(Xo - Yo) ) U(x, y),
(23)
where U(x, y) is the parallel transporter
which involves the integral of the gauge potential along the straight line trajectory of the particle.
J.-P Blaizot/NuclearPhysicsA 606 (1996)347-356
354
The retarded propagator SR(x -- y) is calculated by inserting the expressions (23), (24) of GR (x, y lA) in the functional integral (18). Its Fourier transform with respect to x - y can be written as
SR(t, p) = iO(t)e -it~v'p) A(t),
(25)
where
A(t) -----/[ d a ] U(x,x - vt) exp {-½ (a, Dol a)o)
(26)
contains all the non-trivial time dependence. The functional integral is easily done:
{ i/ g2
A(t) = exp
--~T
dsl
0
ds2viDidJ(v(s2 - s2)) vj 0
)
. (27)
In this equation, Do/(X) is the coordinate space representation of the magnetostatic photon propagator (see Eq. (20)). The Sl and s2 integrations in Eq. (27) can be done by going to the Fourier representation. One obtains thus:
d 3 q b((~q- )~ 2 ( 1 - c o s t ( t , . q ) ) ) , A(t) =exp { -gZT f (27r)3
(28)
where viDoi(q)L,.j =_f)(q). The integral in Eq. (28) has no infrared divergence, but one can verify that the expansion of A(t) in powers of g2 generates the most singular pieces of the usual perturbative expansion for the self-energy. (The integral in Eq. (28) presents an ultraviolet logarithmic divergence. However, one should recall that the restriction to the static photon mode implies that such an integral is to be cut off at momenta q O)pl.) Calculating the integral in Eq. (28), one finds that at times t >> 1/tOpl the function A(t) is of the form ( a = g2/47r) ,-~
A(wplt >> 1) _~ exp (-o~Tt In OJpit) .
(29)
A measure of the decay time ~- is given by 1
o~Tlnwp,7
o~T ( l n - ~
lnln~
+
)
(30)
'7"
Since aT ~ gmpl, 7" l/g2Tln(1/g). This corresponds to a damping rate y ~ l/~g 2 T l n ( l / g ) , similar to that obtained in a one loop calculation with an infrared cut-off ~
g2T. However, contrary to what perturbation theory predicts, A(t) is decreasing faster than any exponential. It follows that the Fourier transform
S8( oo,p)
/ I --oo
dte-i~°tSR(t,p) = i l dte it(~'-v'p+iO) A(t), ,I 0
(31)
.I.-P Blaizot/NuclearPhysicsA 606 (1996)347-356
355
exists for any c o m p l e x ( a n d finite) co. Thus, the retarded propagator SR(w) is an entire function, w i t h sole singularity at Im oJ --~ - ~ . T h e p r e v i o u s analysis can be extended to the d a m p i n g o f collective excitations for which the s a m e infrared difficulty arises. By the same steps as before, we obtain the retarded p r o p a g a t o r for the two f e r m i o n i c m o d e s -4- in the f o r m O~ g.
Sa:(w,p) =iz+(p) /
d t e ''~ . . . . ±cp)+i~)A~(t),
tl
o
A± ( t ) = A ( I v ± l t ) ,
(32)
with the f u n c t i o n A ( t ) given by Eq. ( 2 8 ) . In these equations, the subscripts :1: refer to the t w o m o d e s o f the soft e l e c t r o n [ 2 ] , with energies w+(p) and residues z+(p), and v+(p)
~ Ow±(p)lop as discussed in the previous section.
Acknowledgements I wish to express m y gratitude to E d m o n d Iancu and J e a n - Y v e s Ollitrault with w h o m the results presented here were obtained. I am also indebted to Gerry B r o w n w h o first m a d e m e r e c o g n i z e the p o w e r and beauty o f Fermi L i q u i d theory.
References I 1 ] J.-P. Blaizot, E. lancu and J.-Y. Ollitrault, in: Quark Gluon Plasma 2, ed. R.C. Hwa (World Scientific, 12] [3 ] 141 15] [61 17} I8 ] [91 [ 101 111] [121
[ 13]
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114] G. Baym, H. Monien, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett. 64 (1990) 1867. [ 151 E. Fradkin, Proc. Lebedev Phys. Inst. 29 (1965) 7; J.E Blaizot, E. lancu and R. Parwani, Phys. Rev. D 52 (1995) 2543. [ 16] J.P. Blaizot and E. lancu, Nucl. Phys. B 459 (1995) 559. [ 17] N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, NewYork, 1959).