Quark--Gluon plasma 3

Thermodynamics

Fig. 4.

of the High- Temperature

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71

The lowest order correction to the thermodynamic potential in ^>4 theory.

so that: ^ J 0 (m) + dm2 = 0.

(17)

Then, the lowest-order self-energy correction due to the thermal fluctuations reads: X (T2

A, , ,

m2

mT

I.

m2

where the expansion in the r.h.s. holds when m -C T. In particular, when m — 0, we recover the result II2 = m2D, cf. eqs. (11)—(12). This quantity is generally dubbed a "hard thermal loop" (HTL), because when m
V = 47 <04>o + \Sm2{
(19)

where we have also included the contribution of the mass counterterm and used eq. (17) to verify that the temperature-dependent ultraviolet divergences cancel out, as they should. The second term in the last line in eq. (19) is divergent, but contributes only to the vacuum energy density, so that it can be discarded. For ra = 0, eq. (19) yields: A 1152

Pi = - -

15 /.
.

(f) *»

<20>

in agreement with eq. (8). At the next order in the loop expansion, the self-energy receives contributions from the 2-loop diagrams. However, some of these contributions are infrared divergent in the massless limit m = 0. Consider, e.g., the second diagram displayed in Fig. 5. The upper loop (together with the corresponding mass counterterm, not shown explicitly in Fig. 5) is recognized as the "hard thermal loop" computed previously. Thus, this particular 2-loop contribution reads: n (q)

_

A T

W

d3fc

™D

(21)

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Rebhan

The three-dimensional integral over k in eq. (21) has a linear infrared divergence, coming from the term with uin = 0. (For the other terms, uin ^ 0, the Matsubara frequency u)n acts effectively as an infrared cutoff.) A non-zero mass m would cut off this divergence, and give a contribution ~ \m2D{T/m) ~ g4T3/m. As we shall see, when m
various

2.2.1. The next-to-leading-order

resummation

schemes

thermal mass

The effect of a thermal mass is most directly seen by carrying out perturbative calculations with the dressed propagator

For the calculation of the tadpole, this corresponds to resumming the diagrams of bare perturbation theory shown in Fig. 5 (each of the inserted tadpoles in Fig. 5 is accompanied by a mass counterterm which removes its ultraviolet divergence in the vacuum, cf. eq. (17)).

Fig. 5. Diagrams that are resummed when computing the tadpole in Fig. 3 with the dressed propagator in eq. (22).

Since rho = gT, this resummation involves arbitrarily high powers of g. However, the final result can be read from eq. (18) in which we set m — rho = gT: ntadPoie = g2T2 - -g3T2

+ 0(g4 In g).

(23)

This formula exhibits the next-to-leading order contribution to the thermal mass, of order g3, consistent with eq. (11). This order-g 3 contribution comes

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73

from the term linear in m in the r.h.s. of eq. (18). It thus is related to the non-analyticity of the thermal integral 7r(m) as a function of m 2 and to the infrared divergences which occur when one attempts to evaluate this integral after expanding the integrand in powers of m 2 . These infrared divergences are easily identified in the sum of the diagrams in Fig. 5 with at least two loops:

nnns

=V/i!i/_! 2

^

J

L_

(2TT)3 \ wl + k2 + m2D

Ji

+ k2

where the subtracted term is the massless tadpole. The term with I = 1 corresponds to the 2-loop diagram that we have considered separately in eq. (21). Each of the terms with w2 > 0 in eq. (24) starts to contribute at order g2rh2D ~ g 4 . But the terms with un = 0 are individually infrared divergent for any I, and the divergences cancel only in the sum of all the terms. In fact, the contribution of order g3 arises from the static modes alone, i.e., the modes with zero Matsubara frequency. If we separate their contribution in eq. (24), we obtain indeed

*&-**T I ^ivt^-h)--;*'•

(25)

where we recognize the order g3 contribution in eq. (23). It is instructive to look at this contribution from another perspective. As we have argued before, the dominant contribution to i r ( ^ ) in eq. (18) comes from momenta k ~ T. The subleading contribution is coming from soft momenta k
( 3

1

1\

2

r) \k*+m

mDT

2

D

kJ

2

4TT '

^

3

After multiplication by A/2 = 12# , one recognizes the g term of eq. (23). In writing the approximate equality above, we have used the fact that the dominant contribution in the integral (26) comes from momenta k <§C T, so that we can use nfo) w —

for

£fc
(27)

Note that the above resummation of the thermal mass, focusing on the static mode, hides an ultraviolet problem that we have ignored so far.

74

J.-P. Blaizot, E. Iancu and A. Rebhan

Namely, in order to obtain the finite result in eq. (23), we had to cancel an ultraviolet divergence, cf. eq. (17), which now involves the thermal mass, and is therefore temperature dependent. The T-dependent divergent piece is of order g4 however (since proportional to g2rh2D), and thus does not formally affect the order-g 3 calculation. But it signals that our calculation is not complete. The divergence is identified in eq.(24) in the term with 1 = 1, i.e., in the 2-loop diagram in Fig. 5, after performing the Matsubara sum. The overall divergence of the 2-loop diagram arises as the product of a vacuum divergence in the 4-point function subgraph (the lower loop) times the finite-temperature piece of the upper loop. Thus, the divergence is removed by the vacuum renormalization of the 4-point function, i.e., by adding a 1-loop counterterm to the vertex in the tadpole. 2.2.2. The plasmon effect in the pressure

Fig. 6.

The "ring" diagrams responsible for the order-g 3 effect in the pressure.

The contribution of order g3 to the pressure, also known as the "plasmon effect", comes again from resumming diagrams which are infrared divergent in the naive diagrammatic expansion. These are the "ring" diagrams in Fig. 6, in which the central loop is static (u)n = 0), while each outer loop represents an insertion of rh2D. This series can be easily summed up, with an infrared finite result 5 :

P3

=JI fd3fc 2 J

- ~2j

yc-^-y^V

(2TT)3 L ,

I

U

2

(2^)3 [ l0g {1+J^)~~1^\-

J

127T '

(28)

which is the order-g 3 contribution in eq. (8). 2.2.3. Static

resummation

From the previous examples, we learned that (a) the occurrence of infrared divergences in the diagrammatic expansion calls for resummations and (b)

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75

it appears sufficient, in order to cure these divergences, to resum m^, in the static Matsubara modes alone. Since fr?D summarizes interaction effects, care should be taken to avoid overcounting in carrying out high order calculations within that scheme. One way to proceed systematically is to add and subtract a mass term for the static modes in the (imaginary-time) Lagrangian: £ = i(^)

2

+ i(V0)2 + ^ 4

= \{dr<S>? + \{Vf + \ rhUl

+ j

/ + A£,

(29)

where A £ = —^ritftsfro and o(x) = T J0 dr(r, x). This allows for a reorganization of the perturbation theory in which the tree-level amplitudes are generated by the first three terms in eq. (29) while AC is treated formally as an additional interaction, of order g2, to ensure that the HTL is not double counted in the static sector. This procedure preserves a strict correspondence0 between diagrams (or well identified pieces of it) and powers of g. Thus, it is well suited for weak coupling expansions of static quantities, and it is indeed the scheme in which the higher-order calculations for the pressure and the screening mass alluded to before have been originally performed (for both scalar theory 35 - 36 - 37 and QCD 41 - 42 - 43 ). 2.2.4. Dimensional reduction The special role of the static mode <po may be handled in a more systematic fashion by constructing an effective theory in which the non-static modes uito are 'integrated out' in perturbation theory. Specifically, one can write the thermal partition function as the following path-integral: Z = ^yp(0o)exp{-5eff[^o]},

(30)

where exp{-S eff [<M} = .Af y p ( < A „ # o ) e x p j - J

dr Jd3xCE(x)

1 , (31)

and £eff[0o] i s the effective action. Aside from the direct classical field contribution (i.e., the restriction of eq. (7) to the static mode <j>o), this effective c

This is true, strictly speaking, only if one uses dimensional regularization to control the ultraviolet divergences.

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J.-P. Blaizot, E. Iancu and A.

Rebhan

action receives also contributions which, diagrammatically, correspond to connected diagrams whose external lines are associated to (po, and the internal lines are the propagators of the non-static modes o, which are also non-local. In practice, however, one wishes to expand 5efr[<^o] in terms of local operators, i.e., operators with the schematic structure am,„Vmo with coefficients a m ,„ to be computed in perturbation theory. It is then useful to introduce an intermediate scale A (A A) and soft (k < A) momenta. All the non-static modes, as well as the static ones with k > A are hard (since K2 = w2 + k2 > A2 for these modes), while the static (wn = 0) modes with k < A are soft. Thus, strictly speaking, in the construction of the effective theory along the lines indicated above, one has to integrate out also the static modes with k > A. The benefits of this separation of scales are that (a) the resulting effective action for the soft fields can be made local (since the initially nonlocal amplitudes can be expanded out in powers of p/K, where p -C A is a typical external momentum, and K > A is a hard momentum on an internal line), and (b) the effective theory is now used exclusively at soft momenta. This strategy, which consists of integrating out the nonstatic modes in perturbation theory in order to obtain an effective threedimensional theory for the soft static modes, is generally referred to as "dimensional reduction" 44,45,46,47,48,49,50 As an illustration let us consider a massless scalar theory with quartic interactions; that is, eq. (7) with m = 0. The effective action for the soft fields takes the form:

S\M = W ) + Jd3x {i(V<£o)2 + \M2{K)4>2 + fffWo + •••}, (32) where ^"(A) is the contribution of the hard modes to the free-energy, and the dots stand for higher-order, local operators. We have changed the normalization of the field (o —> VTcfro) so as to absorb the factor /3 in front of the effective action. The parameters of the effective Lagrangian in eq. (32), like M 2 (A), 33(A), etc., are computed in perturbation theory (they involve diagrams in which all the internal lines are hard), and depend on the separation scale A, the original coupling g, and the temperature T (note that the effective theory depends on the temperature only through these parameters). To lowest order in g, g2 w g2T, and M ~ gT, as we shall see shortly. Note that the scale A acts as an explicit ultraviolet (UV) cutoff for the loop integrals in the effective theory. Since it is arbitrary, the dependence on

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77

A of soft loop contributions must cancel against the dependence on A of the parameters in the effective action. Let us verify this cancellation explicitly in the case of the thermal mass M of the scalar field, and to lowest order in perturbation theory. To this order, the mass parameter M 2 (A) in the effective action is obtained by integrating over hard momenta within the one-loop diagram in fig. 3 (cf. eq. (15)). This gives 2 f d 3 f c (1 ~ *n0) + *(fc ~ AMnQ M2(A) = l2g2-rV T ^ J (27r)3 w2+fc 2

The first two terms within the last integral are the same as in eq. (15) with m = 0. The first of them, involving the thermal distribution, gives the HTL contribution rh?D = g2T2, while the second one, involving 1/2/c, is removed by the vacuum renormalization. The third term, involving the ^-function, is easily evaluated. One finally gets: M2(A)

=rh2D-%KT 2 7T

= g2T2 U-\£\. 2 \

(34)

7T T

The A-dependent term above is subleading, by a factor A / T
_ ,. l2gd2/c TJ-^Q(A-k)

6M2(A) =

1

k2 + M2(A)

•K*

\

2A

A /

K*

7T

where the terms neglected in the last step are of higher order in rho/A or A/T. As anticipated, the A-dependent terms cancel in the sum M 2 = M 2 (A)+ 6M2(A), which then provides the physical thermal mass within the present accuracy: M2 = M 2 (A) +SM2(A)

= g2T2 - -g3T, 7T

which agrees again with eq. (23).

(36)

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J.-P. Blaizot, E. Iancu and A.

Rebhan

2.2.5. Screened perturbation theory Still another strategy has been used for taking into account the effects of the screening mass to all orders, rather than perturbatively. It consists of resumming the HTL rh2D in all the Matsubara modes, static and nonstatic (as we did in subsection 2.2.1). It can be formulated via a simple generalization of eq. (29), viz. C = \d^d^

- \ mU2

- ^

4

+ AC,

AC = \ m2D4>2 , (37)

with AC to be treated as a quantity of order g2 in perturbative calculations. This scheme is commonly referred to as screened perturbation theory (SPT) 1 8 ' 2 0 . To appreciate its virtues, and its limitations, it is again best to work out some examples. We have obtained earlier the expression of the tadpole calculated with the dressed propagator (22) (cf. eqs. (23) and (18)): Iltadpoie ^ 12g2IT(mD).

(38)

We have seen that Iltadpoie {i^D = gT), when expanded in powers of g, reproduce the correct perturbative result to order g3 (see eq. (23)). But, as g grows, it deviates rapidly from this perturbative result, as can be seen in Fig. 7. The poor convergence of the weak coupling expansion exhibited here is related to that of the high-temperature expansion in eq. (18) which, as we have seen, involves in fact non analytical terms. By including the screening mass in the tree-level Lagrangian, and not treating it a perturbative quantity of order g, SPT provides a smooth extrapolation to large values of g, which represents a definite improvement over perturbation theory. However, in contrast to the static resummation scheme based on eq. (29), the calculations within SPT generally involve temperaturedependent ultraviolet divergences. Of course, these are compensated by the "counterterm" AC, but only in all-order calculations. At any finite-order, the compensation holds only to the perturbative accuracy d of the calculation. For instance, in order to obtain (38), one has to introduce the mass counterterm dm2 = — (\/2)Io(rhp), which depends on T, via rhn- Specifically (with dimensional regularization, for definiteness, and p,2 = 4ne r/J.2): v

2

,„ 2 e f

d3-£A:

1

3g2m2

dm2 = - 1 2 5 V / / 0 , 33 ££ 0 , . v - -2-rr-2 J (27r) - 2ek{mD) 4n

(2

,

p,2

\

. .

- + In 4 V + 1 , 39 \e m2D J

By the "perturbative accuracy" of an all-order result we understand the highest order in g that is correctly reproduced by the weak coupling expansion of that result.

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of the High- Temperature

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79

m/T

1/2

Fig. 7. The one-loop thermal mass in SPT, mo = n t ^ d l e , eq. (38), normalized to the temperature as a function of g (full line). For comparison, we also present the perturbative estimates to order g2 (long-dashed line) and g3 (short-dashed line).

where £fc(m£))2 = k2 + rh2D. This formula makes explicit the fact that the T-dependent divergence is of order 0(g4), and thus lies beyond the perturbative accuracy of this one-loop calculation, which is 0(g3). As further illustration of the difficulties with SPT, let us compute the pressure to 2-loop order, and compare the result with its perturbative expansion to order g3. To one-loop order in the effective theory, the thermodynamic potential is given by eq. (13) with m —> rho- That is:

II

d3k £k{mD) (2n)3

-

Po(mD)

(40)

with Po(m) given by eq. (14):

« ( * » , - ^ {.-f (J)"+ H(f)' + o w . b „}. («, This overincludes the order-<;2 effect by a factor of 2 (compare to eq. (8)), but includes correctly the term of order g3 (the fact that the plasmon effect is included correctly is somewhat accidental: If we were to resum not just the leading-order thermal mass rh2D, but also the NLO correction to it, Ami, = -(3/7r)g 3 r 2 , one would modify the order-g 3 content of fl0). The first piece in eq. (40), i.e., the sum of the "zero-point energies", is also UV divergent, and the divergent terms depends on T since they depend on rhoTo eliminate these divergences, we may introduce a temperature-dependent

80

J.-P. Blaizot, E. Iancu and A.

Rebhan

counterterm:

i f d3k

sn0

,. .

i r d4k , , l 2

]2

„

2 x

whose dependence upon T starts already at order g2. (This dependence can be pushed to order g4, though, by using dimensional regularization.) This lowest order calculation reveals another unsatisfactory feature of SPT: the tree-level pressure calculated with the Lagrangian (37) fails to reproduce the lowest-order effect of the interactions. Of course these drawbacks are partially corrected when we move to the next order, that is to two-loop order, which we do now. At this level, there are three contributions: the two-loop diagram in Fig. 4, and two one-loop diagrams involving mass counterterms, namely AC in the Lagrangian (37), whose role is to correct for overcounting, and the ultraviolet counterterm 6m2, that has been introduced before, in eq. (39). Altogether: ^2

[/(m D )f + | (Sm2 - m2Dy{mD).

V

(43)

It is convenient to combine the last counterterm with the tree-level expression fio (written as in the first line of eq. (40)): m2D

fto

T--M.I(mD)

2

„

/w{ta("*+k"+*l,-=d$H«&}-

m

Clearly, the main effect of A £ is to cancel out the whole contribution of order g2 that was originally present in the tree-level pressure. This includes the "anomalous" order-g 2 effect within PO{THD), and the divergent contribution of order g2 that was eliminated by the counterterm (42) at one-loop order. The terms in eq. (43) can be simplified as follows: \ \nmD)f

+ ~

I(mD) = | [IT(mD)}2

- ± [I0(mD)f

,

(45)

(cf. eq. (19)). The last term is divergent and depends on the temperature, via rhi>, but this dependence counts only to order g4, or higher. The orderg2 is obtained by replacing Irirho) by ir(0) in the first term in eq. (45). The order g3 receives contributions both from eq. (44) (as the lowest order correction beyond the ideal gas result), and from eq. (45) (as the nextto-leading order contribution). In both cases, the integration involves soft

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81

momenta only. Thus, we can restrict ourselves to the static mode u>n = 0 in eq. (44), and get: 1_, f d3fc f,

/

m2D\

m2n

}

while in eq. (45) we may proceed as follows: \ {lT{mD)f

- ^ (MO)) 2 ^ ~IT(0)

.21

m

/ A

T

(IT(mD)

/

1

- J r (0))

1 \

" -2 i(2^(FT^-^J

(4?)

where we have also used eq. (26). Clearly, after adding the two contributions in eqs. (46) and (47), we are left with the standard expression for P 3 , eq. (28). Now, all the remaining ultraviolet divergences in eqs. (44) and (45) are to be cancelled by appropriate counterterms. One could be satisfied by the fact that the T-dependence of these counterterms starts at order g4, which is beyond the perturbative accuracy of the present 2-loop calculation (namely, 0(g3)). But leaving temperature dependence in UV counterterms is somewhat unphysical and would cause problems in nonperturbative calculations. Moreover, ambiguities arise from the choice of the subtraction scheme: One could e.g. either choose to subtract all the T-independent terms of the thermal pressure before identifying rhp with a thermal mass, or one may only subtract pole terms minimally on the grounds that also the terms not explicitly dependent on T become so after making rho temperature-dependent. Some of these drawbacks may actually be absorbed by the flexibility in the choice of the mass parameter rho in (37). Obviously, the results of SPT would be independent of this mass if they were obtained in an all order calculation. But for a finite-order calculation, the choice matters. One may try to optimize perturbation theory by a principle of minimal sensitivity resulting in a self-consistent gap equation 18,20 . This has been shown to reduce greatly (though not completely) the dependence on the subtraction procedure 51 (see also below, Fig. 9). Indeed, in Ref. 20 the calculation of the pressure of scalar
82

J.-P. Blaizot, E. Iancu and A.

2.3. Self-consistent

Rebhan

resummation

2.3.1. Skeleton expansion for thermodynamical potential A systematic way to take into account in the calculation of the thermodynamics, screening effects, or more generally propagator renormalisations, is to use the representation 52,22 ' 53,54 of the thermodynamic potential fi = — PV in terms of the full propagator D: 3Q,[D) = - log Z = \ Tr log D~l - \ Tr ILD + $[£>],

(48)

where Tr denotes the trace in configuration space, B = 1/T, and $[D] is the sum of the two-particle irreducible (2PI) "skeleton" diagrams with no external legs: -$[£>] =

1/12 C

) +1/8 C^Y^)+1/48

(49)

(where in the <j>A example the first diagram is absent of course). The full propagator D is expressed in terms of the proper self-energy II by Dyson's equation (Do is the free propagator): D =

^

>

( 50 )

and II itself, which is the sum of the one-particle irreducible (1PI) diagrams with two external lines, is obtained from $[D] by: 8$[D]/6D = ^ n .

(51)

By using this relation, one can check that Q.[D] is stationary under variations of D (at fixed Do) around the physical propagator: Sil[D]/6D = 0

for

D = (D^1 + II)-1.

(52)

We shall usually refer to eq. (50) in which II is given in terms of D by eq. (51) as a 'gap equation'. An explicit expression for fl is obtained by performing the summations over the Matsubara frequencies in eq. (48), using standard contour integration techniques. One obtains:

/

d^k -—j 4 n{u>) (Im log(-w 2 + k2 + U) - ImUD) + T$[D]/V (27T)

(53)

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83

where n(w) = l/(e /3<J —1), and the imaginary parts are denned with retarded prescription. For instance: Im D(u, k) = Im D(u> + te, k) = p ^ ' f c ^ ,

(54)

where /o(u>,fc)is the spectral function, which enters the following representation of the analytic propagator D(w,k), valid for frequencies u> off the real axis: K

' '

J_002lT

k0-LJ

(55)

The stationarity condition (52) can be used as a variational principle to deduce non-perturbative approximations for the physical propagator. A convenient approximation scheme of this type is the self-consistent (or "^-derivable") approximation 22 obtained by selecting a class of skeletons in $[D] and calculating II from eq. (51). We shall see also that the stationarity property (52) brings simplifications in the calculation of the entropy. 2.3.2. A simple model As a simple illustration of the general formalism of the previous subsection, we consider here the 2-loop approximation to $, given by the first skeleton in the r.h.s. of eq. (49). e The corresponding skeleton for II is the dressed tadpole (see Fig. 3), and is local. Thus, the self-consistent self-energy is simply a mass, and we shall write II = m2. (The vacuum mass is set to zero.) The gap equation (51) reads then:

m2 = ±I(m).

(56)

In contrast to the previous renormalization of eq. (38) (recall eq. (39)), it is possible to renormalize eq. (56) without introducing a thermal counterterm. This is done through coupling constant renormalization. To proceed, we first rewrite eq. (56) as (with the same notations as in eq. (39)):

^"--^(!+*&+iW<s?!£2+0W-(n> e

T h e self-consistent solution to 3-loop order has been worked out using mass expansions in Ref. 5 S with satisfactory numerical results. However, the negative conclusion therein concerning renormalizability of ^-derivable approximations beyond two-loop order has since been refuted by Refs. 56 > 57 .

84

J.-P. Blaizot, E. Iancu and A. Rebhan

and relate Ao to the renormalized coupling A by:

Then, the equation takes the following, manifestly finite, form (for e —> 0): 2

A f d3k n(ek)

Am2 /

m2

\

One should stress a subtle point in the procedure: expressing the bare coupling Ao in terms of the renormalized one A according to eq. (58) is not sufficient to render finite the r.h.s. of eq. (57) for arbitrary values of m 2 , but only for these values which satisfies the finite equation (59). Note that the 1-loop correction to the 4-point function that is considered here is iterated only in one channel among the three possible ones. Accordingly, the /3-function for the renormalized coupling constant (58): dA _ _ ^ _ dlog/2 167T2'

{

'

(this follows by noticing that Ao is independent of p. in eq. (58)) is only one third of the lowest-order perturbative /3-function for this scalar field theory (see eq. (9)). This is no actual fault since the running of the coupling affects the thermodynamic potential only at order A2 which is beyond the perturbative accuracy of the 2-loop ^-derivable approximation. (In order to see the correct one-loop /3-function, the approximation for $ would have to be pushed to 3-loop order.) By using eq. (60), one can check that the solution m 2 of eq. (59) is independent of ji. By using the previous formulae, one can easily compute the pressure 58 : P =

_T [J*L io10gU — f^L 3 3 ^ M ++ JUL22 K(l - e-^M) + + 'J (27r)3 2 J(2ir) ) 2e -128TT °~ ' 2efek

(en {bi)

which differs from the pressure Po(m) of an ideal gas of massive particles, eq. (14), by the last two terms in the r.h.s. By construction, the 2-loop ^-derivable approximation is perturbatively correct to order # 3 , and, indeed, it can be easily verified that the weak coupling expansions of eqs. (59) and (61) include the correct perturbative effects of order g2 and gs (together with incomplete contributions of higher orders). But the complete, self-consistent results, as obtained by numerical evaluation in eqs. (59) and (61), show a much better behaviour than the respective perturbative expansions to order g3, in the sense of being monotonic with g (unlike the perturbative approximants of order g3),

Thermodynamics

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85

m/T 1 0.8 0.6 0.4 0.2 0.5

1

1.5

2

2.5

3

g(fi = 2wT) Fig. 8. Numerical solution of the mass gap equation (59) with p, = 2nT in comparison with the perturbative result to order g2 and g3 for m (long and medium dashed lines, resp.). The short-dashed line just below the full line corresponds to the quadratic gap equation (65) following from the requirement of only soft self-consistency.

and showing saturation at very large g > 2 (in contrast to the order-g 2 estimates). This behaviour is illustrated in Figs. 8 and 9 for the self-consistent mass (59) and pressure (61), respectively. Also shown in Fig. 9 is the result of SPT at two-loop order when the mass parameter is determined by a principle of minimal sensitivity. It turns out that for a minimal subtraction of the additional UV divergences of SPT and identification of the corresponding regularization mass scale with that used in the (MS) renormalization of the coupling constant, the result of SPT coincides with that of the 2-loop ^-derivable approximation. However, if the additional UV divergences of SPT are subtracted differently, this coincidence no longer occurs. As an example, in Fig. 9 the grey full line displays the SPT result with a "maximal" subtraction of the additional UV divergences where P(T) is made finite by subtracting P(0) when treating rhp as temperature-independent (since it is just a parameter in the Lagrangian), and only afterwards turning it temperature-dependent by fixing rho by a variational principle. Although (for N = 1 scalar theory) one cannot decide how the exact result would look like, the result appears to be less satisfactory in that it exceeds the free pressure for g > 1.5. The prescription that has been used in the application of SPT to QCD is in fact that of a minimal subtraction of the additional UV divergences of SPT 21 ' 59 - 60 . We have previously argued that the expected accuracy of perturbation theory is a priori different in the soft and the hard sectors. Within the present solvable model, it is possible to verify this argument explicitly. To

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Rebhan

Fig. 9. The 2-loop ^-derivable pressure (61) in comparison with the perturbative results to order g2 and g3 (long and medium dashed lines, resp.). The short-dashed line just above the full line corresponds to the pressure evaluated perturbatively for hard modes and self-consistently only for the soft modes, eqs. (62) and (66) with (65). SPT at 2-loop order turns out to coincide with the 2-loop ^-derivable given by the full line, when the former is subtracted minimally, but gives different results (grey full line) when subtracted "maximally" (see text).

this aim, we shall reformulate the self-consistent calculation of the thermal mass and the pressure in an approximate way, exploiting the distinction between the hard and soft sectors. That is, we shall build on the strategy of effective theories to obtain perturbatively, through a calculation in the hard sector, the coefficients of an effective Lagrangian for the soft sector. We shall then look for an "exact" solution in the soft sector, which amounts here to use the two-loop self-consistent approximation. As we shall see, this strategy, which mimics that used in calculations based on dimensional reduction, turns out to be remarkably accurate. It also provides a clearer physical justification for approximations that have been motivated previously by other considerations. Thus, we shall approximate the fully self-consistent 2-loop result for the pressure, eq. (61), by P = P(2) + Ps, where P(2) is the perturbative result to order g2, which is due to the hard modes alone:

P

^ = 90T

8

W

(62)

and P s is obtained via a 2-loop self-consistent calculation within the effective three-dimensional theory with Lagrangian (32). To the order of interest, one can set M2 « rh2D and g2E « g2T in eq. (32), and neglect all the other interaction vertices.

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The corresponding free-energy functional reads then:

k2+m2D+Us'

(63)

'

where fk denotes the momentum integral in 3 — d dimensions. Note that, as compared to Sect. 2.2.4, here we find it more convenient to use dimensional regularization, with the scale /i chosen as the separation scale A between soft and hard momenta. This has the advantage that all the power-like divergences in integrals like those in eq. (63) are now set to zero, so, conversely, the parameters of the effective Lagrangian are independent of A to the accuracy of interest 37 . In eq. (63), IIS is the scalar self-energy within the 3 — d effective theory, and is determined from a gap equation which follows by demanding fis [IIS] to be stationary with respect to II S : n

s = 92 ^ , „ 2J k2,+m / 2 2 |TT , k D+Us

(64)

or, equivalently, with m 2 = m2D + IIS m2 = m2D-

3 -g2Tms.

(65)

This is a second-order algebraic equation for m s , whose r.h.s. is recognized as the expansion of the integral in the general gap equation in four dimensions, eq. (56), up to terms linear in m/T. Note that, to this accuracy, there is no ultraviolet divergence at all, so the coupling constant A (or g2) in the equations above is the renormalized one. The free energy (63) can be easily evaluated in terms of the solution ms to the above gap equation. One obtains (Ps = — £ls/V): „ Ps

Tm3

Tms ,

9

„,N

3 / a\ 2 ,

= 1 2 7 - 1 ^ - ^ -16(f)

Tm

-

,

. , (66)

Note that this contribution starts at order g3, but includes terms of all higher orders, via ms. The sum P = P(2) + Ps of eqs. (62) and (66) should be compared to the fully self-consistent result in eq. (61). Clearly, these two approximations coincide up to order g3, but deviate in higher orders, with the deviations restricted, however, to contributions due to the hard modes. In Figs. 8 and 9, we also compare the numerical solutions to the full gap equation (59) and its "soft" approximation (65), and also the corresponding

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predictions for the pressure. We see that these predictions are extremely close to each other, for both m and P, which confirms that the higher order effects associated with the hard modes are small indeed. 2.3.3. The entropy We have mentioned already that the stationarity property (52) allows one to simplify the calculation of the entropy: S = -d{n/V)/dT

(67)

or, more generally, of any first partial derivative of the pressure. the thermodynamic potential (53) depends on the temperature the statistical factors n(w) and the propagator, or equivalently its function p. Because of eq. (52) the temperature derivative of the density cancels out and one obtains 61 ' 26 : 5 =

f dAk dn(u>) x

,

__••,

-/(2^^ilml°gD

Indeed, through spectral spectral

,.

<"'*>

+ J-0^^ImU(u;,k)ReD(uj,k)

+ S'

(68)

with d{T$) S' = -1dT

D

+

f d*i d4k dn(uj) RelUmD. J (2TT ) 4 dT

(69)

Loosely speaking, the first two terms in eq. (68), which have a one-loop structure, represent the entropy of 'independent quasiparticles', while S' accounts for a residual interaction among these quasiparticles 26 (see also the discussion below). For 2-loop f skeletons we have the following additional simplification61 - 26 ' 25 : S' = 0,

(70)

a property which holds independently of the self-consistency condition (52). The formula (68), together with S' = 0, makes the entropy a privileged quantity to study the thermodynamics of ultrarelativistic plasmas. Let us discuss some of the attractive features. f

For the scalar field theory with 0 4 self-interactions, the property (70) holds also for the 3-loop skeleton in eq. (49).

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First, we note that, for a finite function II(w, k), the integrals in eq. (68) with <S' = 0 are manifestly ultraviolet finite, since dn/dT vanishes exponentially for both w —> ±oo. This reduces the problem of the UV renormalization of the 2-loop ^-derivable approximation for the entropy to that of the renormalization of the gap equation for the 1-loop self-energy. In general, this is still a difficult task (because this is a non-perturbative approximation), but as we have seen in the previous subsection, progress can be made at least in specific cases. In particular, it has been recently demonstrated, in Refs. 56 - 57 , that for scalar field theories at least, the selfconsistent approximations are renormalizable to all orders, i.e., they can be made finite by adding temperature-independent (vacuum) counterterms to the Lagrangian, which are themselves determined in a self-consistent way. Furthermore, as already mentioned, the entropy has a more direct quasiparticle interpretation than the pressure. To see this more explicitly, use the following identity: Im log£>-V,fc) = arctan ( ^ ™ , | - T r e ^ W - R e - D " 1 ) , \ Re D i J to rewrite S as <S = Spoie + Sdamp, with

d3k

r

(71)

1

/ ——3|(l + n fc )log(l-|-n fc ) - nfc log nfc j , (72) where nk = n(ek), with €k solution of Re.D - 1 (a; = £fc,fc) = 0. This is the on-shell energy of the dressed single-particle excitations in the system, or 'quasiparticles'. The quantity <Spo/e is the entropy of a system of 'noninteracting' quasiparticles, while the quantity dAk dn{ui) T „ n „ . imn f dV (73) Sd,•amp Imll Re D — arctan ReD-1 Y dT ~ J (2* which vanishes when Imll vanishes, is a contribution coming from the continuum part of the quasiparticle spectral weights. We see that for the entropy, unlike for the pressure, the main effects of the interactions come via the change in the spectrum of the excitations, and therefore can be taken into account by simply dressing the propagator. The above formulae, which are generic (i.e., they hold for any field theory, including QCD), become even simpler when the 2-loop self-consistent approximation is applied to the scalar theory with
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previous subsection, and therefore <S = Spoie, with S po j e given by eq. (72) with e\ = k2 + m2 and m determined by eq. (59). That is, in this approximation, the entropy of the interacting scalar gas is formally identical to the entropy of an ideal gas of massive bosons, with mass m. As emphasized after eq. (61), such a simple interpretation does not hold for the pressure. 2.3.4. Approximately self-consistent calculations of the entropy In this subsection, the calculation of the self-consistent 2-loop entropy will be considered from a new perspective, which will suggest a different strategy for implementing the self-consistency in an approximate way. This strategy will be especially useful in view of subsequent applications to QCD, where a fully self-consistent calculation seems prohibitively difficult, and not even desirable (see Sect. 4.2). It is first instructive to see how the perturbative effects of order g2 and g3 get built within the self-consistent 2-loop entropy, as given by (cf. eq. (72)): f dAk dn(uj) 2 2 2 S(m) = -J —-- - ^ - Im log(fc - u' + m J )

The first two terms in the expansion in the second line come from integrating over hard momenta (k ~ T) within the loop integral in the first line, while the third term, non-analytic in m 2 , comes from the soft loop (k ~ m). The expansion in eq. (74) must be considered simultaneously with the perturbative expansion of the solution m to the gap equation (59). To order g3, this is the same as the standard perturbative expansion, eq. (23): m 2 ~ 5 2 T 2 ( 1 - 3g/n). If one replaces m in eq. (74) by the LO thermal mass rho — gT, then the resulting approximation for S reproduces the perturbative effect of order g2, but underestimates the correction of order g3 by a factor of 4. But to order g3, the thermal mass itself must be corrected, as discussed above. By replacing m 2 —+ m2D + 8m2D in the second order term of eq. (74), the correction 5m2D to the thermal mass generates the 3/4 of the plasmon effect missing. To conclude, the order-5 3 contribution to the entropy comes up in a subtle way: 1/4 of it comes directly from the soft loop in eq. (74), with m replaced by the LO screening mass % , while the other 3/4 comes from

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S/So 1.2 1.1 1 0.9 0.8 0.5

1

1.5

2

2.5 3 9(A=2irT)

Fig. 10. Comparison of the fully self-consistent 2-loop entropy with the perturbative results to order g2 and g3 (long and medium dashed lines, resp.). The short-dashed line between the order g2 result and the full line corresponds to the "HTL approximation" where the 2-loop entropy is evaluated with just the HTL mass; the short-dashed line almost on top of the full line corresponds to the "NLA approximation" using (76).

the hard loop, via the NLO correction to the dispersion relation of the hard particles. Of course, within the present context, where the self-energy is independent of the external momentum, there is no distinction between the screening mass and the quasiparticle mass: both are solution to the same gap equation (59). But in QCD, these are distinct quantities already at lowest order, and this difference will play an important role in the construction of approximations in Sect. 4. Thus, although this may look at first somewhat artificial here, we shall introduce different notations for the two masses. We shall keep the notation mo for the screening (Debye) mass, and denote the hard quasiparticle mass as m ^ : the latter characterizes the excitation spectrum at large momenta, typically of the form Ek = y/W+m^. Thus the mass m? in the second order term of eq. (74) should be read as m ^ , while TO3 in the cubic term is m|>. That is, we rewrite eq. (74) as

In perturbation theory, mo and rrioo coincide with each other up to order g3 , but differ in order g4 and higher. 8 Within the 2-loop ^-derivable approximation, they coincide if computed exactly, as already mentioned. s

N o t e that only the corrections to mo can be calculated by means of dimensional reduction. Dynamical masses in general require the resummation of the nonstatic hard thermal loops (for simple examples see Ref. 6 2 ) .

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But if the self-consistency is imposed only in the soft sector, the corresponding approximations may a priori look different. We have seen earlier that self-consistency in the soft sector is enough to obtain an excellent approximation for the pressure, and we want to verify here the status of such an approximation for the entropy. In fact we shall examine the accuracy of two non-perturbative approximations to the entropy, that we shall later implement in the case of QCD. In both approximations, the integral in the first line of eq. (74) is evaluated exactly (i.e., numerically), but for values of m which are only approximate solutions to the gap equation. i) In the 'HTL approximation' we set m = rhp. The ensuing approximation for S includes the correct LO effect of the interactions, of order g2, but only 1/4 of the plasmon effect. In Fig. 10, the resulting approximation is compared to the exact (i.e., fully self-consistent) result, and also to the weak coupling expansion. ii) In the 'next-to-leading approximation' (NLA), the mass moo of the hard particles is determined with NLO accuracy, so as to restore the correct plasmon effect in S. That is,

« & —> TT-4 - (3Mg2TmD,

(76)

where mo is the self-consistent mass, given by the solution to the gap equation (65). Of course, the mass m<x> thus obtained coincides with the Debye mass mo, but this 'degeneracy' is specific to the present scalar model, as we have already emphasized. The quality of the NLA based on eq. (76) can be appreciated from its comparison, in Fig. 10, with the "exact" (fully self-consistent) result, and also with the HTL approximation alluded to before. Although the HTL approximation is seen to deviate from the fully self-consistent result at large coupling, it remains smaller than the ideal-gas result despite the presence of a quarter of the plasmon effect, which, when treated purely perturbatively, would result in a nonmonotonic behaviour leading to an excess over the ideal-gas result at large coupling. The ^-derivable 2-loop approximation in fact becomes "exact" in the large-A'' limit, where N scalars are interacting in an 0(A^)-invariant manner according to {2)2- In this limit the exact result is /S-independent, while all approximations to it are renormalization scale dependent. However, as can be seen in Fig. 10 the agreement is almost perfect for NLA when p = 2irT and varies only slightly when varying this scale, which motivates us to adopt p, = 2TTT as optimal renormalization point. However, in what follows we shall always consider variations by a factor of 2 around this central value

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to have an estimate of the inherent uncertainty of our approximations. 3. Quark—Gluon Plasma Before describing how to overcome the difficulties of perturbation theory in the thermodynamics of QCD along the lines sketched for the simple scalar models of the previous section, we shall briefly review the results of perturbation theory obtained so far, which, at least for vanishing chemical potential and high temperature, have been recently pushed to the limits of calculability. 3.1.

Perturbative

results

In QCD with TV/ flavors of quarks, the thermodynamic pressure at high temperature has been computed 41 ' 42 in perturbation theory through order g5: 8TT 2

p = r

nr T 4

1+

45 +30

21

+

i^' 1+

15

4

tm

Nf

I + T^N

as

12

3/2

-{237.2 + 15.977V/ - 0.4132V? + ™ (l + ^ ] In ^ ( 1 + ^-)

+

1+

Nf 6 495

+-

1/2 r

-799.2 - 21.967V/ - 1.9267V?

1+

Nf 33 ')

2irT

m

5/2

+0(.c h i n a s ) >,

(77)

where as = g2 (p) / (4ir). The coefficient of the a3, l n a s term, the last in the pressure at high T and vanishing chemical potential that is calculable in perturbation theory, has recently been determined as 6 3 o 2 r P 6 = — T 4 1134.8 + 65.897V/ + 7.6537V? 3 lng 45 1485

3I l + ^Vl-^7V/Vn4, 6 33 * I 2TTT ( T ) 4-<78> \ it /

a..

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Here p, is the renormalization scale parameter of the MS scheme and a s (/2) is the running coupling whose beta function is known to four-loop order 64 , though we shall restrict ourselves to its two-loop version in the following, because lattice determinations of Tc/A-j^ usually employ only the latter. Evaluating the perturbative expression to order g2, g3, gA, and g5 gives results which do not show any sign of convergence, as depicted in Fig. 11, but rather show increasing ambiguities due to the dependence on the renormalization point, signalling a complete loss of predictive power. (When the g6 In g-term of Eq. (78) is included with a suitably adjusted constant under the log, the latter can at least be chosen such that the lattice results are qualitatively reproduced 63 .)

Fig. 11. The poor convergence of thermal perturbation theory in pure glue QCD. The various grey bands bounded by differently dashed lines show the perturbative results to order g2, g3, g4, and gs with MS renormalization point p, varied between -KT and 4irT. The thick dark-grey line shows the lattice results from the Bielefeld group 6 5 ; the lighter one behind that of a more recent lattice calculation using an RG-improved action from the CP-PACS collaboration 6 6 .

However, the poor convergence behaviour of perturbation theory is not very different from that we have observed above in the case of simple scalar A theory. In the following we shall review the attempts to overcome this impasse by reorganizing perturbation theory on the one hand by a variant of SPT and on the other by an approach which is guided by ^-derivable approximations for entropy and quark number density. In both cases, the resulting expressions go beyond strict perturbation theory in that higherorder terms are kept unexpanded, and the hope is that strictly nonperturbative contributions that appear in the pressure of hot QCD at order g6

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make up only a small, negligible part of what can be captured already by an effective quasiparticle description as soon as the temperature is a few times the transition temperature. When compared with the perturbative results, it is remarkable that the next-to-leading result to order g2 fares rather well at temperatures > 2TC, though the higher-order results prove that perturbation theory is inconclusive. On the other hand, simple quasiparticle models which describe the effective gluonic degrees of freedom by 2Ng (Ng — iV2 — 1) scalar degrees of freedom with asymptotic masses taken from a HTL approximation can be used quite successfully to model the lattice data by fitting the running coupling 27 - 28 ' 29 - 30 .

3.2. Lattice

results

The thermodynamic potential has been studied on 4-dimensional lattices both for pure-glue QCD and QCD with a number of comparatively light (but still massive) quarks by a number of groups. In the case of pure-glue QCD, the temperature dependence of the thermal pressure has been studied in great detail on lattices with varying temporal extent up to 8 x 32 3 for temperatures up to about 4.5TC. The continuum extrapolation using the standard Wilson action of Ref. 65 is given in Fig. 11 by the darker-grey line with the width of the line roughly representing the estimated error. Similar studies have been performed using improved lattice actions 67 ' 66 and the results obtained in Ref. 66 using a renormalization-group improved lattice action are included in Fig. 11 by the lighter-grey line. The latter are systematically higher by about 5 . . . 2% for temperatures 2 . . . 3.5TC compared to the results of Ref. 65 . As can be seen in Fig. 11, the pressure drops sharply as the transition temperature is approached from above. Not shown in the figure is that the pressure in fact drops to virtually zero at and below the transition temperature. There the pressure is exponentially suppressed by the rather large masses of glueballs. In the presence of (light) quarks, lattice calculations are much more difficult to perform and the continuum extrapolation is correspondingly more involved and uncertain. However, through the use of improved lattice action, considerable progress has been made recently 68,69 . When rescaled to the ideal-gas values, the results, as a function of T/Tc with the respective different values of T c , the pattern of the pressure of hot QCD with quarks is quite similar to that of pure-glue QCD, and there is moreover a rather

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weak dependence on the number of flavours if all of them are light. 3.3. Dimensional

reduction

Dimensional reduction in hot QCD (at zero chemical potential) leads to an effective three-dimensional Lagrangian 45 ' 46,47 CE = \ trFl + tr [DUAQ]2 + m2EtvA20 + ^ ( t r . 4 2 ) 2 + . . .

(79)

where the parameters are determined perturbatively by matching 43 ' 70 . In lowest order one has a dimensionful coupling g2E = g2T and 47 }- N m2E = (1 + Nf/6)g2T2, XE = %^942T, (80) 12TT

though XE starts to contribute to the pressure only at order g6. At this order, however, the self-interactions of the massless magnetostatic gluons start to contribute, but these contributions are inherently non-perturbative because the three-dimensional theory for the zero modes Ai(x) is a confining theory.71-72-73 The thermal pressure of the 4-dimensional theory can be decomposed into contributions from the hard modes ~ T, calculable by standard perturbation theory, and soft contributions governed by (79) which involves both perturbatively calculable contributions up to order g5T4 and the nonperturbative ones starting at order g6T4. In Ref. 4 3 the effective theory based on (79) has been used to organize and reproduce the perturbative calculation of the thermal pressure (77) of Refs. 41 - 42 . This turns out to be particularly elegant when dimensional regularization is used to provide both the UV and IR cutoffs of the original and effective field theories. To order g4, the contribution of the hard modes can then be written 43 as

fl-^K-f^^K*' + J244.9 + 17.247V/ - 0.4157V2 + 135 (l + ^

J In

^

In the first logarithm the dimensional regularization scale p, is associated with regularization in the infrared and thus has to match a similar logarithm in the effective theory, whereas the second logarithm is from UV and involves the first coefficient of the beta function.

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Indeed, calculating the pressure contribution of the soft sector described by (79) in dimensional regularization gives, to three-loop order (neglecting Ag-contributions)

(82)

-SMI-T^+H^

All the contributions to the pressure involving odd powers of g in (77) (as well as part of those involving even powers) are coming from the soft sector. Inserting the leading-order value (80) for m j gives the QCD pressure up to and including order g4 \ng; to obtain all the terms to order g5, next-toleading order corrections to the m^-parameter have to be obtained by a matching calculation as given in Ref. 4 3 . In order to obtain the g6 In ^-contribution (78) one also needs g\ to order g4 (given in Ref. 70 ) and above all the four-loop contribution of the effective theory (79) which has recently been calculated analytically as 6 3 ,(4)

N

P£k/T = 9

\6

(Ng 2 (4TT)

4

43 12

1577Ta

768

ln4 +

43

491TT 2 1 ,

p.

\

SE

(83) up to a constant c that is strictly nonperturbative and needs to be determined by three-dimensional lattice calculations. Such calculations have been carried out recently 17 , but they depend on an as yet undetermined 4loop matching coefficient. At the moment the conclusion is that it is at least not excluded that the lattice results based on dimensional reduction can be matched to the full four-dimensional results at temperatures of a few times the transition temperature. For this reason, the most reliable results on the thermodynamics of hot QCD (particularly for pure-glue h QCD) remain to date the four-dimensional lattice data. An interesting feature of the perturbative calculation when organized through the dimensionally reduced effective theory (79) is that the large scale dependences of strict perturbation theory can be reduced when the effective parameters are not subsequently expanded out when their perturbative results are inserted in (82) 63,74 . What is more, considering successively the full two-loop and full three-loop contributions from the soft h

Inclusion of fermions is particularly easy in the dimensional reduction method, which makes a full three-dimensional prediction clearly most desirable.

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sector, the results do not exceed the ideal-gas result (as the strictly perturbative results to order g3 and g4 do), and their scale dependence diminishes by going from two to three-loop order. 74 At three-loop order, it is in fact possible to eliminate the scale dependence altogether by a principle of minimal sensitivity. The result in fact agrees reasonably well with the 4-d lattice results down to ~ 2.5TC. (This is also true when the four-loop logarithms are included provided a suitable value for the unknown constant c in (83) is chosen. 63,75 ) While this goes only minimally beyond a strictly perturbative treatment, it strongly suggests that perturbative QCD at high temperature, when supplemented by appropriate resummation of soft physics, is capable of quantitative predictions at temperatures of interest. 4. Thermodynamics of HTL quasiparticles in QCD 4 . 1 . HTL-screened

perturbation

theory 4

In the simple example of scalar theory, we have seen that a promising strategy for improving the poor convergence of perturbation theory may be screened perturbation theory based on the reorganization of the Lagrangian according to (37). The corresponding resummation changes the UV structure of theory at any finite order of perturbation theory, but removing the additional UV divergences by some minimal subtraction does seem to lead to results which avoid the bad behaviour of conventional perturbation theory. This method can be extended to QCD by employing the gauge invariant HTL effective action 76 ' 77,13 instead of a simple mass term. The HTL effective action is nonlocal and involves nontrivial vertices in addition to self-energies, but it is uniquely determined by only one mass parameter for gluons (and a further one for quarks, if any) 2 1 , 7 8 , 5 9 , 6 0 . This variant of screened perturbation theory has been termed HTL perturbation theory (HTLPT) 1 by their authors, but it should be noticed that it is meant to transcend perturbative HTL resummation, where hard propagators and vertices with hard propagators attached remain undressed 9 . In HTLPT the QCD Lagrangian is improved by writing £QCD = £QCD

+ (1 - <5)£HTL

(84)

as—+6aa

'We would suggest however to spell this out as HTL-screened perturbation theory to recall that this method transcends the original HTL resuramed perturbation theory 9 .

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with the standard HTL Lagrangian 76 ' 77 ' 13

where v^ = (l,u) is a light-like four-vector and (...) represents the average over the directions v. Since this only serves as gauge invariant generalization of the thermal mass term for SPT, i.e. a variationally improved perturbation theory, the parameters mo and M need no longer be restricted to their leading-order HTL values (given in eqs. (91) and (94) below). Technically, HTLPT corresponds to using 8 as an expansion parameter that is set equal to 1 eventually. In Ref. 2 1 , HTLPT has been used to calculate the pressure of QCD at one-loop level. This corresponds to calculating the thermal pressure of a modified free Lagrangian (84) with 6 = 0. If (but only if) the mass parameters mo and M are identified with their leading-order HTL values (91) and (94), this includes the plasmon term ~ g3 of the pressure correctly, but over-includes the leading-order interaction term ~ g2 by a factor of 2. (In Ref. 21 the over-inclusion factor was in fact larger because of an inconsistent usage of dimensional regularization 25 . Corrected numerical results at one-loop level are contained in Refs. 59>60.) Starting at two-loop order, it becomes possible to determine raj and M by a variational principle. The two-loop pressure in HTLPT is in fact almost prohibitively complicated because of the necessity to include the non-local HTL vertices along with the nontrivial HTL propagators. Refs. 59 ' 60 have simplified this task by expanding the resulting sum-integrals in powers of mo/T and M/T such as to include all terms through order g5 if mo/T and M/T are taken to be of order g. (These mass expansions have been checked to be fairly accurate in the applications of interest 79 .) When determining mo/T and M/T through variational equations, the two-loop HTLPT result is equivalent with the perturbative result (77) through order gA log(g), with deviations setting in at order g4. The thermal mass parameter mo coincides with the perturbative result for the Debye mass at leading order, though not beyond, which is not too surprising since it is just a variational parameter and not directly related to the full electrostatic propagator; the quark mass parameter M when determined by a variational principle comes out as ~ g at weak coupling, however with a coefficient that differs from leading-order perturbation theory and is in fact subtraction-scale dependent 60 . When evaluated numerically, the HTLPT results, like those of SPT in the scalar case, turn out to be a clear improvement over the conventional

100

J.-P. Blaizot, E. Iancu and A. Rebhan 1.05 P/Po 1 0.95 09 0.85 0.8 1.5

2

2.5

3

3.5

4

4.5 5 T/Tc

Fig. 12. The two-loop DRSPT result 7 4 for pure-glue QCD (gray band) in comparison with the two-loop H T L P T result of Ref. 5 9 (dashed lines) and the "NLA" result of Ref. 2 5 described in Sect. 4.4 (dash-dotted lines), all with fx varied around 2-KT by a factor of 2.

perturbative results in that (at least for all T > 1.5TC) both the one-loop and the two-loop results are lower than the ideal-gas result while including the plasmon effect ~ g3 which in conventional perturbation theory (see Fig. 11) leads to results much above the ideal-gas value in contradiction with all lattice results. Moreover, the one-loop and the two-loop results do not deviate too strongly from each other so that they seem to have satisfactory convergence. On the other hand, they fall short of describing lattice results quantitatively but remain too close to the ideal-gas value and account for only about half of the deviation of the lattice results from the latter. A hint on the possible reason for this apparent deficiency of HTLPT in the application to the thermodynamics of QCD comes from a simpler variant of SPT, namely when the latter is implemented only in the dimensionally reduced sector (79) by rewriting (79) trivially as £EQCD = ^

^

+ i(A^o) Q (AA 0 ) a +i(m|+5m 2 )ASAg-i<Sm 2 ASAS.(86)

Here m2E is determined by perturbative matching, whereas the additional Sm2 is to be fixed by a gap equation (i.e., a principle of minimal sensitivity). Like any SPT, this dimensionally reduced SPT (DRSPT) introduces additional UV singularities at finite loop order, which should be minimally subtracted to allow comparison with HTLPT. Like HTLPT, this formalism is gauge invariant, but very much simpler and it avoids the intermediate modification of hard modes (where ordinary

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perturbation should be accurate anyway). In Ref. 74 it has been recently shown that the two-loop result of DRSPT is much closer to the lattice results (reproduced in Fig. 12). Because fermions are integrated out in DRSPT, this also avoids the problem HTLPT appears to have with a gap equation for the fermionic mass. (On the other hand and in contrast to HTLPT, DRSPT is clearly restricted to static quantities.) Unfortunately, at three-loop order DRSPT gives rise to a gap equation for Sm2 which does not connect to perturbation theory any longer so that it cannot be viewed as an improvement of perturbation theory. 74 4.2. 2PI formalism

in gauge

theories

In the following we shall describe a different approach in which the 2PI formalism in terms of full propagators is used to construct approximations which avoid the UV problems and ambiguities of SPT and HTLPT. This turns out to amount, at the two-loop level, to a description of the thermodynamics of hot QCD in terms of weakly interacting quasiparticles, whose nontrivial spectral data serve to incorporate (at least in leading and subleading orders) the strong interactions of the elementary degrees of freedom. In QCD, the thermodynamic potential is a functional of the full gluon (£>), quark (S), and Faddeev-Popov ghost {Dgh) propagators, pn[D,S,Dgh]

= ^ T r \ogD~1 - Tr l o g S " 1 - Tr -^TrIW+

T r E S + TrUghDgh

logDj +

^[D,S,Dgh},(87)

where Tr now includes traces over color indices, and also over Lorentz and spinor indices when applicable. The self-energies for gluons, quarks and ghosts are denoted by n , E and Hgh, respectively.

. . • < • . .

Vww\» ' ' • > • • '

(a)

(b)

CO 0 (c)

(d)

Fig. 13. Diagrams for 5> at 2-loop order in QCD. Wiggly, plain, and dotted lines refer respectively to gluons, quarks, and ghosts.

In the following we shall consider a two-loop ^-derivable approximation which amounts to keeping the diagrams displayed in Fig. 13. This again

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J.-P. Blaizot, E. Iancu and A. Rebhan

leads to the important simplification that the entropy is given by a comparatively simple expression

s lr

- /(0^l t o l °^~ 1 - I °' n R « D ] -2tr/^J^[Iml°6s~'-ImSReSl'

(88)

with vanishing interaction term S' (cf. (69)), and this also holds true for fermion number densities: U=

" 2 t r / ( 0 ^ T [ I m l o S 5 _ 1 - ImEReS] *

(89)

where the completely analogous interaction term M' is equally absent at the two-loop level. While ^-derivable approximations are symmetry preserving as concerns global symmetries 22 , local gauge invariance is not respected, so this approximation introduces gauge dependences as long as vertices are not dressed in a self-consistent manner in line with the self-energies3. These gauge dependences are actually parametrically suppressed at the stationary point 81 , where they enter only at twice the order of the truncation. However, in what follows we shall construct further approximations based on the gauge invariant hard thermal loops, and this will entail manifest gauge independence in our applications. In these approximations, where the self-consistency of the two-loop ^-derivable scheme is only approximately realized, but such that the deviations from full self-consistency are of the order of three-loop corrections, only the two physical polarizations of the gluons contribute and the Faddeev-Popov ghosts just cancel the unphysical gauge degrees of freedom of the former so that they can be simply dropped. 4.3. Approximately density

self-consistent

entropy

and quark

As discussed above, conventional thermal perturbation theory exhibits very poor apparent convergence as soon as collective phenomena such as the plasmon effect are included, which involve odd powers in the coupling. The self-consistent entropy and density functionals at two-loop order also contain the plasmon effect, but together with higher-order terms that ordinarily would have been discarded by a strictly perturbative expansion in g because ••On a formal level this has been worked out in Ref.

Thermodynamics

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103

otherwise the renormalization programme could not have been carried out. By contrast, the two-loop entropy and density functionals (88) and (89) are UV finite expressions and can be evaluated in a nonperturbative manner using the high-temperature/density approximation of the gluon and quark propagators, which are finite at leading and next-to-leading order in HTL perturbation theory.

4.3.1. HTL approximation As a first approximation we consider the two-loop entropy and density functionals evaluated completely using HTL (HDL) propagators. Prom the discussion of the scalar model we anticipate that this includes correctly the leading-order interaction contribution oc g2 in the entropy (in contrast to the pressure). k But unlike the scalar <j>A model, where the HTL self-energy is a simple constant mass, cf. (18), the HTL self-energies for gluons and quarks contain two non-local structure functions each. The gluon HTL self-energy contains transverse and longitudinal components given by IlL{u),k)

+ fc" 2fc log —-k

=m2D 1

nT(w,fc) = i

™D

1

+

LJ

2

U

-fc 2 ,o nflLL , fc2

(90)

where n2T2

rhl = (2N + Nf)9-r-

„2 2

+ Nf9-^r

(91)

is the (leading-order) Debye screening mass.1 As is well known 83,84 , the gluon propagator dressed by these self-energies has quasiparticle poles at w ^ i ^ ) with momentum-dependent effective masses and Landau damping cuts for \w\ < k. When k ^> mo, the pole corresponding to the collective longitudinal excitation has exponentially vanishing residue 85 , whereas k

A different procedure for including the full HTL propagators has been proposed by Peshier 8 2 , who considers a (somewhat ad-hoc) modification of the $ functional such as to have correct leading-order contributions and UV finiteness when full propagators are approximated by HTL ones. This turns out to be equivalent to our results when expanded out to order g3. 'For simplicity we shall write out our formulae for only one value of the chemical potential; the generalization to several different chemical potentials is straightforward.

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Rebhan

transverse excitations tend to u> —> ^k2 + m2^, where the asymptotic mass is given by

m20o=nT(k,k)

= ^ml.

(92)

The (massless) quark propagator at finite temperature or density is split into two separate branches of opposite ratio of chirality over helicity with propagators A± = [—u> + k± E ± ] _ 1 . In the HTL approximation, the respective self-energies read 86 : - , ,. M2 E ± (a,,*) = — where (Cf = (N2 -

/

wTfc, w + /c\ l - _ l o g — ,

,_ (93)

1)/2N) M2 =

in.

9

/

,,2

-^l I T* + ^

).

(94)

The dressed fermion propagators have quasiparticle poles at u>± (k) and Landau damping cuts. At large k and positive frequency w + —> \Jk2 + M^ with asymptotic mass M^ = 2M2; the additional branch u;_ which has no analogue at zero temperature and density has exponentially vanishing residue 87 ' 88 . Although the various HTL self-energies constitute a leading-order result only under the condition of soft momenta and frequencies, the result for the asymptotic masses applies also for hard momenta 89 ' 90 . Using these expressions in the entropy formula (88) this takes care of all contributions of order g2, but only part (< 1/4) of the plasmon term ~ g3. However, it also contains infinitely many higher-order terms which despite being incomplete may help to get rid of the pathological behaviour of the perturbation series truncated at low orders in g. Among such higher-order contributions is for instance a g4 contribution reading (for pure glue)

involving a g4 log(l/<7)-term, whose coefficient in pure-glue QCD is 1/12 of that of the complete perturbative result."1 By contrast, simple massive quasiparticle models such as those used in Refs. 27 ' 28 do not have a g4 log(l/g)-term in the entropy at all. m

T h e correct coefficient will be restored by 0{gi l o g ( l / s ) T 2 ) corrections to m^, the constant under the logarithm also receives three-loop contributions.

whereas

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105

SHTL/SSB

1 0.9 0.8 0.7 0.6 0.5

1

1.5

2

2.5

3 mD/T

Fig. 14. HTL entropy per gluonic degree of freedom normalized to its Stefan-Boltzmann value as a function of the Debye mass rhr)(T,n)/T. The full line gives the complete numerical result; the dashed line corresponds to the perturbative result to order (mD/T)3 ~ g3. The dotted line gives the entropy for scalar degrees of freedom with momentum-independent mass m = moo = " I D / V ^ ; its perturbative approximant is given by the dash-dotted line.

In Fig. 14, the numerical result for SHTL/SSB in the case of pure glue is given as a function oirho/T, which is the only independent parameter. The HTL result is depicted by the full line and is found to be a monotonically decreasing function of rho/T. If this were expanded in powers of rho/T and truncated beyond ( r a o / T ) 3 ~ g3 (dashed line in Fig. 14), this property would have been lost at m£>/(27rT) ~ 1/3, where one might still expect a sufficiently clear separation of hard and soft scales which is a prerequisite of the HTL approximation. The numerical result for SHTL is in fact very close to a simple massive quasiparticle model with entropy 2NgSsB(mo0), represented by the dotted line in Fig. 14. This is due to large cancellations between the longitudinal and transverse contributions to the HTL entropy. However, quite remarkably one has SHTL < 2./Vg<SsB(raoo) even though the latter contains 30% less of the plasmon term ~ g3, which would be expected to work in the opposite direction. This is further emphasized by comparison with its perturbative approximation given by the dash-dotted line. In Fig. 15 the HTL entropy of 1 quark degree of freedom at zero chemical potential is displayed as a function of the fermionic plasma frequency M. Like in the pure-glue case, this contains also an (incomplete) g4 \og(l/g)

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Rebhan

contributions that is not present in simpler quasiparticle models:

SfiiTL = NNf^

(log J + 0-22...)

(96)

Sf,HTL does not contain any contributions to the plasmon term ~ g3, which entirely come from NLO corrections to M^,. There is therefore rather little deviation of the full numerical result for the fermionic contribution (solid line) from the perturbative one truncated above order g2 (dashed line). Compared to the entropy of a simple massive fermionic quasiparticle model 5}0)(Moo) (dotted line in Fig. 15), one finds extremely good numerical agreement, which again takes place only after adding up all the quasiparticle ((+) and (—)) and Landau-damping contributions, however.

<5/,ffTi/<5/ 1 0.9 0.8 0.7 0.6 0.2

0.4

0.6

0.8

1

1.2

1.4 M/T

Fig. 15. HTL entropy per quark degree of freedom at fi = 0 normalized to its free value (solid line), the corresponding perturbative order-g 2 result (dashed line), and the entropy of a quark with constant mass Moo (dotted line).

In the case of zero temperature but high chemical potential, the quantity of interest is the quark density. This does not contain any plasmon term ~ g3, but rather ~ gA log(l/g). The HDL approximation does contain some though not all of this term: < k

= ^ / ^ ( l ° g f +

0.35..-)

(97)

Order-<72 log g contributions to the asymptotic masses of the quark and gluon quasiparticles, still within the framework of the 2-loop ^-derivable approximation, are responsible for the remaining contribution to the coefficient of the
Thermodynamics

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107

In Fig. 16 the numerical result for NHDL is given as a function of Mj\i (full line) and compared to that of a simple quasiparticle model -/Vo(Moo)

T=0

1 3^

(M'-M^^/X-MOO)

(98)

as well as to a perturbative approximation truncated beyond (M/n)2 ~ g 2 . Remarkably, the full HDL result drops to zero at almost the same value as the simple quasiparticle model. However, the former becomes negative thereafter, showing that the HDL approximation is breaking down at Moo/fi > 1 at the latest.

MHDIIN^

0.6

0.7

M/lt Fig. 16. HDL quark density at T = 0 normalized to its free value (solid line), the corresponding perturbative order-g 2 result (dashed line), and the free quark density of a quark with constant mass Mew (dotted line).

4.3.2. Next-to-leading

approximations

The plasmon term ~ g3 at high temperatures T 3> rhu becomes complete only after inclusion of the next-to-leading correction to the asymptotic thermal masses ra^ and M ^ . These are determined in standard HTL perturbation theory through Sm200{k)=

Re
Re(

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J.-P. Blaizot, E. Iancu and A.

Rebhan

where thick dashed and wiggly lines with a blob represent HTL propagators for longitudinal and transverse polarizations, respectively. Similarly,

(100) s+s *** = Re ( , f , \, + , f, V ) \ u = l s . The explicit proof that these contributions indeed restore the correct plasmon term is given in Ref. 25 . These corrections to the asymptotic thermal masses are, in contrast to the latter, nontrivial functions of the momentum, which can be evaluated only numerically. However, as far as the generation of the plasmon term is concerned, these functions contribute in the averaged form

,

_ Jdkkn'BE(k)ReSUT(uj

2

6m

°°-

= k) m

Jdkkn'BE(k)

>

and similarly ,

2 _ Jdfcfcn'FD(fc)Re2fc<SE+(^ = fc) °°~ Jdkkn'FD(k) •

l

;

which can be determined analytically 24 , leading to the remarkably simple expressions Sml

= -l-g2NTmD,

6Ml = -±g2CfTmD,

(103) Cf = Ng/(2N).

(104)

These corrections only pertain to the hard excitations; in contrast to the scalar case, in gauge theories the one-loop corrections to the various thermal masses of soft excitations are known to differ substantially from (103) or (104). For instance, the gluonic plasma frequency at k = 0 reads 91 Sm^ Jrh^ w -Q.l&y/Ng, which is only about a third of Srrv^/m2^; the NLO correction to the nonabelian Debye mass on the other hand is even positive for small coupling and moreover logarithmically enhanced 39 5m2D/m2D = +V3A7(27r) x glog(l/g). For an estimate of the effects of a proper incorporation of the next-toleading order corrections we have therefore proposed to include the latter only for hard excitations and to define our next-to-leading approximation (for gluons) through SNLA

=

SHTL

soft

hard

Thermodynamics

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Plasma 109

where m2^ includes (103). To separate soft (k ~ m o ) and hard (k ~ 2irT) momentum scales, we introduce the intermediate scale A = y/2irTiriDCA and consider a variation of CA = \ . . . 2 as part of our theoretical uncertainty. Another crucial issue concerns the definition of the corrected asymptotic mass 77100. For the range of coupling constants of interest (g > 1), the correction l^m^J is greater than the LO value m 2 ^, leading to tachyonic masses if included in a strictly perturbative manner. In the simple scalar model of Sect. 2.3.2 we have seen that the requirement of self-consistency for the soft modes leads to a quadratic gap equation for the asymptotic thermal mass, eq. (76). In QCD, where the (non-local) gap equations are much too complicated to be attacked directly, we simply consider perturbatively equivalent expressions for the corrected m^o which are monotonic functions in g. Besides the solution to a quadratic equation analogous to (76), namely

<-*£±f^-££*.,

(.06)

we have tried the simplest Pade approximant (degree [2,1]), which also gives a greatly improved approximation to the solution of the gap equation of the scalar (large N) model. In QCD, our final results do not depend too much on whether we use the Pade approximant 23,24 or a quadratic gap equation 25 . The main uncertainty rather comes from the choice of the renormalization scale which determines the magnitude of the running strong coupling constant (chosen to satisfy the renormalization group equation to 2-loop order in the following). In Fig. 17, the numerical results for the HTL entropy and the NLA one are given as a function of T/Tc with Tc chosen as Tc = 1.14Aj^g. The full lines show the range of results for SHTL when the renormalization scale p, is varied from irT to ATTT; the dash-dotted lines mark the corresponding results for SNLA with the additional variation of CA from 1/2 to 2. The dark-gray band are lattice data from Ref. 6 5 (the more recent results from Ref. 66 are consistent with the former within error bars and centered around the upper boundary of the gray band for T « 3TC and somewhat flatter around 2TC). Evidently, there is very good agreement for T > 2.5TC. When Nf > 0 massless quarks are included, the next-to-leading corrections to the asymptotic fermionic masses (104) have the same problem as the bosonic ones that for large coupling the asymptotic masses may turn tachy-

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J.-P. Blaizot, E. Iancu and A.

Rebhan

S/SSB

1 0.9 0.8 0.7 0.6 1.5

2

2.5

3

3.5

4

4.5 5 T/Tc

Fig. 17. Comparison of the lattice data for the entropy of pure-glue SU(3) gauge theory of Ref. 6 5 (gray band) with the range of SHTL (solid lines) and SNLA (dash-dotted lines) for p, = -KT . . . 4TTT and

c A = 1/2 . . . 2.

onic. In Ref. 25 they have therefore been treated like the bosonic asymptotic masses, by postulating the simplest quadratic gap equation which is perturbatively equivalent to (104). However, there is an obvious asymmetry in that only gluonic quantities appear on the right-hand side of (103) and (104). Indeed, treating them in the same manner is not consistent in the limit of Nf —> oo, which has been solved recently in Refs. 92 . To be consistent with this limit, the fermionic gap equation has to remain linear as it is in perturbation theory. We therefore restrict our requirement of a perturbative equivalent gap equation with monotonic behaviour only to the bosonic (gluonic) sector. For the fermions we proceed by treating the right-hand sides of eqs. (103) and (104) in the same manner, which amounts to 9 3 Ml

= 9-^f^

- ^g'CfTfh^.

(107)

There is then no negative feedback from the fermion mass itself, it only inherits higher-order terms from the solution to rhoo • This does not avoid the appearance of tachyonic masses when both Nf and g are large, but it turns out that for Nf < 3 one still has a monotonic behaviour of M^ as a function of g. In fact, in the case Nf = 3 the solutions to the correspondingly corrected gap equations happen to coincide precisely with those of the (structurally rather different) decoupled quadratic equations adopted in Ref. 25 . As a result, the changes on the results of Ref. 2 5 are rather minimal for phenomenologically interesting values of Nf. (In the limit Nf —> oo, there are more important effects, such that the picture of stable quasi-particles

Thermodynamics

of the High-Temperature

Quark-Gluon

2

4.5

Plasma

111

S/SSB

0.94 0.92 0.9 0.88 0.86 0.84 0.82 1.5

2.5

3

3.5

4

5

T/Tc Fig. 18. Comparison of the HTL entropy (solid lines) and the NLA estimate (dashdotted lines) for Nf = 0, 2, 3 with the estimated continuum extrapolation of Nf = 2 lattice data of Ref. 9 4 . (The gray dash-dotted line corresponds to using a quadratic fermionic gap equation, which only has an effect for NLA Nf = 2, which is minimally higher than the result obtained with a linear one required for consistency with the largeNf limit 9 3 ).

seems to have to be abandoned above some large coupling, but up to this point the quantitative agreement with the exact result is quite good 93 .) In Fig. 18, the (updated) results for the entropy with Nf < 3 massless quarks are compared with a recent estimate 94 of the continuum limit of lattice results for Nf = 2 (gray band), but now with SHTL and SNLA evaluated for the central choice of p, = lixT and c\ = 1 (with unchanged Tc/h.y£s). When Nf is increased, there are competing effects of larger thermal masses versus slower running of a s , which result into a rather weak dependence of our results on Nf as a function of T / A ^ s as it is in Fig. 18. This seems to be consistent with the very small Nf dependence observed on the lattice (when the thermodynamic potentials are normalized by their ideal-gas limit and expressed as a function of T/Tc). It should be understood, however, that the simple quadratic gap equation (106) and its extension to the fermionic sector, eq. (107), are at present only an estimate of how the next-to-leading order corrections (99) and (100) in actuality modify the spectral properties of the hard quasiparticles. Moreover, the former are included so far in the averaged form (101) and (102). A possible refinement is to include the non-trivial momentum dependence of (99) and (100) which we expect to render a separation scale between hard and soft quasiparticles superfluous because the corrections (99) and (100) appear to grow with momentum so that their incompleteness at soft

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J.-P. Blaizot, E. Iancu and A.

Rebhan

momenta may be negligible. When doing so we also hope to be able to come to a better understanding of the mechanism which provides the nonlinear feedback preventing tachyonic masses from appearing. 4.4.

Pressure

In the above, we have concentrated on the entropy (and quark number density) as the thermodynamic potential which is most suitable for a (HTL) quasiparticle description. Were it not for the trace anomaly 95 (Tjf) = £—3P ^ 0 the corresponding pressure would be simply given by P = (T5+/jV)/4 (in the case of massless quarks). The correct relation is instead provided by an integration such as P(T,

M)

= /

S(T', n) dT' + P{T,, n).

(108)

Up to a single integration constant, entropy and density evidently contain all the relevant information. This integration constant, however, represents strictly nonperturbative information, because it amounts to fixing the bag constant. Since QCD is asymptotically free, but confining at low temperature, this constant cannot be fixed by requiring that the thermal pressure as obtained from entropy and density vanishes at zero temperature. It is also not possible to determine P(T\) from the requirement that both p(as) = P{T)/P0{T) and s(as) = S{T)/S0{T) approach 1 in the limit of as —» 0, because the differential equation p(as) + j/3(as)p'{as) = s(as) with (3(as) = — Poa^ — (3\az3 — . . . has as homogeneous solutions p(as)hom. = Cexp \

[4PQ1 + 0(as)}

I,

and this vanishes at as = 0 together with all its derivatives. Instead we can fix the integration constant by comparison with lattice data for the pressure at fi = 0. A simple possibility for our purposes is to take Ti = Tc and P(TC, 0) = 0. For T substantially larger than Tc this does not influence much the results derived from entropy and density, since in the ratio P/PQ the (bag) constant term decays like T - 4 . The result obtained from integrating the entropy in the case of pure-glue QCD with boundary condition P(TC) = 0 is shown in Fig. 12 and compared with available lattice data as well as the results obtained recently in 2-loop HTLPT by Andersen et al.59 and in a dimensionally reduced variant of SPT 7 4 .

Thermodynamics

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113

4.4.1. Finite chemical potential For finite chemical potential, the entropy and quark density expressions satisfy the Maxwell relations

as

dT

(109)

only to the same order in g to which they are accurate when expanded out perturbatively. In simple quasiparticle models 27 ' 28 one therefore has to allow for a bag function (instead of a bag constant) to restore thermodynamic consistency. This can in fact be utilized for matching lattice results and mapping them in a phenomenological manner to nonvanishing chemical potential 29 , where recently significant progress has been made on the lattice side 96 . This includes by now concrete results on the thermodynamical potentials 97 , which have been found to agree well with the simple quasiparticle models 98 . A similar programme has been carried through with the HTL expressions for entropy and density in place of those of the simpler quasiparticle models with constant masses, and the results are, like those of the simpler models, encouraging 99 . An extension which includes next-to-leading corrections to the asymptotic thermal masses is work in progress.

4.5. Quark number

susceptibilities

Quark number susceptibilities (QNS) are defined as the second derivatives of the pressure with respect to the chemical potentials of the different quark flavors:

_dAfi

d2P

*« = wr d^r3 = x*

(110)

where i, j are flavor indices. These quantities are of direct interest for studying the transition to the quark-gluon plasma phase as they can be related to fluctuations in conserved charges which can discriminate against a purely hadronic phase 100 . On the theoretical side, there are new results on QNS from lattice gauge theory 101 ' 102,103 ' 104 , extending them to higher T/Tc which can be compared against conventional perturbative results and those resulting from resummation schemes aiming at overcoming the poor convergence of the former.

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J.-P. Blaizot, E. Iancu and A. Rebhan

4.5.1. Diagonal quark number susceptibilities The diagonal QNS, which for degenerate quark flavors are all equal and denoted simply by x m what follows, are found to increase sharply at the deconfinement phase transition toward a large percentage of the ideal gas value xo = NT2/3 (for SU(iV) and massless quarks). In perturbation theory, the series expansion of x/Xo is now known 105 through order g6\n(l/g) and the coefficient under the log at order g6 is in fact computable in perturbation theory, in contrast to the one appearing in the thermal pressure, though not yet available. The first few terms of this expansion read 106 ' 5 :

J. Xo

=

i_IA^£/T\2 +!^£ lE + UkfiY 2N 8 W

N 8 V3

_ ^ ( £ ) \ g l + ev>

6

W

(in)

(for the more lengthy results on the subsequent coefficients we refer the reader to Ref. 1 0 5 ). In these strictly perturbative results the plasmon term overcompensates the leading-order interaction term, much like in the case of the pressure, though less severely so (for Nf = 2 the diagonal QNS fall below the idealgas value only when T > 40TC, and only for T > 700TC does one find a growth of x/Xo with temperature as observed on the lattice — in the case of the pressure these temperatures below which perturbation theory clearly fails are orders of magnitudes higher). The ^-derivable 2-loop expression (89) captures both the leading-order term oc g2 and the plasmon term ex g3 accurately, together with a subset of higher-order terms that are usually expanded out in 2-loop perturbation theory. As additional approximations we may start from the HTL quark density MHTL where (89) is evaluated with HTL propagators. Differentiation with respect to the chemical potential yields the corresponding approximation for the diagonal" QNS. While this approximation contains the leading-order term oc g2 accurately, it does not include anything of the plasmon term. In the QNS, the plasmon term arises completely from corrections to the asymptotic thermal masses of quarks according to (100). "In the HTL approximation the QNS are purely diagonal.

Thermodynamics

X/Xo

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Quark-Gluon

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115

!

0.95 0.9 0.85 0.8

1.5

2

2.5

3

3.5

4

4.5

5

T/Tc Fig. 19. Comparison of our results for x/xo m the formal limit Nf=0 continuum-extrapolated lattice results for quenced Q C D 1 0 3 .

with two recent

Numerically, the HTL approximation gives results somewhat above the strictly perturbative result to order g2, and, using the same gap equations for the asymptotic thermal masses of quarks as in the previous application, the result is halfway towards the ideal-gas result, whereas a strictly perturbative inclusion of the plasmon effect would have resulted in a result in excess of the free one. Fig. 19 shows the results for quenched QCD together with recent results for two different continuum extrapolations 103 (both are higher than the previous lattice results 108 ). Fig. 20 shows our prediction 107 for Nf = 2 in comparison with the lattice data of Ref. 101 , for which a continuum extrapolation is not yet available, however. It would be interesting to compare our predictions with HTLPT to twoloop order; the one-loop HTL-resummed results from charge correlators 109 suffer from severe overcounting while missing out the plasmon term 110 so that one should not perform a comparison yet.

4.5.2. Off-diagonal quark number susceptibilities Since lattice results are starting to become available also for the off-diagonal QNS, which in the limit of massless QCD we denote collectively by x, we briefly discuss this quantity, too. In contrast to the diagonal QNS, the idealgas value of off-diagonal QNS is zero, so this directly probes the interactions in the system. The leading-order contribution to off-diagonal QNS require two fermion loops connected by gluon lines. The diagram with just one gluon exchange

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Rebhan

4.5 5 T/Tc Fig. 20. Comparison of our results for x/xo in raassless Nf=2 QCD with the lattice results of Ref. 1 0 1 (no continuum extrapolation). Full lines refer to the HTL approximation, dash-dotted lines to NLA, when the MS scale is varied from p, = nT to p. = 4nT.

vanishes by colour neutrality. The one with two gluon exchange is non-zero, but because the fermion loops are then even functions of /x, it contributes to \ only when fx ^ 0, starting at order g3, namely

g4(N2 -

\)Tinm

for i ^ j . (112) 16ir5m,£) This contribution is indeed contained in the next-to-leading approximation discussed above, but only after the corrections to the asymptotic thermal masses have been included. When all chemical potentials vanish (which is the case most easily accessible to lattice studies), the above approximations of the QNS become strictly diagonal. In this limit the lowest-order diagram contributing to offdiagonal QNS is the 4-loop "bugblatter" diagram shown in Fig. 21. 1 0 7 In this diagram each quark loop produces an effective C-odd vertex for electrostatic gluons 111 which in turn give rise to a logarithmically enhanced contribution at order g6, reading 107 Xij —

X Xo

(iV 2 - 1)(AT2 - 4) 128JV2

6

il o g -l .

(113) 9 This vanishes in SU(2) gauge theory (though not in Q E D 1 0 7 ) , and in (massless) Q C D this leads to 6 5 10 ,l o g -1 ^log(l/Q;s). (114) 144 '9TT 3 Xo N=3 9 Assuming that the undetermined (though perturbatively computable) constant under the log is of order one, the natural order of magnitude of

2L

(?)

Thermodynamics

of the High-Temperature

Quark-Gluon

Plasma

117

Fig. 21. Lowest-order diagram in the thermodynamic potential that contributes to offdiagonal susceptibilities dAfi/d/Xj at fj, = 0.

X/X0 = x/T2 is ~ 10" 4 for T ~ 3TC. Recent lattice studies in Nf = 2 QCD 1 0 1 ' 1 0 3 have found only values consistent with zero so far, with errors claimed to be below 10~ 6 , but this perhaps remains to be confirmed by independent groups before it can be viewed as a new puzzle arising from the comparison of lattice and perturbation theory. 5. Conclusion and perspectives In this paper, we have reviewed progress made recently in the calculation of the high temperature quark-gluon plasma using weak coupling techniques. Such calculations meet with difficulties, many of which are not typical of gauge theories, but also occur in scalar field theories. In fact, what determine the validity of a weak coupling expansion in thermal field theories is not only the strength of the coupling, but also the magnitude of the thermal fluctuations which varies according to the relevant momentum scales. The thermodynamical functions are dominated by hard degrees of freedom, carrying momenta of the order of the temperature T. Soft modes, with typical momenta of order gT, contribute a correction, which however turns out to be difficult to calculate, for essentially two reasons. First, the dynamics of the soft modes receive contributions from the hard modes which can only be taken into account through all order resummations. Second, although the interactions among the soft modes can be treated perturbatively, the corresponding perturbation theory is an expansion in powers of g rather than g2, as it is for the hard modes. The latter feature is what causes the apparent poor convergence of perturbation theory which has been observed in calculations up to order g5, both in the scalar case and in QCD. In QCD, because of the unscreened magnetic fluctuations, perturbation theory eventually breaks down. But this occurs at order g6 only, and accordingly one could expect the contribution of the magnetic fluctuations to the pressure to be numerically small. This is in fact suggested by detailed calculations based on dimensional reduction 63 . Therefore most of this re-

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view has concentrated on the difficulties which are common to all thermal field theories, and which involve the momentum scale k ~ gT. The various resummation techniques which have been proposed to overcome the difficulties of perturbation theory rely to some degree on a separation of scale between hard and soft excitations, and follow the general pattern of effective theories. This is obviously the case for the approaches based on dimensional reduction, but is also true of screened perturbation theory and ^-derivable approximations, although somewhat more indirectly. In dealing with the soft sector, most resummations use the hard thermal loops as common building blocks. Non-linear "gap equations" for the calculation of "thermal mass" parameters emerge in various forms, and the difficulties associated with the ultraviolet renormalizations are dealt with in different fashions, depending on the specific approximations. In "Screened Perturbation Theory", and its extension to QCD referred to as "HTL-perturbation theory", the use of the HTL Lagrangian as a basis for the resummation has the advantage to preserve manifest gauge symmetry, but also the drawback to lead to rather complex calculations beyond the lowest order. Besides the HTLs are used for all momenta, soft and hard, while they are accurate only at soft momenta. At any finite order in this scheme, one meets with temperature-dependent UV divergences whose renormalization requires the use of "thermal counterterms". Although they get in principle corrected gradually as one includes higher and higher orders, these features remain unsatisfactory in the low order calculations that one is able to perform. At least at the level of the formalism, ^-derivable approximations provides a conceptually clearer perspective on the general problem of propagator renormalization than the simple "add and subtract" approach of SPT. The overcounting is automatically avoided by using skeleton expansions, and the gap equations emerge naturally, via the variational principle for the thermodynamic potential. The HTLs appear as lowest-order approximations to the solutions of the gap equations, and are used only in the kinematical domains where they are legitimate approximations. However, the practical implementation of $ derivable approximations is not exempt of difficulties: In general, the gap equations are non-local, which complicates both their UV renormalization, and their numerical resolution. A further complication comes from the fact that the ^-derivable approximations violate gauge symmetry. (However, this violation is parametrically under control, which leaves the possibility to construct approximate solutions to the gap equations which are gauge-invariant.)

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One of the most obvious advantages of ^-derivable approximations remains the simplifications they allow in the calculation of the entropy: The two-loop calculation for the pressure turns into an effectively one loop calculation for the entropy, which furthermore is finite once the self-energy involved in the calculation has been made finite. The results reported in Refs. 23,24,25 have been obtained however after additional approximations. In the scalar 2-loop model, which is exactly solvable, it has been possible to test these approximations by comparing against the respective exact results. In QCD, where such a comparison is not possible, further refinements should be studied, in particular to deal with non-local effects in the gap equations. Also, the calculation of the entropy is built on real-time information, namely the dispersion relation for the hard quasiparticles. Understanding the relation with purely Euclidean formalisms, such as that of dimensional reduction where this information does not enter in an explicit fashion, is certainly worth further investigations. In closing, one should perhaps emphasize that most of the calculations reported in this review point to the validity of the quasiparticle picture of the quark-gluon plasma at high temperature. That is, they support the idea that, by suitably dressing the elementary degrees of freedom, one could minimize the effects of their residual interactions. This is particularly transparent in the entropy calculation. The results obtained in this manner for the thermodynamic potentials and also for quark susceptibilities seem to encourage this physical picture, which gives hope for the eventual applicability of correspondingly improved perturbation theory to dynamical quantities. Acknowledgments This work has been supported by the Austrian-French scientific exchange program Amadeus of APAPE and OAD, and the Austrian Science Foundation FWF, project no. P14632. References 1. 2. 3. 4.

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JET Q U E N C H I N G A N D R A D I A T I V E E N E R G Y LOSS IN D E N S E N U C L E A R M A T T E R

Miklos Gyulassy1, Ivan Vitev2, Xin-Nian Wang3 and Ben-Wei Zhang4 1 Department of Physics, Columbia University, 538 W. 120th Street, New York, NY 10027 2 Department of Physics and Astronomy, Iowa State University, Ames, IA 50010 3 Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China We review recent finite opacity approaches (GLV, WW, WOGZ) to the computation of the induced gluon radiative energy loss and their application to the tomographic studies of the density evolution in ultrarelativistic nuclear collisions. 1. Introduction Since June 2000 measurements of Au + Au reactions at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory (BNL) at y i = 56,130,200 AGeV (GeV per nucleon pair) have revealed a variety of novel multiparticle phenomena not observed previously in e + e ~ , ep, pp collisions at any energy nor in nuclear collisions at lower (SPS/CERN and AGS/BNL) energies (y/s = 17,5 AGeV). While the bulk global observables such as the rapidity dependence of hadron multiplicities and transverse energy scale geometrically from elementary p + p collisions, the most striking new phenomena were discovered in rare high transverse momentum observables 1-9 . The preliminary RHIC data are discussed extensively in the Quark Matter 2001 and 2002 proceedings 10 . At RHIC collider energies the hard pQCD rate of rare high pr parton scattering becomes sufficiently large that jets can be used to probe the dense quark-gluon plasma formed in nuclear collisions. In 1982 Bjorken proposed that elastic final state energy loss of partons may "extinguish" jets in high

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energy p + p collisions11. However, the elastic energy loss of partons in a QCD plasma of temperature T ~ 300 MeV turned out to be too small (dEei/dx < 500 MeV/fm) 12 ' 13 for jet extinction. The data on p + p jets up to Tevatron energies show no sign of deviations from unquenched factorized pQCD. Di-jet acoplanarity was proposed as another possible manifestation of elastic final state interactions 14 ' 15 ' 16 . Prior to current RHIC data on nuclear collisions at 200 AGeV, the data showed no hint of this effect. In some early studies 17 * 18 ' 19 it was suggested that induced radiative energy loss in nuclear collisions could be much larger than the elastic energy loss and jet quenching should become observable at least in collisions of heavy nuclei. With the development of the Monte Carlo HIJING event generator 20 ' 21 ' 22 , predictions for the magnitude of the suppression pattern high pr hadrons were made in a first study 23 as shown in Fig. 1. The middle panel shows that up to an order of magnitude suppression

dN^/d?) L

; Au+Au(b=0)

R P A (PI)

R"(PT)

Au+Au(b=0)

p+Au(b=0)

with shadow and quenching no shadow no quenching with shadow no quenching no jets 1 2

3 PT

4

5

6

Fig. 1. HIJING predictions 2 3 of the inclusive charged hadron spectra in central Au+Au and p + Au collisions at T/S = 200 AGeV. The competing effects of minijet production (dash-dotted), gluon shadowing (dashed) (assuming that gluon shadowing is identical to that of quarks), and jet quenching (solid) with dE/dx = 2 GeV/fm are shown. RAB(pr) is the ratio of the inclusive PT spectrum of charged hadrons in A + B collisions to that of p + P- In contrast to Au + Au, no significant quenching is expected in p + Au (or d+Au) since only ~ 20% initial state shadowing and Cronin effects modify the pQCD spectrum at high PT-

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of charged hadrons was expected in the moderate pr ~ 5 GeV range. The input assumption was that the gluon energy loss in the plasma was dE/dx = 2 GeV/fm due to induced gluon radiative energy loss. The HIJING model is a two component model with semi-hard and hard pQCD jet above pr > Po = 2 GeV computed via the PYTHIA code and the "soft" beam jet fragments computed via a hybrid LUND and Dual Parton model algorithm. See the original article 20 for a detailed discussion and references. A selection of recent data from RHIC that confirm strong nuclear attenuation of moderate high transverse momentum charged hadron and 7r° production are shown in Fig. 2 through Fig. 5. In order to appreciate just how remarkable and different this experimental discovery is relative to previous lower energy data from the SPS, the quenching pattern of n° measured at RHIC is compared in Fig. 4 to the strong enhancement of high px 7r° measured in Pb + Pb at 17 AGeV at the SPS 2 4 .

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• • •

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-

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The enhancement of high pr hadrons at SPS is an amplified version of the well known Cronin effect first observed in p + A collisions. As we elaborate in later sections, it is due to multiple initial state interactions and completely masks any possible energy loss effects at the SPS. In contrast, at RHIC the high energy loss overwhelms the Cronin enhancement in our present interpretation of the data. As discussed for example in Refs. 25 ' 26 , it is useful to decompose the nuclear geometry dependence of invariant hadron distributions produced in A + B —»ft+ 1 at impact parameter b into a phenomenological "soft" and perturbative QCD calculable "hard" components as E

dNAB(b) d3p

Npart(b)

dN3oft(b) 1 dahard{b) + Ncou(b) 2 dyd pT TZ dyd2pT

(1)

where Npart(b) is the number of nucleon participants and Ncou(b) = af%TAB{b) is the number of binary NN collisions at impact parameter b. The nuclear geometry of hard collisions is expressed in terms of the Glauber profile density per unity area T ^ B ( 6 ) = f d2r TU(r)TB(r — b) where T^(r) = f dz PA{Y,Z) (see Fig. 6). The hard part scales with the number of binary collisions oc A4/3 because the probability of high pr pro-

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A+A collision geometry

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Fig. 6. Illustration of key aspects of the relation between the geometry of nuclear collisions and the participant Npart(b) = 2 / 2(PT) (see Refs. 2 7 , 2 8 ' 2 9 , 3 0 ) used to constrain the geometry experimentally are also illustrated.

cesses is small and built up from all possible independent parton scattering processes. The soft part scales with only Npart oc A because the probability of low transverse momenta processes is bounded by unitarity. This is sometimes referred to as Glauber shadowing or saturation depending on the calculational frame. The "soft" part is actually the overwhelming bulk of the produced hadron distribution. It is expected to reflect the collective hydrodynamic properties of the produced plasma in A + A collisions. Relativistic hydrodynamics predicts specific flow patterns, such as hadron mass dependent transverse elliptic flow 31 ' 32 ' 33 ' 34 , which may provide direct constraints on the QCD equation of state. Unfortunately, low transverse

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momentum processes are not directly computable via QCD and many competing phenomenological models can be adjusted to fit the data not only at RHIC but SPS and AGS as well. The great advantage of RHIC over the previous AGS and SPS explorations of nuclear collisions is that the computable high pr pQCD processes are sufficiently abundant, and that they can be used as effective "external probes" of the quark-gluon plasma that is produced. High pr and heavy mass partons are produced first, at time ST ~ l/rnr fm/c, while the most of the partons of the plasma with temperature T form and equilibrate at later times ~ 1/gT ~ 0.5 fm/c. Hard jets propagate along approximate straight eikonal lines through the plasma until T ~ R ~ 5 fm/c. The energy loss and transverse momentum broadening suffered by the jet prior to hadronization provides the tomographic handle to probe the opacity of the evolving plasma. Since non-central collisions have a well known geometrical asymmetry, the azimuthal distribution of final high pr hadrons provides even more information 35 ' 36 . This is illustrated in Fig. 7 by the normalized average thickness of nuclear matter as a function of the angle with respect to the reaction plane that a fast parton sees on its way out. The azimuthal distribution can be Fourier decomposed into dN(b)

1 dn(b) „

„

.

L

+ v2(y,b,pT)cos(2(<j)-&))

,

.,

, .

+ •••) ,

(2)

where fa is the azimuthal angle of the impact parameter b. In Fig. 8 the large elliptic flow component v2 is seen to extend up to the highest pr in noncentral collisions at mid-rapidity. Another critical new experimental test for jettiness at moderate pr ~ 5 GeV is provided by two particle correlation. The data from both STAR and PHENIX show clear evidence for backto-back correlations predicted by pQCD (and checked via the PYTHIA generator). However, there appears to be a difference between the correlation pattern of charged and neutral hadrons as seen in Figs. 9 and 10. The STAR data are in accord with predictions in Refs. 17,18 ' 37 that the two jet structure should be quenched by an order of magnitude relative to jet correlations in p+p due to the high opacity of the plasma. Similar jet structure in high px neutral hadron correlations is also found in the preliminary PHENIX data 4 . Whether the suppression of away-side jet seen in STAR data also manifests in the PHENIX experiment has yet to be addressed. For our review, the most important point about both data sets is that narrow azimuthal near side and away side correlations do exist. This is a

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1.4 Woods-Saxon geom. Sharp cylinder geom.

Au+Au

1.2

b = 2fm

1.0 ^

0.8

*

0.6

/ ^ > 0.4

/

0.2

b = 6fm

\

\

/

\

b = 10fm

Ov-

,''

/'

,-''

^ -

0.0

o.o

2.0

4.0

6.0

4) [rad] Fig. 7. The average normalized optical depth (R from Fig. 2 of Ref. 3 6 ) seen by a parton propagating through a plasma in azimuthal direction for different impact parameters and different density profiles.

H >

0.3

STAR Preliminary

M-

0.1

-0.1 -0.2 0.3 0.2 0.1 0 -0.1 -0.2

l

/^Hcentrality: 10-30%

centrality: 0-10%

r

•>+-f"

ZT^i centrality: 0-50%

centrality: 30-50%

0

2

4

6

8

10 12 0

2

4

6

8

10 12

pT (GeV/c) Fig. 8. Preliminary ^/s = 200 AGeV STAR 5 on high pr elliptic flow is shown. A value of t>2 = 0.2 corresponds to a 2-to-l azimuthal asymmetry of hadrons.

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

• Au+Au data

131

o p+p data + flow

l i i i i l ' i ' i I

-2

-1

1

2

3

A<|> (radians) Fig. 9. STAR data on high p y azimuthal correlations of charged hadrons at 200 AGeV from Ref.6 showing clear evidence near side and away side correlations characteristic of two jet production in elementary p + p. However, in Au + Au reactions the away side jet correlations are reduced in central collisions. The Back-to-Back jet correlations are attenuated by an order of magnitude in the most central collisions as expected if the plasma produced is o p a q u e 1 7 , 1 8 , 3 7 .

necessary (though not sufficient) conditions to enable us to interpret moderate pr hadronic quenching patterns as due to jet energy loss. In this review we focus on the rare hard pQCD tail of the hadron yields where all these interesting new nuclear phenomena have been discovered at RHIC. The central element of the theory that will be elaborated in the following sections is the predicted dependence of the induced QCD radiative energy loss on the jet energy, the plasma density, and the expansion properties of the plasma. We restrict the discussion to two complementary approaches, the GLV and WW/WOGZ formalisms, that are best suited in our opinion to applications to nuclear collision problems where the opacity is necessarily finite and the energy range of minijets accessible experimentally is typically 3 — 15 GeV. In the absence of final state interactions we recall the well-known lowest

132 M. Gyulassy et al.

^.0.25 x10'

: PHENIX Preliminary

Z

- Yield - P Y T H I A + cos(2A(|>)

0.2

•• cos(2A(|>)

•o

-0.1

J

i

1

1

1

i

I

i

I

i

I

.

I

3

A<|> Fig. 10. Preliminary PHENIX data 4 on high px azimuthal36 correlations of neutral pions also show clear jet structure amid an elliptic flow component.

order invariant pQCD differential cross section for inclusivep+p given by £|

a

"hard

d3p

= KY1 -I

abed

dx dx

" bfa/p(Xa,

h+X

Ql)fb/p(Xb, Qb)

1"

D (z ,Ql)

I u x—^(ab cd)- h/c c (3) TTZ dt where xa = Pa/PA, %b ~ Pb/PB are the initial momentum fractions carried by the interacting partons, zc = Ph/Pc is the momentum fraction carried are by the final observable hadron, fa/p(xa,Q^) the parton distribution functions (PDFs), and Dh/c{zc,Q2c) is the fragmentation function (FFs) for the parton of flavor c into h. The phenomenological K factor is introduced to mimic next-to-leading order (NLO) corrections. One finds that Eq.(3) tends to over-predict the curvature of the inclusive hadron spectra in the pr < 4 GeV range. This can be partially corrected via the intrinsic A^-smearing of partons associated with vacuum radiation and generalized parton distributions fa(x,kT,Q2)For the corresponding modification of the kinematics in (3) in addition to the /d 2 kj. f d2kj. {•••) see Refs. 38,39 . The generalized parton distributions d

a

r

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

133

are often approximated as -fc 2 / ( f c 2 ) 2

2

e

g(fer)= Jfc2)T

fa(x,kT,Q )*fa(x,Q )g(kT),

,

(4)

where the width (kj,) of the Gaussian is related to initial state vacuum radiation. Discussion of some qualitative features of {kj)pp is given in Refs. 39 ' 40 . For comparison to experimental data the reader is referred to Refs. 39 ' 41 ' 42 . Fig. 11 shows that this parton model approach provides a good description of the high pr data on hadron production in the range above a few GeV at all center of mass energies. In order to calculate the effects of parton energy loss on the attenuation pattern of high px partons in nuclear collisions, we must modify the free space fragmentation functions 39 ' 43 . Energy loss of the parton prior to hadronization changes the kinematic variables of the effective fragmentation function. In the first approximation, this effect can be taken into account by replacing the vacuum fragmentation functions in Eq.(3) by effective quenched ones 44 ' 45 zcD'h/c(zc,Q2c) J = c

= z'cDh/c(z'c,Q2c,)

Ph

+ NgzgDh/g(zg,Q2g) Ph

z =

pc-AEc(pc,cj>)'

9

; (5)

AEc(pc,cj>)/Ng

'

where z'c,zg are the rescaled momentum fractions. The first term is the fragmentation function is of the jet c after losing energy AEc(pc, <j>) due to medium induced gluon radiation. The second term is the feedback due to the fragmentation of the Ng(pc, ) radiated gluons. The modified fragmentation function satisfies the light-cone sum rule Jdzc zcD'h,c(zc,Ql) = 1. Eq.(5) takes into account the dependence of the energy loss on the parent parton energy and the possibly azimuthally asymmetric region of high density nuclear matter. In the above first approximation, only the average value of the energy loss is used. One can also include multigluon fluctuations 46 ' 47 of the energy loss via an energy loss distribution P(e, E) where e = ^ i Ui/E is the fractional energy loss of a jet of energy E in the rest frame of the plasma. The mean energy loss in the first approximation is related to P via deP(e,E)e

= AE/E

.

(6)

Jo The invariant hadron distribution attenuated by fluctuating energy loss

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\\J.\J •

GRV98 LO

UA2 data, s" 2 = 540 GeV •

= 1.8 GeV 2

*^ 3 xfl.O

&.£&<£; *

•:I7.^..Z...,::II/T..^Z-;

CO

Q K

NLO =

1

"

p - b a r + p - > jt° + X

2

r\ -\

8

12

16

PT [GeV] 10.0 •

GRV98 LO Q

CDF data, s1/2 = 1800 GeV

= P T

= 1.8GeV 2

* 1.0 \:

« • *

CO

13 Q K

NLO =

1

p-bar + p -> (h++h")/2 + X

5

0.1

4 6 PT [GeV]

10

Fig. 11. Comparison of the pQCD model 4 2 with intrinsic (fcf,) = 1.8 GeV 2 and GRV 98 PDFs to p+p data at -/s = 540,1800 GeV. The normalization K factor simulating higher order corrections is fit at each energy. A ± 2 5 % error band is also included.

in A + A collision is then given by d3p

= TAA(b) J2 fdxidx2 fd2kad2kb g(ka)g(kb) abed X SA(xa,Q2a)SB(Xb,Qb) da ab—*cd pi

dt

fa/A(Xa,Q2a)fb/B(Xb,Qb)

2 {z*cDh/c(z*c,Q c) / deP(e)'-z Jo c nzc

(7)

where z* = zc/(l - e). Here, TAA(V) is the Glauber binary collision profile

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

135

density at impact parameter b. It is found numerically 47 , that fluctuations tend to reduce the magnitude of the attenuation by a factor about two at RHIC. It is important to include this effect when inverting the attenuation pattern in jet tomography to determine the initial plasma density. In B + A reactions isospin effects can be accounted for a nucleus with Z protons and N neutrons by fa/A(xa,Ql) = (Z/A) fa/p(xa,Q2a) + (N/A) fa/n(xa,Ql). Nuclear modifications to the PDFs are estimated in our applications by using the shadowing function SA(xa,Q^) proposed by EKS'98 48 . 2. GLV Theory of Radiative Energy Loss In this section we turn to the theoretical problem of computing the energy loss of a fast parton penetrating a finite, expanding quark-gluon plasma. High px many body pQCD is a new frontier of research at RHIC and eventually LHC. The needed new theoretical development is the non-Abelian analogue of the radiative energy loss theory familiar from classical E&M. An important practical problem in QCD is that there exists no external beam of isolated colored quarks or gluons, and that the jets are always produced in hard processes just before the plasma is created. In addition, the nuclear medium has a small dimension compared to the jet fragmentation coherence length. Therefore, the basic formation time physics of LandauPomeranchuk-Migdal 49,50,51 (LPM) is expected to lead to strong destructive interference effects that have to be carefully taken into account. In this section we review one of those approaches 52 ' 53 ' 54,55 developed by Gyulassy-Levai-Vitev (GLV) based on an algebraic reaction operator formalism. The expansion parameter is the opacity \ — L/\ = aph of the system and the result is presented to all orders in powers of x or equivalently to all twist parton-parton correlations. Other approaches relying on asymptotic techniques include BDMS 56 ' 57 ' 58 ' 59 , Z 60 - 61 ' 62 ' 63 , and SW 6 4 , 6 5 , 6 6 , 6 7 have been reviewed elsewhere68. In the second part of this report the twist expansion approach 69,70,71 for parton energy loss in nuclear matter is reviewed. 2.1. The GW Plasma

Model

The GLV approach is built around a simple model of multiple scattering in a plasma formulated in GW 72 . However, the results of GLV 52,53 are more general since even out of thermal equilibrium the effective in-medium interactions are of finite range R ~ /x - 1 . Consider the sequential elastic scattering of a high energy (jet) parton in the random color field produced by an ensemble of m static partons located at Xj = (zi,x_u) such that

136

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z»+i > Zi and (zi+i — z,) > /x _1 , where fi is the color screening mass in the medium. As a simplified model of multiple scattering in a color neutral quark-gluon plasma, we assume a static Debye screened potential for each target parton: V;»(q)=ff(7?)c,c,-jl-jC-^

,

(8)

where T? is a dj-dimensional generator of SU(N) corresponding to the representation of the target parton at Xj. The initial and final color indices, c, c', which refer to the target parton, are averaged and summed over when computing the ensemble averaged cross sections. With V? oc T? the ensemble averaged potential vanishes everywhere, (TAa) oc TrT? = 0. However, since TrT^

= SabSii[dildA)C^

,

(9)

the diagonal mean square fluctuations and the cross sections are finite. Recall that for SU(N) the second order Casimir, C 2 i = (N2 - 1)/2JV = CF for quarks in the fundamental (di = N) representation, while Cq_u) with each component being much less than the incident energy, EQ. The assumption that the potentials are static is approximately valid in a high temperature plasma of massless quarks and gluons in the following sense: As T —> oo, the effective coupling g —* 0 (albeit very slowly). The perturbative Debye screening mass \i ~ gT limits q±£gT. The typical thermal energy ET ~ 3T of the plasma constituents is therefore large compared to fi. Consequently, the average energy loss per elastic collision, —q° « — qz w q\/2Ex oc g2T, is ~ g times smaller than the average transverse momentum transfer. Because we are interested in relatively low momentum transfer scattering (AQCD "C Q± ~ gT <^C T), the spin of the partons can be neglected. The jet parton is allowed, however, to be in an arbitrary d-dimensional representation of SU{N) with generators, Ta, satisfying TaTa = ^ l d The Born (color matrix) amplitude to scatter from an incident four momentum p^_1 to pf in the potential centered at X* is then given by Mi(pi,Pi-1)

= 2n6(P0i-p°i_1)Mqi)e-i**<

,

(10)

where q* — p , — P i - i , and A\ is shorthand for Mm)

= TaAU
.

(11)

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

137

The differential cross section averaged over initial and summed over final colors of both projectile and target partons reduces to the familiar form for low transverse momentum transfers: 47ra2 d<Ti/dq±i « C,%lrfi (12) (Qli+V2) 2\2 ' where the color factor is 1 d = ^-Tr{TaTb)Tr{T«T*) = C2C2i/dA (13) adi

For SU(3), the number 2Cj gives the usual color factors 4/9,1,9/4 for qq, qg, gg scattering respectively. In our notation, the angular distribution is given by dtTi/dSli

2.2. GLV

(14)

= ^-rr|Ai(q<)|2/(47r)2 •

Formalism

In Refs. 54,55 a systematic recursive graphical technique was developed and translated into an algebraic operator method. The goal was to compute medium induced gluon radiation amplitudes of the type shown in Fig. 12. The exponential growth of the number of graphs with the number of in-

q l5 a!

q 2 ,a 2

q 3 ,a 3

q 4 ,a 4

q 5 ,a 5

Fig. 12. Induced radiation amplitude 5 4 contributing to fifth order and higher orders in the opacity expansion of QCD energy loss in the GW model 7 2 . The crosses denote color screened Yukawa interactions on a scale fi. The blob is the initial hard jet amplitude.

teractions makes it very tedious to go beyond order three in opacity \ or twist 8 (2 + 3 x 2) from the parton-parton correlations in the medium. In

138

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GLV 52,53 the combinatorial problem is solved by summing the inclusive radiative gluon distributions recursively. The first step in the approach is to compute the three direct (single Born) and four surviving virtual (contact double Born) diagrams shown in Figs. 13 and 14 that contribute to the first order in opacity induced radiation. Detailed instructive calculation of these amplitudes is given in 52 . k,c

Mi >(U JZQ

Z

ZJ

Fig. 13. Three first order (singe Born) direct amplitudes that serve to define the D„ component of the reaction operator in Eq.(21) are derived in Refs. 5 2 , 5 3 .

For scattering of a jet off of n scattering centers located at depths z, in a homogeneous medium of large area fx2A± S> 1, we can write the inclusive radiative gluon "probability", P„(k, c), as a sum over products of partial sums of amplitudes and complementary complex conjugate amplitudes 52 ' 53 . Every term in the sum contributes to the same 0(g4n+2) in the strong coupling constant g, where g2 comes from the radiation vertex and (
Ail...in{x,'k,c)

= n

y>o,im +Si,imDm

+ S2,irnVm)Go(x,'k,c)

.

(15)

ro=l

Here Go is the initial hard {q, g} + g color matrix amplitude, x = k+/p+ is the gluon lightcone momentum fraction, k is the gluon transverse momentum, and c is its color index. In the inclusive probability each class contracts with a unique complementary class, P„(a;,k) = Ail"-in(x,k,c)Ail...in(x,'k,c).

(16)

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

139

M2,0,3 s,z0

z

z

l

k,c qJ2.a2

Qi.ai

k,c M2,o,o

k,c

^

M2, 0 ,0

q"i.ai®

^ /

q"2>a2

M:2,0,2 /$&?° f

z

>

Zl

1 •» *% o o o o u o . » P I k,c q,,a,||§jj| & q 2 .a 2

Fig. 14. Diagrams with two momentum exchanges that contribute to first and second orders in opacity. In the contact (22 = 21) limit the (double Born) virtual (contact) amplitudes define the V„ components of the reaction operator in Eq.(21) 52 > 53 . Note that two of the diagrams vanish due to zero measure JZ1 dz • • • = 0 of the radiation interval and two are topologically indistinct 5 2 ' 5 3 .

140

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Elastic (Zj)

Z2

(Z3)

I

Z4

(Z 4 )

Z3

Z2

Zj

Fig. 15. Example of graphs constructed via £>i,Vi that contribute to the 4-th order in opacity (or twist 2+8 in-medium parton-parton correlations) in elastic and inclusive inelastic final state interactions. The longitudinal depths of active scattering centers are denoted by z, and inactive (created with lj) by (zi). The form of £>i,Vi depend on the process type but the tensorial bookkeeping of partial sums of amplitudes is the same.

The complementary class is constructed to satisfy the fixed opacity power oc x " requirement as n

^- < "Or,k,c) = Gj( a : ) k ) C )II (s0,imVl + ShimDl + d2iim) .

(17)

Fig. 15 shows an example of how this formalism works at 4th order in opacity for elastic and inelastic inclusive distributions.

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

141

Detailed diagrammatic calculations that illustrate the color and kinematic modifications to the amplitudes for both direct and double Born momentum and color exchanges with the propagating jet+gluon system are given in Refs. 52 ' 53 . Direct single momentum transfer interactions enlarge rank n — 1 class elements as follows: -^'n-4ii---i„_i \X, K, C) = [dn + Jn + Bn)Ai1...in_1

= a,nAi1...in_1(x,k,c)+e

"

n

(X, K, C)

.4j1...jn_1(:E,k — q n , [c, an\)

B B e <W0 *-[ C ,a n ]T e ,(A 1 ...i B _ 1 ) ,

-(-2j

(18)

where B„ = H —C„ = k / k 2 — ( k - q „ ) / ( k —q n ) 2 ) is the so-called BertschGunion amplitude for producing a gluon with transverse momentum k in an isolated single collision with scattering center n. The momentum transfer to the jet is q n . The notation wn = (k — q n ) 2 /2u> is for a gluon with energy UJ and an is the color matrix in the <1R dimensional representation of the jet with color Casimir CR. NV = Y^!n=i ^im,2 counts the number of virtual interactions in Ai1...in_1. Tei(Ai1—in_1) is the elastic color factor associated with all n — 1 momentum transfers from the medium to the jet line. Unitarity (virtual forward scattering) corrections to the direct processes involve the sum of four surviving double Born contact diagrams in Fig. 13 that enlarge rank n — 1 classes via the action of Vn = --{CA

+ CR) - anSn - anBn = -anDn

- -{CA - CR) .

(19)

This key operator relationship between direct and virtual insertions discovered in Refs. 52,53 reduces the problem to recursive algebra. More specifically: KiAi-i„_i(i;.k.c) = =

n i)

[~2^CA

Ai'"i n _i (x,k,c)

+CR

)

~anSn

—e

-anBn\

Ai1..-in_1(x,k,c)

an^Ki1 ...in_1 [x, k

q„, [c, an\)

"^Bne^»ca|r_-11..-oi1 .

(20)

The tensorial bookkeeping of classes of diagrams makes it possible to construct the distribution of radiated gluons in the case of n interactions, Pn(x,k), recursively from lower rank (opacity) classes via a "reaction" operator Pn = Ail-i-iRnAl-in^

,

Rn = Dlbn

+ Vn + V* .

(21)

142

M. Gyvlassy et al.

Using the key identity (19), the reaction matrix simplifies to Rn = (Dn-

an)\Dn

- an) - CA= (Sn + £„)+(£„ + Bn) - CA .

The next major simplification occurs because both S and B involve the same gluon color rotation through ifcand. This fact reduces the color algebra to simple multiplicative Casimir factors in the adjoint representation ^
C A ) ^ - ^

= CA (P„-i(k - q„) - Pn-i(k)) = C^(ei«»--6-l)pn_i(k)

,

(22)

where b = iVk is the transverse momentum shift operator. We have proved in Refs.52-53 that Ail"-in-1BlBnAi1...in_1=0

.

(23)

The interference term is also found to be color trivial (containing only powers of the Casimir in the adjoint representation CA) and can be expressed in recursive form: 2Re^il-i'1-lS+5„A1...in_1 = - 2 C ^ B „ - ( R e e - i ^ z " e i q » £ I n _ i )

. (24)

Eqs.(22)-(24) present a clear proof that only the gluon mean free path which is proportional to CA (or equivalently the gluon elastic scattering cross section) appears in the induced non-Abelian bremsstrahlung formulas. The transverse vector amplitude I n above obeys a recursion relation I„ = C A ( e i ( ^ - w " ) z " e i q " - 6 - l ) l r i _ 1 - < 5 n , 1 C ^ C f l B 1 e i W 0 Z l

,

(25)

where Io = — CRRelWoZ°, and the Gunion-Bertsch amplitude can be written as •BXeiU0Zl

= - ( e *(«>o-"i)2i e iqi-b _ {\

Hetwo*i

_

(26)

For n > 1 Eq.(25) can be solved in closed form In —

CRCA

TT

fei(wo-U>m)Zmgiqm-b _

]}

H(eiW0Zl

-eibJ0Z0)

,

(27)

.m=l

where the product is understood as ordered for left to right in decreasing order in operators labeled by m. The inclusive radiation "probability" is then found to obey the soluble recursion relation P„(k) = CA{Pn-l{^ ~ q„) - Pn-l(k)) - 2CABn • ( R e e - i w " z " e i q " b I „ - i ) +

S^CACR^2

. (28)

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

143

The initial condition for this recursion relation is the initial hard vertex radiation amplitude without final state interactions that is given by PQ = CRH2 = CR/k2. The solution to the problem for any order n in mutiparticle interactions can therefore be expressed in closed form as n

I

n

I

P„(k) = - 2 C * C 2 R e ] T \ \{ ( e ^ 6 - 1) \ ® B i »=1 [j=i+l J

e *q>-bg-»wozi

i-l

x I Yl (ei{u"'-u'm)Zmei^-b

- 1) V
.

(29)

This expression is very general and is suitable for any distribution of interaction centers as well as any distribution of 2, dependent transverse momenta exchanges that can arise due to expansion of the medium. Therefore, it can be directly applied to realistic expanding plasmas where the distance between adjacent centers (the mean free path) varies along the medium as does the local screening scale, fi(z). This form is also well suited for possible future Monte Carlo implementation for arbitrary q^, Zi medium ensemble averages. The final complete solution to the inclusive induced gluon radiation valid to all orders in opacity can be expressed in terms of the following infinite series: dN"

CRas ^lfj([°° n=l

\d2ag(Zi)

f2 \Jzi-i

i=l

J

L

M

/

2

*

( X Pn{zi,---

,Zn)

X

^

COS

w

I - 2 C ( i | . . . i „ ) • 2_^ ^{m+l,\ m=l

,n)(m,-

,n)

(fc,--' ,n)

Azfcj-cos _

^W( f c i ... i n )Az f e j

j

, (30)

(k - q,, - qi2 - •


K

Bj = H — Cj

**(hi2---i-m.)(jlJ2---in)

(k-q„ W

{ m , •••,«)

qj 2xE

(

=

^(il

*2 "-jm)

~

^(jlj2

•jn)

2

(31)

144

M. Gyulassy et al.

The infinite opacity series can be understood as a sum over all twist partonparton correlations, {• • • (2nFF) • ••), in the nuclear matter. We emphasize that Eq.(30) is not restricted to uncorrelated geometries as in Refs. 5 6 - 6 6 . It also allows the inclusion of finite kinematic boundaries on the q* as well as different functional forms and elastic cross sections
= n\p% 0(L-zn)O(zn-zn-i)---6{z2-zi)

,

(32)

where the mean density is po = NS/LA±, leads to an oscillatory pattern that is an artifact of the assumed sharp edges. Here A± is the transverse area of the interaction region and Ns is the total number of scattering centers. A somewhat more realistic model may be an exponential longitudinal distribution of scattering center separations

This converts the oscillating formation phase factors in Eq.(30) into Lorentzian factors f ( m \ / dpncos ]Tw(fc,...,n)A;zfc

m Nn = —^-Re TT ——

1 —— .

In order to fix Le(n), we can impose (zk — ZQ) = kL/(n constrains 52 ' 53,54 Le(n) = L/(n+l). 2.3. First Order Non-Abelian

Energy

(34)

+ 1). This

Loss

The simplest and fortunately dominant (a posteriori) application 52 ' 53 of our general solution to the energy loss problem was to calculate numerically the total radiated energy loss as a function of jet energy E, plasma depth

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

145

L, and the typical transverse momentum transfer /x. In the absence of a medium, the gluon radiation associated with the parton jet (in the small x approximation, where we do not distinguish between quark and gluon parents) is distributed as dNi0)

Cn**

x

1 f35)

where x = k+ /E+ « w/E, and CR is the Casimir of the jet in the CIR dimensional color representation. The differential energy distribution outside a cone defined by k 2 > p? is given by ax

fj,

n

where the upper kinematic limit is k j ^ = min [4E2x2,4E2x(l The energy loss outside the cone in the vacuum is then given by A #

=

^ i £ i

o g

! .

— x)] .

(37)

Beyond the small x approximation, the splitting functions Pgq(x) and Pgg(x) can give rise to some small corrections. While Eq.(37) overestimates the radiative energy loss in the vacuum (self-quenching), it is important to note that AE^ /E ~ 50% is typically much larger than the medium induced energy loss. However, the vacuum energy loss is included in the DGLAP evolution of the fragmentation functions D^/c(z,Q2). To compute the medium induced radiation we focus on the density of the Ns scattering centers given by Eq.(33). Averaging over the momentum transfer qi via the color Yukawa potential leads to a very simple first order opacity result for the i « l gluon double differential distribution dN™ dxdk2

dN^ L f>™* ,2 Me/7 H1 drr.dk2 Xg J0 7r(q2 + /x2)2 2k-qi(k-qi)2L2 " 16a: 2 £ 2 + ( k - q 1 ) 4 L 2 '

(38)

where the opacity factor L/Xg = Nsa^'/A±_ arises from the sum over the Ns distinct targets. Note that the radiated gluon mean free path Ag = (CA/CR)\ appears rather than the jet mean free path. The upper kinematic bound on the momentum transfer is q^^ = s/4 ~ 3Efi and 1/Me// — l//i 2 — l/(/x 2 + ^max). For SPS and RHIC energies this finite limit cannot be ignored. Numerical results comparing the first three orders in opacity corrections to the hard distribution Eq.(35) were presented in Refs. 52 ' 53

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M. Gyulassy et al.

and the opacity series was shown to converge fast is dominated by its first order term for realistic nuclear opacities. Fig. 16 illustrates the convergence properties of Eq.(30) in the example of the mean energy loss AE, as well as the quadratic dependence of AE on the size of the nuclear matter for static media. 100.00

: />

10.00

AFH>

V^JCT^

0 — - 0 AE"*2'

-

1

2 3

J. AE '* * '

i ^

1.00

-

3

c

%

-

SPS LHC: EJa/\l = 1000

0.10 r :

RHIC; Eja/n = 100

Q ^

SPS: EtJ\l = 10 1

1

0.01

1

L/X s Fig. 16. The radiative energy loss of a quark jet with energy Ejet = 5, 50, 500 GeV (at SPS, RHIC, LHC) is plotted as a function of the opacity L/\g (A9 = 1 fm, \i = 0.5 GeV) for a static medium from Ref. 52,53 . Solid curves show the first order in opacity results. The dashed curves show results up to second order in opacity, and two third order results are shown by solid triangles for SPS energies.

In Ref.73 we considered various asymptotic limits of the first order energy loss in 1+1D and 1+3D expanding plasmas. In the case that the mean density decreases as p(z = T)=PQ

(-^J

(39)

due to a longitudinal Bjorken expansion, it is possible to obtain a closed analytic formula36 under the strong assumption that no kinematic bounds need to be considered. For a hard jet penetrating the quark-gluon plasma, dAE^ dx

_

2CRa6 E n

f

J To

dr f(Z(x,r)), A(r)

(40)

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147

where x ~ u/E is the momentum fraction of the radiated gluon and the formation physics function f(Z(x,r)) is defined36 to be

Kx>T) = f°° TTTTTT I 1 Jo

u l

\

i

u

cos

< u Z ^ T ))l •

( 41 >

)

With Z(X,T) = (T — TQ)II2{T)/2XE as the local formation physics parameter, two simple analytic limits of Eq.(41) can be obtained. For x » xc = M W V L1—?/2E = Ln2{L)/2E, in which case the formation length is large compared to the size of the medium, the small Z(X,T) limit applies, leading to f(Z) « nZ/2. The interference pattern along the gluon path becomes important and accounts for the the non-trivial dependence of the energy loss on L. When x -C xc, i.e. the formation length is small compared to the plasma thickness, one gets f(Z) ~ logZ. In the x~5> xc limit 36 , the radiative spectrum Eq.(40) becomes d*E£jXe dx In the opposite

I < I

C

^ CRas M (ro) 2 r 0 "L 2 -" 1 ~ 2(2 - a) A(r0) x '

{

'

limit we have

(43)

- & — w=^) ~ ^ r s — ^ E —

and the intensity spectrum has an integrable divergence at x = 0. Higher orders in opacity do change the x-dependence of the radiative spectrum relative to Eqs.(42) and (43), but this change is small 52,53 for L ~ RA — 5fm. The mean energy loss (to first order in x) integrates to A£(D - C»a° M(ro) 2 r 0 "L 2 - / "2(2-a) A(r 0 ) {**

K7t))2T*

2_E »-*

+

\ ) '

^

The logarithmic enhancement with energy comes from the xc < x < 1 region 52,53 . In the case of sufficiently large jet energies (E —> oo) this term dominates. For parton energies < 20 GeV, however, corrections to this leading log l/xc expression that can be exactly evaluated numerically from the GLV expression and are found to be comparable in size. The effective AE/E in this energy range is approximately constant. In order to study non-central collisions and azimuthal anisotropy we also solved the analytically tractable case of a sharp expanding elliptic cylinder. We approximate the assumed <j>o independent screening /x(r) ss gT(r) = 2( / o(r)/2) 1 / 3 since g ~ 2 and p = ( 1 6 C ( 3 ) / T T 2 ) T 3 ~ 2T 3 for a gluon plasma.

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We define T(4>O) as the escape time for the jet to reach the expanding elliptic surface from an initial point Xo = (xo,yo) in the azimuthal direction (po: (xp +TCQSQ)2

(y0 + T sin 0)2

2

(RX

+ VXT)

=

VyT)2

(Ry+

l

'

'

We take uj((f>o) = 2 T ( ^ ) (P(T((J)O))/2)2/3 to estimate an upper bound on the logarithmic enhancement factor. Performing the remaining integrals one gets: „3

4

dy

w((/>o) 7 0

ai1_

(xo + Tcos(j>o) (RX

4 RxRy

Rx + WxT -Ry + VyT 2

dy

2

(yo+Tsincpo)2

+ VXT)

(Ry +

ax — ay

w(o)

VyT)2

where ax = vx/Rx,ay = vy/Ry. This expression provides a simple analytic generalization that interpolates between a pure Bjorken 1+1D expansion for small ax,yr, and a 3+1D expansion at large aXtVT. In the special case of a pure Bjorken (longitudinal) expansion with vx — Vy = 0,

9CRa3sdN9 ... , E AC,(i),, , AE(Bj (0o = j ^ - ^ -7- T{4>) log — T - T . •> iRxRy dy w(0o)

,„„, 47

In this case, the energy loss depends linearly on r().

2.4. Centrality Effect at

and Rapidity RHIC

Dependence

of the

Cronin

In the subsections that follow we will describe some of the important applications/predictions of the GLV reaction operator formalism to initial and final state multiparton interactions. We emphasize that these calculations can provide complementary information about the properties of cold and hot nuclear matter such as the transport coefficients ^ 2 /A q , /i 2 /A 9 and the effective initial gluon number density and energy density created in relativistic heavy ion reactions. In Refs. 40 ' 74 the explicit solution for the transverse momentum distribution of partons that have traversed cold nuclear matter has been found

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

149

using the GLV approach: 00

d3Nf(khk±) dk\\d2k.i

vn

n=0 "" ^

r

n

1

i=l

dael(R,T)

(Tel

g - q i X - V k i . (g) e -9*ll a *||

d2<\i J_

_

X

This leads to nuclear induced broadening (A/c^ )PA = (J?x£ f° r opacity X = (L)/X and typical transverse momentum squared /i 2 . Beyond the naive Gaussian approximation harder fluctuations along the projectile path lead to a logarithmic enhancement (Afc^ )PA, i.e. £ = ln(l+cp?r). The question of the effective longitudinal momentum shift associated with this broadening has also been addressed 40 ' 74 :

Afc =

(49)

- » ^4-

To implement initial state elastic and radiative energy loss we focus on the large Q2 ~ p?r partonic subprocess a + b —> c + d, where ka, kb are the initial momenta involved in the hard part da/dt of Eq.(3). If partons a and b have lost fractions ea ( a = a,b) of their longitudinal momenta according to a probability distributions P Q (e), then ka = ka/l — eQ and fa/P(xa,Q2)

-> / deaPa(ea)fa/P

(xa = — — , Q 2 J 0(xa < 1) ,

(50)

at asymptotic t = — oo . Eq.(50) provides a simple modification to the factorized pQCD hadron production formalism. For bremsstrahlung processes, P Q (e) are sensitive to multiple gluon emission47. For the simpler case where one considers only the mean energy loss, P(e) = S(e — (Ako)/ko). More specifically, for the elastic longitudinal shift that we consider here, Pa(e) = s(e- »2Xat/2k2a „) and fa/P(Xa,

Q2) -

fa/p (Xa + ^

1

1, Q2\

6(xa

< 1) .

(51)

The observable effects of Eqs.(50) and (51) can be very different for valence quarks, sea quarks, and gluons due to the different z-dependence of the PDFs. A natural application of these results is in the quantitative description of the Cronin effect. Perturbative calculations that include initial state parton broadening remain its only successful explanation to date 3 9 ' 4 0 ' 7 5 , 7 6 ' 7 7 ' 7 8 . Comparison to low energy data on proton-nucleus collisions 40 ' 75 allows

M. Gyulassy et al.

1.4

1

™

1

1 —

— -

0.5{re%*'), EKS'98 shadowing

—

0.5(n*+n"), no shadowing

Y=0

mm

•*

1.1

1.5 a.

n

1.4

•o

< 1.3 < 1? «j 1.1 K> •a

•a

1.0

|

"

I

i

1

1

i

•

' - - - • ,

-

i: 1 • 1 i 1 i I i 1 i 1 i

1.6

-

mm

1

j

-

•

i

i

i

i

1

M

All panels:

Z3

-'-*.-._ —

1

perfect binary scaling

1

0.9

•

1

Y = +3

0.5
••

--.

•

0.5(n* +TC"), EKS'98 shadowing

• ^ . j . , . , ! ^

~ •

1

i

1.0

1

•_

! 1

r_

1.2

1

Total invariant cross section = p p + p A

• .

1.3 -

—1

1 All panels:

1 , 1 ,

1.5

, ! , , , , ! ,

150

1

1

1.2

ft 1.1 o T) 1 I)

<< 0.9 < T3

•

— -

0.5(ir «i"). EKS'9S shadowing

^—

0.5(n*+n'), no shadowing

Y = -3

.J

. _ ..

r -

•

^

3

S

0 T3

OH 0./

0.6

——1 Upper band: no initial state E-loss

4 pT[GeV]

Fig. 17. Rapidity dependence of the Cronin effect in d + Au reactions at y/sNN = 200 GeV with and without antishadowing or EMC effect from Ref. 40 . The result of switching off elastic energy loss is also shown via the upper bands for y = ± 3 .

us to extract the transport coefficients of cold nuclear matter, p?/Xq = 0.06 GeV 2 /fm and p?j\g = 0.14 GeV 2 /fm. These estimates include the effective elastic initial state energy loss and are ~ 25% bigger relative to the case when the longitudinal momentum shifts have been neglected 75 ' 79 . The

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

151

transport coefficients of cold nuclear matter change only slightly 40 with y/s and can be used to predict the centrality and rapidity dependence of the Cronin effect in d + Au reactions at RHIC. Fig. 17 shows the computed rapidity dependence of the Cronin effect in d + Au reactions at RHIC. At midrapidity, y = 0, in the computed transverse momentum range, we find slight Cronin enhancement RdAu max = 1-2 relative to the binary collision scaled p+p result even with initial state elastic energy loss. Including strong antishadowing (shown with dashed lines) leads to RdAu max = 1.3. The effect of antishadowing becomes increasingly important at higher pr- Experiments at RHIC can thus help constrain the poorly known nuclear modification to the PDFs for gluons. At forward y = +3 rapidity (in the direction of the deuteron beam) the Cronin intercept {RdAu = 1) and maximum RdAu max are both shifted to slightly higher PT- The most distinct prediction at forward rapidities is the significantly larger Cronin enhancement region extending to high px- This is understood in terms of the softening of the hadron spectra away from midrapidity, as predicted by perturbative QCD. Steeper spectra tend to enhance the effect of otherwise identical transverse momentum kicks. At backward y = — 3 rapidity there is no enhancement since the nucleus does not scatter multiply on the deuteron. This region is shown to be sensitive to the initial state energy loss 40 . For further details on the systematic understanding of the Cronin effect versus i/s the reader is referred to Refs. 75 ' 76 . Recently it has been suggested by models based on gluon saturation, final state hadron absorption, and coherent nucleon scattering, that hadron cross sections in d + Au reactions will be suppressed by about 30% for charged hadrons relative to the binary collision scaled p + p result. Experimental data at RHIC will make possible a critical test of the validity of different theoretical models.

2.5. Jet Tomography of d + Au and Au + Au at SPS, RHIC, and LHC Some of the most important applications 47 ' 75 of the GLV results on medium induced gluon bremsstrahlung are related to jet tomography, the study of the properties of matter through the attenuation pattern of fast particles that propagate and lose energy as a result of multiple elastic and inelastic scatterings. Under the assumption of local thermal equilibrium all relevant scales in the problem, such as the Debye screening scale fi, can be related to dN9 /dy - the initial effective gluon rapidity density of the bulk soft

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background matter (the quark-gluon plasma).

10

n r r . dN g /dy=200-350

SPS

• WA98 7t°(17.4AGeV) n^r.

dN°/dy=800-1200

• PHENIX n (130 AGeV) rn^r. dN°/dy=2000-3500 T PHENIX n (200 AGeV)

1

* STAR h* (200 AGeV)

T^do"

Q.

DC

RHIC

0.1

Preliminary PHENIX and STAR data at s =200 AGeV Au+Au at s 1/2 =17, 2 0 0 , 5 5 0 0 A G e V

U.U I

i

2

i

i

i i i111

i

10 PT [GeV]

i

i

i i i 111

100

Fig. 18. The suppression/enhancement ratio RAA{VT) (A = B — Au) for neutral pions at i/sNN = 17, 200, 5500 GeV from Ref. 75 . Solid (dashed) lines correspond to the smaller (larger) effective initial gluon rapidity densities at given ^/s that drive parton energy loss. Data on ir 0 production in central Pb + Pb at \/sNN = 17.4 GeV from WA98 2 4 and in central Au + Au at \fsNN = 130 GeV, as well as preliminary data at 200 GeV from PHENIX 3 ' 1 and h± central/peripheral data from STAR 5 are shown. The sum of estimated statistical and systematic errors is indicated.

Our main results for central Au + Au reactions, which includes three important nuclear effects, Cronin+Shadowing+Quenching, are presented in Fig. 18. Jet tomography consists of determining the effective initial gluon rapidity density dN9 /dy that best reproduces the quenching pattern of the

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

153

data 1 ' 3 , 5 ' 2 4 . At SPS the large Cronin enhancement 75 is reduced by a factor of two with dN9/dy = 350 but the data are more consistent with a smaller gluon density < 200. Unfortunately at this low energy the results are very sensitive to the details of the model. At RHIC, for px > 2 GeV jet quenching dominates, but surprisingly the rate of variation with PT of the Cronin enhancement and jet quenching conspire to yield an approximately constant suppression pattern with magnitude dependent only on the initial dNs/dy. At higher px > 20 GeV the softening of the initial jet spectra due to the EMC modification of the PDFs compensates for the reduced energy loss. This unexpected interplay among the three nuclear effects at RHIC was our main prediction in Ref.75. Preliminary y/s = 200 AGeV PHENIX and STAR results included in Fig. 18 support the predicted magnitude and approximate pr dependence of the suppression. At LHC energies the much larger gluon densities dN9/dy ~ 2000 — 3500 are expected to lead to a dramatic variation of quenching with PT, as shown. In nuclear media of high opacity the mean fractional energy loss (AE)/E of moderately hard partons can become of the order of unity. For LHC this may be reflected in the pr < 10 GeV region through deviations from the extrapolated highPT suppression trend. Hadronic fragments coming from energetic jets would tend to compensate the rapidly increasing quenching with decreasing transverse momentum (seen in Fig. 18) and may restore the hydrodynamic-like participant scaling in the soft regime.

2.6. Enhanced

Baryon/Meson

Ratios

One of the unexpected results reported during the first year RHIC run at ^NN = 130 GeV was that, in contrast to the strong 7r° quenching for 2 GeV < pr < 5 GeV, the corresponding charged hadrons were found to be suppressed by only a factor ~ 2 — 2.5. Even more surprisingly, the identified particle spectra analysis suggests that RB(PT) = P/^~, p/ir+ > 1 for PT > 2 GeV. Thus, baryon and antibaryon production may in fact dominate the moderate to highpr hadron flavor yields 80 ' 81,82 ' 83 . These and other data point to a possible novel baryon transport dynamics nucleus-nucleus reactions. More recent results corroborate the non-perturbative baryon production hypothesis through equally abundant A and A production 84,85 . It has also been observed that the mean transverse momentum ( p r ) s for various baryon and antibaryon species is approximately constant and deviates from the common hydrodynamic flow systematics of soft hadron production in A + A collisions. Identified particle analyses from the recent

154

M. Gyulassy et al.

y/sNN = 200 GeV RHIC run find similar puzzling features of moderate-px baryon spectra 3,4,5 . In Refs. 86,87,88 the GLV jet energy loss 52,53 was combined with a topological non-perturbative baryon production and transport mechanism 8 9 - 9 3 to gain insight into the anomalous anti+baryon behavior at RHIC. Phenomenological applications are currently based on Regge theory where a Regge trajectory J = a(0) + a'(0)M 2 is specified by its intercept a(0) and slope a'(0). It has been argued 91,92 that aj(0) ~ 0.5 and a'j(0) ~ l / 3 a ^ ( 0 ) . Regge theory gives exponential rapidity correlations, which in the presence of two sources (at ±Ymax) lead to net baryon rapidity density in central A + A collisions of the form:

H£± = (Z + W)(l - ^(PM.y-?.{?»» -

(52)

dy sinh(l-aj(0))Ymax It is evident from Eq.(52) that the net baryon distribution integrates to 2A and in going to peripheral reactions scales as Npart. At RHIC energies of •^/s = 130(200) GeV, corresponding to Ymax = 4.8(5.4), in central reactions dNB'B/dy = d{p-p)/dy + d(n-n)/dy + d(A-A)/dy + -- • ~ 18(13.5). The relative contribution of each baryon species can be evaluated from isospin symmetry and strangeness conservation via comparison to midrapidity kaon production. The high pr part of the hadron spectra is computed from Eq.(7). Hadronic transport in small-to-moderate pr is effectively controlled by the slope of the Regge trajectory. This would suggest that the baryon/meson mean inverse slopes in a phenomenological p^-exponential (~ e~ PT//T ) soft particle production model are related as TB '• TM — V3 : V2. Soft pion production, however, is largely dominated by resonance decays, where the cumulative effect from the random walk in pr due to string breaking is destroyed. This leads to the relation (PT)TT : (PT)K '• {PT)B — 1 : (l^-V^) : \/3 (220 MeV : 275 MeV : 380 MeV) 86,87 . One also notes that in the limit of pair production dominated by junction-antijunction loops (which we consider here) the transverse momentum distribution of antibaryons closely resembles that of baryons (with (PT)B = {PT)B)We find an enhanced baryon/meson ratio RB(PT) > 1 that decreases with centrality in a finite px window as illustrated in Fig. 19. For pr > 5 GeV the ratio is reduced below unity and approaches the perturbative calculation. The centrality dependence of RB(PT) is largely driven by the non-Abelian energy loss that suppresses perturbative pion production for PT > 2 GeV. It is important to note that an enhanced pj-K ratio is also

Jet Quenching

2.5 2.0

•

i

and Radiative

|

r— • ,

i

Energy Loss in Dense Nuclear Matter

|

r- •

|

. Au+AuSs'^ISOAGeV bar/it" . P ^^^

•

1

*—* *—* H—h & o •

T - "

'

1

'

•

1

155

'

0-10% Central 2 _30% 0 Central 70-80% Central Boosted Th. vT=0.6 PHENIX 0-5% Central

•

1.5 a. Sf

1.0 0.5 -K.^

tip 5 ''' 0-0 ^ _ _ _ _ _ _ _ 0 1 2 3 •

*^j*-—i

i ,

4 5 pT[GeV]

6

7

m s i

,

8

Fig. 19. The p / 7 r _ ratio versus px for 3 centrality classes at y/sNN = 130 GeV from Refs. 8 6 , 8 7 . ATpart(solid) versus iV(,i n (dashed) geometry is shown. Boosted thermal source computation and ratio of fits to PHENIX data are shown for comparison.

observed in peripheral reactions and is consistent with measurements in p+p. In Fig. 20 baryon enhancement is reflected in the different suppression factor RAA{PT) for neutral pions and inclusive charged hadrons. At high transverse momenta the suppression ratio is expected to be similar and driven by fragmentation of quenched jets. The rate at which this common RAA is approached depends on the fragmentation functions into baryons that are currently poorly constrained or unknown. 2.7. High-pT

Azimuthal

Asymmetry

A new way to probe the energy loss AE in variable geometries was recently proposed in Ref.35'36. The idea was to exploit the spatial azimuthal asymmetry of non-central nuclear collisions. The dependence of AE on the path length L{4>) naturally results in a pattern of azimuthal asymmetry of high pi hadrons which can be measured via the differential elliptic flow parameter (second Fourier coefficient), W2(PT) : V2(PT)

= (cos 2(f))

+

In Ref.36 we predicted

Jo" Jo V2{PT)

h d< _dN_ P\dy px dp-r d

(53)

for two models of initial conditions 25

156

M. Gyulassy et al.

2.00 1.75

i

.

•

s" =130AGeV

*—xGVW,

1.50 ^

i

i

• PHENIX0-10%7t°

2

dN7dy=750±100

• PHENIX0-10%0.5(h*+h") * — * G V W , dN°/dy=750±100

1-25

" '_

1

T^cr * Binary scaling

&

1.00

DC 0.75

---Z4L-

. ^

,

|

0.50 0.25 0.00

I

I

.

I

.

3 pT[GeV]

1

Fig. 20. Suppression of neutral pions and inclusive charged hadrons relative to the binary collision scaled p + p result from Refs. 8 6 ' 8 7 . Central Au + Au at y/sNN = 130 GeV.

(specified by dN3/dy) which differ by almost an order of magnitude. The longitudinal expansion of the plasma and the corresponding mean energy loss were given by Eqs.(39),(44) with a = 1. A novel element of the analysis was the discussion of the interplay between the azimuthally asymmetric soft (hydrodynamic) and hard (quenched jet) components of the final hadron distributions. In non-central A + B reactions the low pr hadrons are also expected to exhibit azimuthal asymmetry caused by the hydrodynamic flow31'94. We therefore modeled the soft component with the following Ansatz: dNs{b) dyd2pT

dns dy

pT T e- / ° ^ T (1 + 2i; 2s (p T ) cos(20) + • • •)

(54)

Here we took To « 0.25 GeV and incorporate the hydrodynamic elliptic flow predicted in Ref.31 and found to grow monotonically with pr as V2S(PT) « tanh(pr/(10 ± 2 GeV)) 31

(55)

The hydrodynamic elliptic flow was found to be less sensitive to the initial conditions than the high pr jet quenching studied in Ref.36. We showed that the combined pattern of jet quenching in the single inclusive spectra and the differential elliptic flow at high pr provide comple-

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

157

mentary tools 36 that can determine the effective density of gluons created in the early stages of relativistic heavy ion reactions.

0.30 0.25

1

'

1

'

1

•

1

1

1

'

1

r

• — • Hydro+GLV quench., dN9/dy=1000 . • — • Hydro+GLV quench., dN9/dy=500 . • — » Hydro+GLV quench., dN9/dy=200

0.20

3 °-15 CM

>

0.10 0.05 0.00

J

0

1

I

I

1 2

I

I

I

I

3 pT[GeV]

I

4

I

I

5

L

6

Fig. 21. The interpolation of V2(PT) between the soft hydrodynamic 3 1 , 9 4 and hard pQCD regimes is shown for 6 = 7 fm adapted from Ref. 36 . Solid (dashed) curves correspond to sharp cylindrical (diffuse Woods-Saxon) geometries.

Fig. 21 shows the predicted pattern of high-p^ anisotropy. Note the difference between sharp cylinder and diffuse Woods-Saxon geometries at 6 = 7 fm approximating roughly 20-30% central events. While the central (b = 0) inclusive quenching is insensitive to the density profile, non-central events clearly exhibit large sensitivity to the actual distribution. In particular, the sharp elliptic geometry also simulates one of the possible scenarios of large energy loss and predominant surface emission95. We conclude that V2(PT > 2 GeV,b) provides essential complementary information about the geometry and impact parameter dependence of the initial conditions in A + A reactions and the magnitude of jet quenching. In particular, the rate at which the i>2 coefficient decreases at high px is an indicator of the diffuseness of that geometry. Minimum bias STAR data at RHIC 8 now confirm the predicted saturation of v2 {pr)• For pr > 6 GeV data is still inconclusive due to the large error bars and can still accommodate different scenarios: constant versus slightly decreasing V2(PT)- High statistics measurements in the future RHIC runs can help to experimentally resolve this question

158

M. Gyulassy et al.

and further test the predicted 35 ' 36 slow decrease of v-x at large transverse momenta. In Refs. 96 ' 97 , in an independent analysis the GLV radiative energy loss 52 ' 53 of fast partons was coupled to a dynamically evolving soft hydrodynamic background to obtain a quantitative description of the disappearance of the back-to-back jet correlations 6 . 3. The W W Approach: Parton Energy Loss with Detailed Balance The above studies of parton energy loss have concentrated on gluon radiation induced by multiple scattering in a medium. Since gluons are bosons, there should also be stimulated gluon emission and absorption by the propagating parton because of the presence of thermal gluons in the hot medium. Such detailed balance is crucial for parton thermalization and should also be important for calculating the energy loss of an energetic parton in a hot medium 98 . 3.1. Final-state

Absorption

Let us assume that the hot medium is in thermal equilibrium shortly after the production of the hard parton. One should then take into account of both stimulated emission and thermal absorption in the final state radiation of a jet in a thermal medium with finite temperature T. One has then the probability of gluon radiation with energy u, dP(°) dw

asCF'F f dz f dk2., rAT . _,.., N 2n J ~ J -^r[ 9(^)S(u

„.

+ zE)

+ (1 + Ng(zE)) 5{w - zE)6(l - z) i > ( | ) ,

(56)

where, JVg(|k|) = l/[exp(|k|/T) — 1] is the thermal gluon distribution. We have define the splitting function Pgq(z) = P(z)/z = [1 + (1 — z)2)/z for q —> gq. The first term is from thermal absorption and the second term from gluon emission with the Bose-Einstein enhancement factor. For E 3> T, one can neglect the quantum statistical effect for the leading parton. Note that the vacuum part has a logarithmic infrared divergence while the finitetemperature part has a linear divergence, since JVg(|k|) ~ T/|k| as |k| —* 0. These infrared divergences will be canceled by the virtual corrections which also contain a zero-temperature and a finite-temperature part. In addition, the virtual corrections are also essential to ensure unitarity and momentum

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

159

conservation in the QCD evolution of the fragmentation functions. However, they do not contribute to the effective parton energy loss. The remaining collinear divergence in the above spectrum can be absorbed into a renormalized fragmentation function that follows the QCD evolution equations. Subtracting the gluon radiation spectrum in the vacuum, one then obtains the energy loss due to final-state absorption and stimulated emission,

(0)

AE. abs

I

du>u)

asCF

(dP^ \ dui

„

dPW du>

f _,_ f^max

T=O ^

ki +P(z)Ng(zE)e(l

P(-z)Ng(zE) (57)

- z)

Even though the stimulated emission cancels part of the contribution from absorption, the net medium effect without rescattering is still dominated by the final-state thermal absorption, resulting in a net energy gain, i.e. a negative energy loss. For asymptotically large parton energy, E ;» T, one can complete the above integration approximately and have,

(0)

AE abs E

irasCF T2 In —;r- + 2 3 E2

7E +

6C'(2)

(58)

where, 7 E W 0.5772 and £'(2) « —0.9376. The quadratic temperature dependence of the leading contribution is a direct consequence of the partial cancellation between stimulated emission and thermal absorption, each having a leading contribution linear in T.

3.2. Induced

Absorption

As in the case of final-state absorption, one can also include stimulated emission and thermal absorption when calculating the induced radiation

160

M. Gyulassy et al.

probability at the first order in opacity, dPW dw

fdz fd2^ y d V 2 v {
C2 8-Kd,Ad,R

J TJ w J W

N j - (Tr [|i?(5)|2 + IRe (RWRM)]) (1 + Ng(zE)) 6(LJ - zE)9(l -z) + Ng(zE)5(w + zE) asC2CFCA

f dk\ f d2q±

fdz

2 N D

,w

J 7 JV J W" ^^

dAit

kx qx N (Re(l (kx - q x ) 2 ^

- e^ 1 " 1 0 )) \N9(ZE)5(LJ

+ (l + AT 3 (z£O)<5(w-z£)0(l

•

*

+ zE)

(59)

)

The factor 1 — exp(iu>iyio) reflects the destructive interference arising from the non-Abelian LPM effect. The target distribution is assumed to be an exponential form p(y) = 2exp(—2y/L)/L. As in the final-state absorption, the contribution from thermal absorption associated with rescattering is larger than that of stimulated emission, resulting in a net energy gain. However, the zero-temperature contribution corresponds to the radiation induced by rescattering which will lead to an effective energy loss by the leading parton. This is the energy loss obtained the previous sections and we denote this part as A£y a d . The remainder or temperature-dependent part of energy loss induced by rescattering at the first order in opacity is then defined as AE

fJ

I

abs

(dpM du

dpw du

dcjcj

(60) T=o

which mainly comes from thermal absorption with partial cancellation by stimulated emission in the medium. According to Eq.(59), AJB:(1)

7T A„

E

Wf/

dVW
Mf/

A±Nqx)|2

i0J

kx qx (kx - qx) 2

w

x P(z) (Re(l - e ™ )) 9(1 - z); A

(i) ^

=

oticLL_ 7T

x Ng{zE)

Xg

P(z) (Re{l -

- P(-z) (Re(l - eiunvw))

iu;iVl0

))

e

.

(61) kx qx (kx - qx) 2

6(1 - z) (62)

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

161

In the limit q±max —> oo, the angular integral can be carried out by partial integration. These contributions to the energy loss become AE, rod

A^l

CXSCF L 2TT

asCF 2-K

where, 7 = fi2L/(4zE)

Ki)

A,

-E [ dzP(z)h(i)9(l J

- z) \

L E / dzNg(zE)h(j) Xa

(63)

P{z)6(l -z)-

P{-z)

, (64)

and

ivSy[f-arcsinN]

7


^=Tln[27+V4?^T],7>l/2.

(65)

One can approximate h{^y) with 7T7 + (11/4 — 27r)72 + (5/2)7 3 for 7 < 1/2 and ln(47) + O.I/7 + O.O28/72 for 7 > 1/2. In the limit of EL » 1 and E 3> /i, One can then get the approximate asymptotic behavior of the energy loss, raa

OLaCF\l2L2 r 2

4XgE AE. abs E

9 C1

TTOCSCF LT2

3

(66)

In - y - - 0.048 \x*L

Ag£

2

2

In

M

L

1 + 7E-

6C'(2)

(67)

Again, the thermal absorption results in an energy gain ( or negative energy loss). This result is accurate through the order of 1/E. In Eq.(67), we have assumed p?LjT 3> 1 and kept only the first two leading terms. In this limit, the average formation time for stimulated emission or thermal absorption is much smaller than the total propagation length. Therefore, the energy gain, A-E^fcs, by thermal absorption (with partial cancellation by the stimulated emission) is linear in L, as compared to the quadratic dependence in the zero-temperature case. However, the logarithmic dependence on p,2L/T, as compared to the factor ln(4£T/p, 2 ) in Eq.(58) for no rescattering, is still a consequence of the LPM interference in medium. A quadratic L-dependence of AE^bl will arise when p?L/T < 1. To evaluate the effect of thermal absorption numerically, we assume the Debye screening mass to be fx2 = 4naaT2 from the perturbative QCD at finite temperature". The mean-free-path for a gluon Ag in the GW model

L-l

{(TqgPq) + (
2™L

„

J3

— ^ 9 x 7 C (/ 30 ) ^ , V M'

(68)

162

M. Gyvlassy et al.

Fig. 22.

The ratio of effective parton energy loss with (AE = AE^b\ + A - E ^ +

and without (AE^.Jd) absorption with (AE^bs)

absorption as a function of E/fi. and without (AE^bs)

AE^)

Inserted box: energy gain via

rescattering.

where £(3) « 1.202. With fixed values of L/Xg and as, AE/fx should be a function of E//J, only. Shown in Fig. 22 are ratios of the calculated radiative energy loss with and without stimulated emission and thermal absorption as functions of Ej\x for L/Xg = 3,5 and a$ = 0.3. Shown in the inserted box are the energy gain via gluon absorption with ( A . E ^ ) and without ( A £ ^ ) rescattering. For partons with very high energy the effect of the gluon absorption is small and can be neglected. However, the thermal absorption reduces the effective parton energy loss by about 30-10% for intermediate values of parton energy. This will increase the energy dependence of the effective parton energy loss in the intermediate energy region. The observed flat or slightly pr dependence of the hadron spectra suppression indicates a strong energy dependence of the parton energy loss. This might indicate the effect of the thermal absorption, in particular in the intermediate pr region.

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

163

4. The WOGZ Approach: Partem Energy Loss and Modified Fragmentation Functions in Nuclei So far we have reviewed the parton energy loss of an energetic parton induced by multiple scattering in a hot QCD medium. The effective parton energy loss is shown to be proportional to the gluon density. Therefore measurements of the parton energy loss will enable one to extract the initial gluon density of the produced hot medium within our assumed screened potential model for multiple parton scattering in a dense gluonic medium. Such an exercise will be more robust if one can also measure the parton energy loss and the gluon density inside a cold nuclear medium. For this purpose, one has to rely on other complementary experimental measurements such as parton energy loss in deeply inelastic scattering (DIS) off nuclear targets. One can then study parton energy loss and extract the initial gluon density in high-energy heavy-ion collisions relative to that in a cold nucleus 71 . Extending the screened potential model of multiple parton scattering to a cold nuclear medium is somewhat problematic because it applies mainly to a medium where colors are screened but not confined. In this section we will study parton multiple scattering inside a nucleus where colors are confined to the size of a nucleon within generalized factorization theory of pQCD. In this case, parton propagation will be different from that in a partonic medium and one should expect the parton energy loss to be related to the nucleon size or confinement scale which, as we shall show, can be related to the gluon density inside a cold nucleus. In addition, one can calculate directly the modification of the fragmentation functions due to multiple scattering and induced gluon bremsstrahlung. This section reviews developments published in Refs. 69,70,101 ' 104 , which are referred to here as WOGZ and are applied in Ref.71 to cold and hot nuclei. In contrast to the situation in QED, the energy loss of a parton cannot be directly measured because partons are not the final experimentally observed particles. The total energy of a jet, traditionally defined as a cluster of hadrons in the phase space, will not change much due to medium induced radiation, because a jet so defined contains particles both from the leading parton and from the radiated gluons. This is particularly the case if multiple scattering and induced radiation do not dramatically change the energy profile of the jet in phase-space. It is also virtually impossible to determine the jet energy event by event because of the large background and its fluctuation in heavy-ion collisions. One then has to resort to parti-

164

M. Gyulassy et al.

cle distributions within a jet and study the effect of parton energy loss by measuring the modification of the fragmentation function of the produced parton, Da->h(z, /J.2), where z is the fractional energy of the parton a carried by the produced particles h. Since the produced quark in DIS is far off-shell, the final-state radiation leads to the scale dependence of the fragmentation functions as given by Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) 100 QCD evolution equations. When a parton is produced in a medium, it will suffer multiple scattering and induced radiation that will give rise to an additional term in the DGLAP evolution equations. This then leads to the medium modification of the DGLAP evolution of the parton fragmentation functions. As a consequence, the modified fragmentation functions become softer. This can be directly translated into the energy loss of the leading quark. 4 . 1 . Generalized

Factorization

To study parton energy loss in eA DIS, we consider the semi-inclusive processes, e(Li) + A{p) —> e(L2) + h(£h) + -X", where L\ and L-i are the fourmomenta of the incoming and the outgoing leptons, and Hh is the observed hadron momentum. The differential cross section for the semi-inclusive process can be expressed as

where p = [p + ,0,Ox] is the momentum per nucleon in the nucleus, q = L>2 — Li = [—Q2/2q~,q~,0±] the momentum transfer, s = (p + Li)2 and «EM is the electromagnetic (EM) coupling constant. The leptonic tensor is given by L^v = l/2Tr(7 • I
lM^

=

\T,
x2nS4(q+p-px-th)

(70)

where Ylx r u n s o v e r a u possible final states and JM = J2q egV"g7^« is t n e hadronic EM current. In the parton model with the collinear factorization approximation, the leading-twist contribution to the semi-inclusive cross section can be factorized into a product of parton distributions, parton fragmentation functions and the partonic cross section. Including all leading log radiative correc-

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

tions, the lowest order contribution (0(a°)) tering can be written as

165

from a single hard 7* + q scat-

dWs ,-^ r - j ^ = E e ' _/ dxfgA(x,ri)H$(x,p,q)Dq^h(zh,n2); 1

H{°J(x,p, q) = - T r ( 7 • ^7^7 • (q + xp)lv)

(71)

9-7T

^ — ^ - xB),

(72)

where the momentum fraction carried by the hadron is denned as Zh = £h/q~ and XB = Q2/2p+q~ is the Bjorken variable. /i 2 and fi2 are the factorization scales for the initial quark distributions /^(:r,/z 2 ) in a nucleus and the fragmentation functions Dg^hizh,!^2), respectively. Considering the leading logarithm approximation of the radiative correction to the fragmentation process as shown in Fig. 23, the renormalized quark fragmentation function Dq->h(zh, /i 2 ) satisfies the DGLAP 100 QCD evolution equations, dDq^h(zh,fi2)

_ a3(fx2)

fl dz ,

2

+ 7q->gq(z)Dg^h(zh/z,ti2)]

,

(73)

where Z

7?-? 9 ( ) = CF

l + z2

3 . + -6(1 - z)

(74) (75)

7g-OT(*)=7»-.w(l-*) are the splitting functions. The ' + ' function is defined as

Jo

(l-*0+

Jo

F(z)-F{l) 1

(76)

with F(z) being any function which is sufficiently smooth at z = 1. In a nuclear medium, the propagating quark in DIS will experience additional scatterings with other partons from the nucleus. The rescatterings may induce additional gluon radiation and cause the leading quark to lose energy. Such induced gluon radiations will effectively give rise to additional terms in the evolution equation leading to the modification of the fragmentation functions in a medium. These are the so-called higher-twist corrections since they involve higher-twist parton matrix elements and are power-suppressed. We will consider those contributions that involve twoparton correlations from two different nucleons inside the nucleus. They are proportional to the size of the nucleus 101 and thus are enhanced by a

166

M. Gyulassy et al.

Fig. 23. The hard partonic part of the first order radiative correction to the fragmentation process.

nuclear factor A 1 / 3 as compared to two-parton correlations in a nucleon. As in previous studies 69 ' 70 , we will neglect those contributions that are not enhanced by the nuclear medium.

Fig. 24. A typical diagram for quark-gluon re-scattering processes with three possible cuts, central(C), left(L) and right(R).

We will employ the generalized factorization of multiple scattering processes 102 . In this approximation, the double scattering contribution to radiative correction from processes like the one illustrated in Fig. 24 can

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter 167

be written in the following form, dWh dW£

fdz

f dy~ _

2

. - d ™ y» T j 2 ,

d

V2 J^d

kT

x e _ l k 7 " y T tf M J / (?r, j / r , y 2 , kT,p, q, z); 1 x 2 ^ l ^ ( ° ) 7 + A+(y2 , 0 r ) A+(yf ,yr)Mv~)\A)

•

*D

Here H ^{y ,y1 ,y2 ,kr,p,q,z) hard part H^x,x\,x2,kx,p,q,

(77)

is the Fourier transform of the partonic z) in momentum space, jdx^^ei^+y-+i^+y:

HUy^y^y2^T,P,q,z) =

xei(^i-^)P^H^(x,xux2,kT,p,q,z)

,

(78)

where kr is the relative transverse momentum carried by the second parton in the double scattering. Values of the momentum fractions x,x\ and x2 are fixed by J-functions and poles in the partonic hard part. They normally depend on krIn order to pick up the next-leading-twist contribution, we expand the partonic hard part around kx = 0, H

uv(v~ > Vi ,y2>kT,P, q, z) = H dH

+

D

dkS

(y~, j/f ,y2,kT

= 0, p, q, z)

2 D

fcT=0

1 dM Huv k% + =: 2 u ,dkZdkt v TwT

fc£ k?T + . . . . (79)

fcr=o 103 This is known as the collinear expansion . On the right-hand-side of Eq.(79), the first term gives the eikonal contribution to the leading-twist results. It does not correspond to the physical double scattering, but simply makes the matrix element in a single scattering gauge invariant. The second term for unpolarized initial and final states vanishes after being integrated over kr- The third term will give a finite contribution to the double scattering process. Substituting Eq.(79) into Eq.(77), integrating over d2kr and d?yr, we obtain dW, ^ dzh

= J2 j -jDq^h{zh/z)

l+Fa+{y2)F^{y^)i>q{y-)\A)

x ^W*(0) *\—29

a/3

dy

J ^2TT_ dVl dy2

d2

2dk%d4

— M HuAy "

D

_

_

_

,(80)

,S/i .2/2 >kT,P,q,z) fcT=0

168

M. Gyulassy et al.

where kj.A+k^A+ integrations.

are converted into field strength Fa+Fl3+

4.2. Double Parton

by partial

Scattering

There are all together 23 cut-diagrams that contribute to the leading twistfour corrections to the quark fragmentation function in eA DIS. For simplification of the calculation, one can use the helicity amplitude approximation by Guo and Wang 69,70 in the limit of soft gluon radiation. Such an approximation enables one to simplify the calculation of the radiation amplitudes. However, a complete calculation of the all cut-diagrams were recently done by Zhang and Wang 104 . Before we list the results, we first consider the contribution from Fig. 24 in detail. Using the conventional Feynman rule, one can write down the hard partonic part of the central cut-diagram of Fig. 24 6 9 ' 7 0 , Hc»Ay~,yI,y2,kT,p,q,z)

dx

-^--^eiXlP+y

= /

x2-K5+(f)8(l-z-^). ft tlap

_ CF 4 7-(q + SiP) ~ 2NC9 (q + xlP? - itla x £^fw fl

7-(g + *P-l)

(q + xp - £)2 + it

(81) l-(q + xlP-£) (q + xlP - £)2 - i t l a l w

"q lp

J L i i +2 ^ L 2*M3), (82) +K K {q + xp) + it

"'

'

where ea/3(£) is the polarization tensor of a gluon propagator in an axial gauge, n-A = 0 with?z = [l,0 - ,Oj_], and^, lq = q+(xi+X2)p+kr—^ are the 4-momenta carried by the gluon and the final quark, respectively, z = £g/q~ is the fraction of longitudinal momentum (the large minus component) carried by the final quark. To simplify the calculation, we apply the collinear approximation to complete the trace of the product of 7-matrices, p'H^Kj-e^Tr^-pOH^pr]

.

(83)

After carrying out momentum integrations in x, x\, X2 and ^ ± with the help of contour integration and (5-functions, the partonic hard part can

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

169

be factorized into the product of 7-quark scattering matrix H^J (x, p, q) [Eq.(72)] and the quark-gluon rescattering part H ,

Ko-.£.*.*.w)-f*w™) x H (y ,y1,y2,kT,x,p,q,z).

(84)

Contributions from all the diagrams have this factorized from. Therefore, we will only list the rescattering part H for different diagrams in the following. For the central-cut diagram in Fig. 24 it reads 69 ' 70 , H

C(Fig.2i)(V

,Vl ,2/2 ,kT,X,p,q,Z)

= I

di\ as „ 1 + z 2 -JT^CFY

^-T^-Ic{Fig.24)(y~,yT>y2^T,kT,X,p,q,z),

7c
=

x % ~ - Vi)0- ~ e-iXLP+y^){l

(85)

+

ei(x+*L)p

v-+™°p+(y;-y;)e{^)

- e-iXLP+to~-Vi))

.

(86)

Here, the fractional momentum is defined as X

, _ ,, r , xD = \ , _ , (87) 2p+q z(l - z) 2p+q z and x = XB = Q2/2p+q~ is the Bjorken variable. The above contribution resembles the cross section of a dipole scattering and contains essentially four terms. The first diagonal term corresponds to the so-called hard-soft process where the gluon radiation is induced by the hard scattering between the virtual photon and an initial quark with momentum fraction x. The quark is knocked off-shell by the virtual photon and becomes on-shell again after radiating a gluon. Afterwards the on-shell quark (or the radiated gluon) will have a secondary scattering with another soft gluon from the nucleus. The second diagonal term is due to the double hard process where the quark is on-shell after the first hard scattering with the virtual photon. The gluon radiation is then induced by the scattering of the quark with another gluon that carries finite momentum fraction XL+XD • The other two off-diagonal terms are interferences between hard-soft and double hard processes. In the limit of collinear radiation {xi —» 0) or when the formation time of the gluon radiation, 77 = 1/XLP+, is much larger than the nuclear size, the two processes have destructive interference, leading to the LPM interference effect. L

-

0

170 M. Gyulassy et al.

One can similarly obtain the rescattering part H diagrams (a-d) in Fig. 25: -rjD , - - - , HC(a)(y ,Wi ,y2,kT,x,p,q,z)

s

f = J

x-j^—Ic(a){y

1

T

_ T ^

,2/i ,2/2

2

-

C

A

T

—

,iT,kT,x,p,q,z),

7c(a)(y-,yT,y;jT,kT,x,P,q,z) =

e^+x^y'+^^^-v^ei-y;) _ e-ixLp+(y~-Vi

X,| e ix D p+(3/--yf)/(l-z)

Hcib)(y ,yi,y2,kT,x,P,q,z)

as „ l + z2

d# {

of other central-cut

)]

/gg}

'2 1 + Z2 _ _ ^C {eT {1 z)kT)2 F 2

=J

• z)kT)

2?r

1- z

x^j—ic(b){y ,yi,y2JT,kT,x,p,q,z), e^+x^+y-+ix^^-v^6{-yz)

~ic{b){y-,y^,y2 ,tT,kT,x,p,q,z) =

x % ~ - y-)e-ixi
jfD , - - _ , , fja ^ ( c ) ( 2 / ,2/ 1 ,2/ 2 ,fcr,x, P , g ,.) = y ^ ( a s CU 1 + 2 2 27ras T X

2TT T ~ T

C ( C )

ic{c)(y-,yT,y2JT,kT,x,p,q,z) = x6(y~ + xreixDp (v~

(*T £ T

'

!

,

(89)

~ k x ) • (*r - (1 - *)k T ) _kT)2(^_(1_z)kT)2 _ _ _

' ,Vl +x +

,V2

^T,kT,x,p,q,z),

+ix +

e^ ^ y~ ^ ^-y^e(-y^)

-yi)e-iXLP+y* -Vi)/(l-z) _ e -ix£,p+(y--yf )i

/QQ\

-irD _ _ f j a ( l T - k T ) • ( I T - (1 ~ *)k T ) ^c ( d )(2/ » y 1 ^ . ^ » ^ g . * ) = < / d g r ( / T _ k r ) 3 ( / T _ ( 1 _ z ) k r ) a X

as CAl + z22irasl: , _ _ _ . . . 2 ^ T l - Z ~]y~JC(d)(2/ ,2/i ,2/2 ,(T,kT,x,p,q,z),

lcid)(y-,yT,y2 JT,kT,x,P,q,z) = x9{y-

e^x+x^+y-+ix^+^-y^e(-y^)

-yi)e-iXLP+{y~-y^)

xreixDp+yJ/(l-z)

_ g-ixtp+y-j _

/gj\

To complete the calculation we also have to consider the asymmetricalcut diagrams(left-cut and right-cut) that represent interferences between

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

171

single and triple scatterings. They can be obtained with similar procedures. We refer readers to Ref.104 for a list of the the rescattering part H of all those asymmetrical-cut diagrams.

\ _ /ty y xp

.i

I *

Ap V

(a)

Fig. 25.

(b)

Four central-cut diagrams that contribute to the final results.

To obtain the double scattering contribution to the semi-inclusive processes of hadron production in Eq.(80), one will then have to calculate the second derivatives of the rescattering part H . After a closer examination of these rescattering parts, one can find that all contributions from the asymmetrical-cut diagrams have the form -jrD Uasym

I T • (IT - /(*)k T ) p ^ _ /( z )k T )2

e

iXp+Y-

(92)

where f(z) = 0, 1, 1 — z, z, X is the longitudinal momentum fraction and Y~ represents the spatial coordinates. One can prove that the second derivative of the above expression vanishes at kr = 0, 72

kT

I T • (*r - / ( * ) k T )

4 ( ^ T - /(*)k T ) 2

0.

(93)

172

M. Gyulassy et al.

Therefore, all contributions from the asymmetrical-cut (right-cut and leftcut) diagrams will vanish after we take the second partial derivative with respect to kr when we keep only the leading terms up to 0(XB/Q2^T),

V2fcTtffU|fcT=o = 0 + 0{xB/QH2T).

(94)

In the same way we find that some of the central-cut diagrams will not contribute to the final results, either. In fact, after taking the second partial derivative with respect to kr only four central-cut diagrams shown in Fig. 25 will contribute to the final result. Including only those contributions that do not vanish after the second derivative with respect to kr, we have

+

4Cjr(l — Z)

c-ixLp

(y-—u-)c-ixi.v+y*

Kjrp

_^_ ^ O A U ~ Z) c-ixLp+y^

+

2CA(l

Q __ e-ixLp+(y~-Vi

~ z)c-ixLp+(y--y-),1

_

))

e-ixLP+y-\

X,rp

+ 0{xB/QH2Tj\

.

(95)

The first term at the right-hand side in Eq.(95) comes from the contribution of HCra\ which is the main contribution in the helicity amplitude approximation 69 ' 70 . It contains hard-soft, double hard processes and their interferences. The other three terms come from diagram (b),(c),(d) of Fig. 25 respectively. They constitute corrections to the first term in powers oil — z. The second term that is proportional to (1 — z)2 is from the final state radiation from the quark in the double hard process in Fig. 25(b). The third and fourth terms are the results of the interference of the final state radiation from the quark and other radiation processes (initial state radiation and radiation from the gluon line). They contain both double hard processes and interferences between hard-soft and double hard processes in Fig. 25(c) and (d). Substituting Eq.(95) into Eqs.(84) and (80), we have the semi-inclusive tensor from double quark-gluon scattering including the contribution be-

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

173

yond the helicity amplitude approximation, WD'q dzh

£

JdxHJ2}(xp,q)£ d£T

2-KOLS

^Dq^h(zh/z)^CA\

TqAJx,xL)

(l-z)TqAM(x,xL)

+

+ ^a-*W a) (*.**)'

(96)

where

TqAJx,xL)

I ^ r - dy7dyo{\ J

- e-iXLP+y*

)(1 - e~iXLP+iy

i(x+x )p+y

p 1L x e-^ """ »

l

O(-y2)0(y~-vr)

-(A\i,q{o)i K+{y^)F+°{y;)My-)\A)

x

T£1\x,xL) =

+

+

J^dy^dy^ei(x+x )p y' L

— 2e~iXLP+<-v

~Vl

+y2

^

,-ixLp+(y

0{-y2)Hy~

, -y1 ) _|_

XL) =

j

(97) +

e-ixLp

y2

-vD

x -{A^^^F^iy^F^iy^^y-^A) TqT{x,

""i>)

27T

^ dy.dy^y-^^^Oi-y^-

-(A\jiq(p) 7 + ^ + ( % ) ^ ' ( i / D ^ ( y - ) | A )

,

(98)

- *) (99)

are twist-four parton matrix elements of the nucleus. Evidently these parton matrix elements are not independent of each other. TA(x, xi) has the complete four terms of soft-hard, double hard processes and their interferences. Therefore it contains essentially four independent parton matrix elements. Tqg (x,xi) and Tgq \X,XL) are the results of the corrections beyond the helicity amplitude approximation. But these two matrix elements are already contained in T^(X,XL). During the collinear expansion, we have kept IT finite and took the limit kx —> 0. As a consequence, the gluon field in one of the twist-four parton matrix elements in Eqs.(97)-(99) carries zero momentum in the soft-hard process. However, the gluon distribution xfg{x) at x = 0 is not defined in QCD. As argued in Refs. 69 ' 70 , this is due to the omission of higher order terms in the collinear expansion. As a remedy to the problem, a subset of the higher-twist terms in the collinear expansion can be resummed to

174

M. Gyulassy et al.

restore the phase factors such as exp(ixTP+y~), where XT = (kT)/2p+q~z is related to the intrinsic transverse momentum of the initial partons. As a result, soft gluon fields in the parton matrix elements will carry a fractional momentum XTUsing the factorization approximation 69 ' 70 ' 102 ' 101 , we can relate the twist-four parton matrix elements of the nucleus to the twist-two parton distributions of nucleons and the nucleus, TqAg(x,xL) = —(l-e-*l'*'*){tf(x +

XA fqA(x)(xL

+

+ xT)f^{xL

xL)xTf?(xT)

+ xT)},

(100)

A

where C is a constant, XA = 1/MRA, f (x) is the quark distribution inside a nucleus, and f^(x) is the gluon distribution inside a nucleon. A Gaussian distribution in the light-cone coordinates was assumed for the nuclear distribution, p(y~) = poexp(y~ I2RTA ), where RA = \/2RAM/P + and M is the nucleon mass. We should emphasize that the twist-four matrix element is proportional to 1/XA = RAM, or the nuclear size 101 . Notice that the off-diagonal matrix elements that correspond to the interferences between hard-soft and double hard processes are suppressed by a factor of exp(—x L /x A ). This is because in the interferences between double-hard and hard-soft processes, there is actually momentum flow of XLP+ between the two nucleons that the initial quark and gluon come from. Without strong long range two-nucleon correlation inside a nucleus, the amount of momentum flow XLP+ should then be restricted to the amount allowed by the uncertainty principle, 1/-R^ ~ P+/RAM. Similarly, the other two-parton matrix elements in Eqs.(98) and (99) can be approximated as T£»(x,

XL) = £ - {[fqA(x + xL) + fA(x)(xL A

- 2f (x)(xL T$2\x,

xL) = —fA{x){xL

xTf»(xT)

+ xT)f»(xL + xT)f?(xL + xT)f»(xL

+ xT)] + xT)} + xT).

e"*^ ,

(101) (102)

XA

Prom the above estimate of the matrix elements, both Tqg{x,xi,) and Tqg" '(X,XL) contain a factor 1 — e~XL^A because of the LPM interference effect. Such an interference factor will effectively cut off the integration over the transverse momentum at XL ~ %A in Eq.(96). As we will show later in the calculation of the effective energy loss, the integration with such a restriction in the transverse momentum due to LPM interference effect will

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

175

give rise to a factor 1/XA in addition to the coefficient fqA(x)/xAConsequently, contributions from double scattering in Eq.(96) that are associated with TqAg(x,xL) and T^g{1){x,xL) will be proportional to R\f£{x). These are the leading double scattering contributions in the limit of a large nuclear size. On the other hand, the third term Tqg \x, XL) in Eq.(96), which does not contain any interference effect, will only contribute to a correction that is proportional to RA/^(X). In the limit of a large nucleus, A1/3 3> 1, we will neglect this term in our study of the double scattering processes. 4.3. Virtual

Corrections

So far we have not considered virtual corrections which will ensure the final result to be infrared safe. The calculation of the virtual corrections can be calculated similarly as the radiative corrections. On the other hand, they can also be obtained via unitarity requirement. When cast into the DGLAP evolution equation as Eq.(73), the real corrections can be interpreted as the probability for the quark to radiate a gluon with momentum fraction 1 — z. Then one must also take into account the probability of no gluon radiation in the evolution to ensure unitarity. Such unitarity requirement gives rise to the same virtual correction as calculated from the virtual diagrams. The virtual contribution to the quark fragmentation in double scattering processes is, for example,

dzh

- £ J dxH$(xp,q)Dq^h(zh)

where T 4 < m \ x , xL) = TqAJx, xL) + (1 - z)T^\x,

xL).

(104)

Assuming a function F(z) which is sufficiently smooth at z = 1, One can single out the infrared divergent part of the following integral, r1

i _i_ z 2

ri

/ dz ——F(z) Jo \- z AF=

2

= F ( l ) / dz J0 1- z [ dz-^—\2F(l)-{l Jo 1- z

AF; + z2)F(z)]

.

The second term AF is finite since F(z) is a smooth function of z.

(105)

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We can apply this procedure to the integral in the modified fragmentation function. The divergent term can be combined with the radiative contribution in Eq.(96) to cancel the infrared divergency. With the help of the '+'function 100 , the final result can be expressed as

£ - E / « - > ^ / f £ £ww.)£<*

dzh

1 + z2 T$mHx,xL) + 6(z - l)AT_f "•>(*,#)

(l-z)+

(106)

<">

/V

AT£m\x,%)

Z L

2T£mHx,xL)\z=i (l + z2)T^m\x,xL)}

. (107)

Here the implicit z-dependence of Tqg (x, XL) plays an important role in the final result. The above integrand will be proportional to the splitting function for a single scattering if one ignores the z dependence of Til (x, XL ) • Similarly, the final result for contributions from gluon fragmentation is WD<9

dzh x

= £

f dxH$)(xP,q)2-^

o ^

11 + (1

2TT

J | r £

z)2TMm){XyXL)

+

*jLDg^h(zh/z) 6{z)ATMm){Xt4)

, (108)

z+

where we have used the fact that xi in Eq.(87) is invariant under the transform z —> 1 — z and so is T^(x, XL).

4.4. Modified Loss

Fragmentation

Function

and Parton

Energy

Including these virtual corrections and the single scattering contribution, we can rewrite the semi-inclusive tensor in terms of a modified fragmentation function Dq^h{zh,^2), dWLILV = £ dzh

J dxfqi(x,iij)H^(x,p,q)Dq^h(zh,^)

(109)

where fq{x, fij) is the quark distribution function which in principle should also include the higher-twist contribution 105 of the initial state scattering.

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter 177

The modified effective quark fragmentation function is defined as Dq->h{Zh,lJ?) =

Dq^h(zh,fl2) r2 d£las

f1 dz

,

rA

„2. „

eT)Dg^h{zh/z)]

.

,

.. (110)

where Dq^.h{zh,H2) and Dg^h(zh,fJ-2) are the leading-twist fragmentation functions. The modified splitting functions are given as ^lq->qg(z,X,XL,(T)

1 + * _TA(m){XtXL)

_(l-z)+ x ~z

&lq->gq{z,X,XL,t2T)

qg

^—^

+ s{1

" '"'

v

_

z)ATMm){xJ2T)

'

qg

,

(HI)

- Z,X,XL,e?n).

(112)

PTNcff{x,tf) = Ajq->qg(l

To further simplify the calculation, we assume XT <§C XT,
I # m \ x , x L ) » —(1 y

XA

e-*l'x*)tf(x) *

i-i^

(113)

where C = 2CxTfgN(xr) is a coefficient which should in principle depend on Q2 and XT- Here we will simply take it as a constant. In the above matrix element, one can identify 1/XLP+ = 2q~z(l — z)/n2 as the formation time of the emitted gluons. For large formation time as compared to the nuclear size, the above matrix element vanishes, demonstrating a typical LPM interference effect. This is because the emitted gluon (with long formation time) and the leading quark are still a coherent system when they propagate through the nucleus. Additional scattering will not induce more gluon radiation, thus limiting the energy loss of the leading quark. The reduction of phase space available for gluon radiation due to the LPM interference effect is critical for applying the LQS formalism to the problem in this paper. In the original LQS approach 102 , the generalized factorization for processes with a large final transverse momentum l\ ~ Q2 allows one to consider the leading contribution in 1/Q2, which is enhanced by the nuclear size RA ~ A1/3. For large Q2 and A, the higher-twist contribution from double parton rescattering that is proportional to OLSRAIQ2 will then be the leading nuclear correction. One can neglect contributions

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from more than two parton rescattering. In deriving the modified fragmentation functions, we however have to take the leading logarithmic approximation in the limit l\ C Q 2 , where IT is the transverse momentum of the radiated gluon. Since the LPM interference suppresses gluon radiation whose formation time (T/ ~ Q2/(-TP+) i s l a r g e r than the nuclear size MRA/P+ in our chosen frame, £T should then have a minimum value of £T ~ Q2/MRA ~ Q2/A1/3. Here M is the nucleon mass. Therefore, the logarithmic approximation is still valid for large nuclei {MR A ^> 1)In the meantime, the leading higher-twist contribution proportional to asRA/(-T ~ otsR2A/Q2 can still be small for large Q2 so that one can neglect processes with more than two parton rescattering. The parameter for twist expansion of the fragmentation processes inside a nucleus is thus asA2/3IQ2 as compared to asA1^3/Q2 for processes with large final transverse momentum as studied by LQS 102 . This is why the nuclear modification to the fragmentation function as derived in this paper depends quadratically on the nuclear size RABecause of momentum conservation, the fractional momentum in a nucleon is limited to XL < 1- Though the Fermi motion effect in a nucleus can allow XL > 1, the parton distribution in this region is still significantly suppressed. It therefore provides a natural cut-off for XL in the integration over z and IT in Eq.(llO). With the assumption of the factorized form of the twist-4 nuclear parton matrices, there is only one free parameter C{Q2) which represents quark-gluon correlation strength inside nuclei. Once it is fixed, one can predict the z, energy and nuclear dependence of the medium modification of the fragmentation function. Shown in Figs. 26 and 27 are the calculated nuclear modification factor of the fragmentation functions for 14 7V and MKr targets as compared to the recent HERMES data 1 0 6 . There are strong correlations among values of Q2, v and z in the HERMES data which are also taken in account in our calculation. The predicted shape of the z- and ^-dependence agrees well with the experimental data. A remarkable feature of the prediction is the quadratic A2!3 nuclear size dependence, which is verified for the first time by an experiment. This quadratic dependence comes from the combination of the QCD radiation spectrum and the modification of the available phase space in £T or XL due to the LPM interferences. Note that the numerical results shown here are obtained with the original helicity amplitude approximation 69 ' 70 . The numerical calculation beyond the helicity approximation differs only about a few percent 104 . By fitting the overall suppression for one nuclear target, we obtain the only parameter in our calculation, C{Q2) = 0.0060 GeV2 with

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

179

Fig. 26. Predicted nuclear modification of jet fragmentation function is compared to the HERMES data 1 0 6 on ratios of hadron distributions between A and d targets in DIS.

o ,4N HERMES "Kr HERMES (Preliminary) 8

10

12

14

16

18

20

22

24

v (GeV) Fig. 27. Energy dependence of the nuclear modification compared with the HERMES data 1 0 6 .

»s(Q 2 ) = 0.33 at Q2 w 3 GeV 2 . This parameter is also related to nuclear broadening of the transverse momentum of the Drell-Yan dilepton in pA collisions107, ( A Q J J « C-KOLS/NCXA- With an experimental 108 value of (Aql) = 0.016A 1 / 3 GeV2 and as{M%) = 0.21 (< M2- > « 40 GeV 2 ), one finds C(M2-) = 0.013 GeV 2 , which is about a factor 2 larger than the value obtained in our fit to the HERMES data. The value of C determined

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from nuclear broadening in photo-production of a di-jet is even larger 102 at Q2 — 4p?r « 64 GeV 2 . Such a strong scale dependence of C(Q2) is in line with one's expectation, since it is related to gluon distribution xg{x, Q2) at small x in nuclei 101 . In principle the modification of the fragmentation functions would be the only experimental effect of induced gluon radiation via multiple scattering. One can never directly measure the energy loss of the leading quark. The net effect of the energy loss is the suppression of leading particles on one hand and the enhancement of soft particles on the other, leading to the modification of the fragmentation functions. One can then experimentally characterize the parton energy loss via the momentum transfer from large to small momentum regions of the fragmentation functions. Upon a close examination of Eq.(llO), we see that the first term is the renormalized fragmentation function in vacuum. The rest is particle production induced by the rescattering of the quark through the nuclear medium. In particular, the last term is particle production from the fragmentation of the gluon which is induced by the secondary scattering. Such particle production is at the expense of the energy loss of the leading quark. We can thus quantify the quark energy loss by the momentum fraction carried by the radiated gluon, {Azg)(xB,fJ.z)

O

«*£

/

= I

I

"<<

,

~gf Jn

4 Jo

Jo

~N~ JVC

2*

A

/

„_

,,2'

_

dz—zA-yq^gq(z,xB,xL,e^)

dz ^ \fdz[i+a /f ~w ^ + (l-z->zf J 7A, — t r ~ *-T JO

JO

./n

lT

J0

fq(xB,Hi)

(114)

Using the approximation for the modified twist-four parton matrix elements in Eq.(113), we have \< 2x ^CA^XB f1 . 1 + ( 1 - * ) 2 {Azg)(xB,fJ. ) = C—— ——j dz2 / /A

Nc xAQ h

•*(!-*)

>< r^a-^a-c-^), (us) JO

2

X

L

l

+

where a;M = [i /2p q~z(l — z) = XB/Z(1 — z) if we choose the factorization scale as /i 2 = Q2. When x^
-,CAa2s xB _ n n \-2xB )«C——-_-r50r[ln—;—

1, -].

(116)

Jet

Quenching and Radiative Energy Loss in Dense Nuclear Matter

181

Since XA — ^-/MRA, the energy loss (Azg) thus depends quadratically on the nuclear size. In the rest frame of the nucleus, p+ = TUN, Q~ — v, and XB = 2 Q /2p+q~ = Q"1 j2m^v. One can get the averaged total energy loss as AE = v(Azg) &C{Q2)a2(Q2)mNRA(CA/Nc)3\n(l/2xB). With the determined value of C, (XB) « 0.124 in the HERMES experiment 106 and the average distance (LA) = RA\[^T^ f° r the assumed Gaussian nuclear distribution, one gets the quark energy loss dE/dL « 0.5 GeV/fm inside a Au nucleus. 4.5. Energy Loss in Hot Medium

at

RHIC

To extend our study of modified fragmentation functions to jets in heavyion collisions and to relate to results obtained in the opacity expansion approach, we can assume (kj,) « /z2 (the Debye screening mass) and a gluon density profile p(y) = (TQ/T)6(RA — y)po for a 1-dimensional expanding system. Since the initial jet production rate is independent of the final gluon density which can be related to the parton-gluon scattering cross section 57 [osXrG(sr) ~ ^2(Js]> o n e has then

aST

ff(XB)L) ~ "3 /^ P(tf)11 " C0S(j//T/)1'

(U7)

where T/ = 2Ez(l — z)jl\ is the gluon formation time. One can recover the form of energy loss in a thin plasma obtained in the opacity expansion approach 36 ,

= ^^fdzP / JoV JO ^ T fR\

x J 'TO

1 + (1

-*)2

u(l + u)

( ^ [l

d^9P(r)

({T-TQ)U^\

[I-oos

^2Ez{1_z))

(118)

Keeping only the dominant contribution and assuming ag « Ca27ra2//U2 (C Q =1 for qg and 9/4 for gg scattering), one obtains the averaged energy loss, dE {

TTCOCA^ )K

dL —RA—JT0

(RA

IE <MT)(T-TO)

In ^ .

(119)

Neglecting the logarithmic dependence on r, the averaged energy loss in a 1-dimensional expanding system can be expressed as dE {

^dE02r0

M)ld~-dTR-A'

(120)

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where dEo/dL oc PQRA is the energy loss in a static medium with the same gluon density po as in a 1-d expanding system at time TQ. Because of the expansion, the averaged energy loss {dE/dL)\d is suppressed as compared to the static case and does not depend linearly on the system size. This could be one of the reasons why the effect of parton energy loss is found to be negligible in AA collisions at v ^ = 17.3 GeV 109 . An effective model of modified fragmentation functions was proposed in Ref.44'45: D^h{z) « j^D^

(^)

,

(121)

with Az to account for the fractional parton energy loss. This effective model is found to reproduce the pQCD result from Eq.(llO) very well, but only when Az is set to be Az « 0.6{zg). Therefore the actual averaged parton energy loss should be AE/E = 1.6Az with Az extracted from the effective model. The factor 1.6 is mainly caused by the unitarity correction effect in the pQCD calculation. A similar effect is also found in the opacity expansion approach 47 . Both PHENIX and STAR experiments have reported 1,5 strong suppression of high pr hadrons in central Au + Au collisions at ^/s = 130 and 200 GeV, indicating for the first time a large parton energy loss in heavy-ion collisions. To extract the parton energy loss, we compare the data with the calculated hadron pr spectra in heavy-ion collisions using the above effective model for medium modified jet fragmentation functions 39 . Shown in Fig. 28 are the nuclear modification factors RAA(PT) as the ratios of hadron spectra in AA (pA) and pp collisions normalized by the number of binary collisions25. Parton shadowing and nuclear broadening of the intrinsic kr are also taken into account in the calculation which decrsibes pA data for energies up to yfs = 40 GeV 39 . The nuclear fcT-broadening gives the Cronin enhancement at large pr in pA collisions, where there is no parton energy loss induced by a hot medium. Fitting the PHENIX data yields (dE/dL)ld « 0.34(lnE/ln5) GeV/fm, including the factor of 1.6 from the unitarity correction effect. We consider only 7r° data here, since at large pr the charged hadrons are dominated by baryons, which could be influenced mainly by non-perturbative dynamics 36 . Taking into account the expansion, the averaged parton energy loss extracted from the PHENIX data would be equivalent to (dE/dL)0 = 0.34(i?^/2ro) In S / I n 5 in a static system with the same gluon density as the initial value of the expanding system at To- With RA ~ 6 fm and T0 ~ 0.2 fm, this would give (dE/dL)0 ss 7.3 GeV/fm for a 10-GeV parton,

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

1.2

•

183

£HENIX Au+Au( 10% central) rs= 1 30 GeV

1

d+Au (min)

^,0.8 a.

Au+Au (central)

10.6 0.4

0.2 --J

" 0

2

4

6

rs=200GeV I

I

I

I

8

I

I

,

I

10

L_

12

pT (GeV/c) Fig. 28. Calculated nuclear modification factor of 7r° p r spectra for d + Au and central Xu + Au collisions at i/s = 130 (solid) and 200 GeV (dashed) as compared to PHENIX data 1 The lower dashed line uses the energy loss as given in Eq.(122).

which is about 15 times of that in a cold Au nucleus, as extracted from the HERMES data. Since the parton energy loss is directly proportional to gluon density of the medium, this implies that the gluon density in the initial stage of Au + Au collisions at TQ = 0.2 fm/c is about 15 times higher than that inside a cold nucleus. We can predict the 7r° spectra at *Js = 200 GeV as given by the dashed lines in Fig. 28, assuming that the initial parton density in central Au + Au collisions at y/s = 200 GeV is about 10% higher than at 130 GeV. In Fig. 28, we have shown the predicted suppression factor at large pr with the same logarithmic energy-dependence of the energy loss as in the cold nuclear matter which gives a suppression factor that increases with prHowever, new RHIC data 3 at 200 GeV show a constant suppression factor at larger pr- This might be the indication of the importance of the detailed balance 98 which gives much stronger energy dependence of the energy loss. We show as the lower dashed line in Fig. 28 the result for an effective parton energy loss A £ a (£?//*-1.6) 1 - 2 0 /(7.5+ £//*)

(122)

which is parameterized according to the result from Ref.98 where both stimulated gluon emission and thermal absorption are included in the calculation of the total energy loss. The detailed balance between emission and absorption both reduces the effective parton energy loss and increases the

184

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energy dependence. The threshold is the consequence of gluon absorption that competes with radiation that effectively shuts off the energy loss. The parameter /x is set to be 1 GeV in the calculation shown in Fig. 28.

5. S u m m a r y In this report we reviewed two recent approaches to the problem of nonAbelian radiative energy energy loss in dense but finite QCD matter. In the first section, we highlighted some of the striking new high px phenomena observed for the first time in A + A reactions at RHIC with y/s > 100 AGeV. An interesting pattern of hadron suppression, already beginning at moderate px > 3 GeV, was seen in the hadron flavor dependence of single inclusive spectra, in the large azimuthal asymmetry, and in the preliminary two-hadron correlations. We interpret these phenomena as manifestations of jet quenching in ultra-dense matter produced in such reactions. Our predictions for these phenomena in both approaches are reviewed in later sections and depend on the energy loss, AE = J dx dE/dx, of fast quarks and gluons propagating through rapidly expanding QCD matter. The two approaches, GLV and WW/WOGZ reviewed here, provide a systematic way to compute AE via an opacity or higher twist expansion in finite nuclear matter. Elsewhere reviewed asymptotic approaches such as the BDMS/Z/SW 6 8 , 1 1 0 are designed for applications to "thick" or macroscopic media at asymptotic energies. An analytic approximation to the sum of the infinite multiple collision series is obtained through an approximate color dipole quantum diffusion analogous to the Moliere series in electrodynamics. The complications due to finite kinematic bounds are neglected. As shown in 111 , the phenomenological applications of the asymptotic expressions tend to overpredict quenching at RHIC and lead to a too rapid variation of the suppression factor with px, inconsistent with the RHIC data. In our approach, on the other hand, the opacity series is computed to arbitrary order in opacity for applications to finite opacity systems where the non-Gaussian (Rutherford) tails of distributions are not yet eclipsed by the approximate Gaussian small px component. In addition, our expressions can be applied to arbitrary 3D expanding and time dependent media such as created in nuclear collisions. The GLV reaction operation approach discussed in Section 2 is based on a general algebraic recursive method that describes the propagation and interaction of systems through dense nuclear matter, taking into account the severe destructive LPM interference effects due to the long formation times

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

185

of ultra-relativistic partons. It provides a means to compute the multiple collision amplitudes to any order \ n in the opacity \ = J dz a p. So far it has been successfully applied to obtain solutions for the nuclear broadening 74 and the final state medium induced radiation 52 ' 53 resulting from multiple elastic and inelastic projectile scatterings. The range of applicability of the calculations can be significantly extended by careful treatment of kinematic bounds. The final state double differential distributions of jets and gluons are presented as an infinite series in powers of the mean number of scatterings x- F° r elastic scatterings this series can be summed to all orders (or equivalently all twist parton-parton correlations) to reproduce the standard Glauber theory result, but more general than the simple color dipole approximation used in asymptotic analyses. For inelastic processes, a closed sum to all orders can unfortunately only be carried out in the dipole approximation as in BDMS/Z/SW (see 110 for most recent developments). However, our analytic 52 ' 74 expressions at any finite order in opacity can be evaluated numerically 52 ' 53 . Numerical evaluation of the expressions through the first three orders (up to twist 8) has been carried out for phenomenological applications. Each power in opacity adds a twist 2 parton-parton correlation. Future work via this approach includes computation of multiple gluon emission beyond the Poisson approximation and the broadening and radiation of dipole-like, possibly heavy, qq systems. Applications to heavy quark energy loss including both the Ter-Mikayelian gluon dispersion effects as well as the "dead cone" effect are also underway 112 . The WW approach reviewed in Section 3 extends the calculation of energy loss to include the positive feedback (jet acceleration) due to absorption of thermal gluons in the medium. Absorption counteracts the induced energy loss for jet momenta less than the typical thermal energy scale ~ 3T. In an expanding hydrodynamic medium with transverse boost rapidity T)T we expect that this absorption feedback contribution is blue shifted to higher momenta pr ~ 2>TeT,T. Thus jet quenching cannot suppress the spectrum below the local equilibrium hydrodynamic limit. In Section 4, several recent works were combined (referred to as the WOGZ approach 69 ' 70,101 ' 104 ) to elaborate on a general twist-expansion of multiple parton scattering beyond the GW model of the medium. In that approach one can calculate explicitly the modified parton fragmentation function up to twist 4 thus far. The LPM interference effect is then embedded in the twist-four parton matrix elements of the nucleus which also contains the fundamental properties of the nuclear medium - parton den-

186

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sity correlations inside a nucleus. These matrix elements replace the Debye screened interactions used in the GLV approach. One can demonstrate explicitly that the quadratic dependence of the modification of fragmentation functions and the effective parton energy loss on the nuclear size RA is caused both by the LPM interference and the specific form of gluon radiation spectra in QCD. The predicted nuclear modification of the fragmentation function, both the energy and nuclear dependence, is found to agree well with the recent experimental data 1 0 6 on jet fragmentation in e + A. This is an important test of both the twist or opacity expansions since it shows the dominance of the twist 4 (first order in opacity) contribution to the energy loss in finite nuclear systems as found numerically in the GLV expansion up to twist 8. Extending the results to a parton propagating in a hot QCD medium, we have shown that the twist expansion to order 4 is equivalent to the first order opacity GLV result under certain simplifying assumptions about the form of the twist 4 matrix elements. The phenomenological application of WOGZ to both DIS on cold nuclear targets and high-energy heavy-ion collisions suggests that the parton energy loss in an expanding system at RHIC would be equivalent to (dE/dx)o « 7.3 GeV/fm in a static medium, which is almost 15 times higher than that in a cold Au nucleus. If the jet quenching pattern is confirmed by further measurements and theoretical refinements, current RHIC data may have already provided the first tomographic evidence that initial parton densities on the order of 100 times nuclear matter density were achieved in Au + Au collisions. The full analysis of the flavor composition, shape, and azimuthal moments of the high pr spectra appears to be a promising diagnostic probe of the evolution of the produced quark-gluon plasma. However, in spite of the consistency of our jet tomography analysis with current RHIC data, it is still too early to draw definitive conclusions. The main uncertainty is the magnitude of gluon shadowing 113 in the initial nuclear wavefunction. While estimates of shadowing vary wildly, unfortunately nothing is yet known experimentally on this important question. QCD analysis 114 suggests that non-linear corrections to DGLAP evolution may cause the gluon density to saturate at small x. Saturation is found in classical Yang-Mills models 115 ' 116,117 ' 118 as well. In Ref.119 it was proposed that gluon saturation or shadowing phenomena alone may account for a significant part of the observed high pr hadron suppression pattern. Our estimates of shadowing based on ESK'98 48 do not support this idea and neither do recent calculations 120 ' 121 in the MQ approach 114 of the upper limit

Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter

187

of the saturation scale and the rate of disappearance of its effects on the small and moderate px spectra. Fortunately, a decisive experimental test is now under way at RHIC via d + Au reactions. If our theory is correct, and the observed suppression pattern in Au + Au is due to jet quenching via final state interactions in the dense QCD matter produced, then instead of suppression, the Cronin effect is predicted to dominate over shadowing in the x > 0.01 range accessible at RHIC. We therefore predict as an upper limit an enhancement of the moderate pr hadrons by ~ 10 — 30% in d + A relative to binary scaled p + p. On the other hand, if the saturation model is correct, and the suppression pattern in Au + Au is a consequence of deep gluon shadowing in the initial state, then a suppression about 30% of moderate and high px hadrons is predicted to occur in d + Au 119 . The experimental answer to this decisive question may be known by the publication time of this review. In either case, the answer will be exciting. N o t e added in proof: On June 18, 2003 the four RHIC experimental groups announced the final results 122 ' 123 ' 124 of the d + Au run at RHIC for RdAuiPr)- The data confirmed our predictions that RdAu > 1 in this reaction and the back-toback jet structure remains intact. The data rule out deep gluon shadowing and the predictions of the extended saturation model in the x > 0.01 kinematic range. This is a major milestone at RHIC because the d + Au results prove that the phenomenon of jet quenching discovered in Au + Au must be due to final state interactions. The initial density of gluons at time T0 = 0.2 fm/c that we infer from the Au+Au jet quenching pattern is P(TO) « 1000/(7ri? 2 r 0 ). This exceeds 30 times the nuclear matter ground state density. One can reach the same conclusion by comparing the effective parton energy loss in Au + Au to that in DIS off a cold nucleus 125 . The d+Au data have provided the critical control calibration measurement of the jet tomographic techniques that we developed and reviewed in this article. The combined p 4- p, d+ Au, Au + Au data on jet production and quenching above px > 3GeV together with the observed bulk collective hydrodynamic flow pattern that was predicted assuming the same initial density in Au + Au provide compelling evidencce that a a new state of matter- the quark gluon plasma - was in fact produced in central Au + Au at RHIC. Future experiments on direct photons, tagged jets, heavy quark propagation, and two and higher particle correlations will help to map out its detailed properties and further test the scope and limitations of the

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emerging QGP paradigm. Acknowledgments M.G. and I.V. gratefully acknowledge extensive collaboration with Peter Levai on the GLV approach. X.N.W. and B.W.Z. would like to thank Enke Wang on extensive collaboration on the WW approach with detailed balance and many fruitful discussions. X.N.W. would also like to acknowledge collaborations with X. F. Guo and J. Osborne on the twist expansion approach. This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, and by the Office of Basic Energy Science, Division of Nuclear Science, of the U.S. Department of Energy under Contracts No. DE-FG0287ER40371, DE-FG02-93ER40764, and DE-AC03-76SF00098 and by National Natural Science Foundation of China under project No. 19928511 and No. 10135030. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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G L U O N RADIATION A N D P A R T O N E N E R G Y LOSS

Alexander Kovner a ' c and Urs Achim Wiedemann ' c a

Department of Mathematics and Statistics, University of Plymouth, Drake Circus, PL4 8AA, Plymouth, UK b Theory Division, CERN, CH-1211 Geneva 23, Switzerland 0 Institute for Nuclear Theory, University of Washington, Box 351550, Seattle, WA 98195, USA

The propagation of hard partons through spatially extended matter leads to medium-modifications of their fragmentation pattern. Here, we review the current status of calculations of the corresponding medium-induced gluon radiation, and how this radiation affects hadronic observables at collider energies.

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Contents 1 Introduction 2 Gluon Bremsstrahlung in the Eikonal Approximation 2.1 N = 1 opacity approximation (single hard scattering limit) . . . . 2.2 Multiple soft scattering limit (dipole approximation) 3 Gluon Radiation Beyond the Eikonal Approximation 3.1 Wilson lines for non-eikonal trajectories 3.1.1 Non-abelian Furry approximation (target field Ao) 3.2 Target averages for non-eikonal Wilson lines 3.3 The medium-induced gluon radiation spectrum 4 Properties of the Medium-Induced Gluon Radiation Spectrum 4.1 Multiple soft scattering 4.1.1 The harmonic oscillator approximation (static medium) . . 4.1.2 Qualitative estimates of w^ vs. quantitative calculations . 4.1.3 Angular Dependence of the gluon energy distribution . . . 4.1.4 Harmonic oscillator approximation (expanding medium) . 4.1.5 Properties of the transport coefficient 4.2 Opacity expansion of the radiation cross section 4.2.1 Expansion to order N = 0 and N = 1 4.2.2 Qualitative estimates of w ^ vs. quantitative calculations . 4.2.3 Parameters in the opacity expansion 5 Applications 5.1 Properties of quenching weights 5.1.1 Discrete part of the quenching weight 5.1.2 Continuous part of the quenching weight 5.2 Quenching factors for hadronic spectra 5.3 Medium-modified fragmentation functions 6 Appendix A: Eikonal Calculations in the Target Light Cone Gauge . . . 7 Appendix B: Path Integral Formalism for the Photon Radiation Spectrum References

194 195 199 203 204 204 207 210 211 214 215 216 217 220 222 224 225 226 228 231 231 232 233 234 235 238 240 242 245

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Wiedemann

1. Introduction In the QCD improved parton model, inclusive cross sections for high-px partons (observed as leading hadrons or jets) can be calculated through collinear factorization. This formalism convolutes the perturbatively calculable parton-parton cross sections with non-perturbative but processindependent ("initial state") parton distribution functions and ("final state") fragmentation functions. What additional physics input is needed if one extends the application of QCD to the description of inclusive high-px parton cross sections in nucleus-nucleus collisions at RHIC or LHC, where hadronic projectiles have significant spatial extension oc A1/3 ? The current discussion of this question focuses in particular on the following effects: (i) The density of the incoming parton distributions increases for given x and Q2 by a factor w A1!3. This implies that with increasing A1/3 the non-linear modifications 21.47,11,29,35 0 f ^he QCD evolution equations become relevant at lower center of mass energies (or larger momentum fractions x). (ii) The QCD evolution of the projectile wave function may be altered due to the presence of a spatially extended "target" through which it propagates 13 . This goes under the name "initial state partonic energy loss". It is discussed e.g. in Drell-Yan production in p — A where a quark in the proton may undergo multiple scattering in the nucleus on its route to annihilation 14 ' 32 . (iii) The hard partonic collision is well-localized on a scale \/Q which is much smaller than the diameter of a nucleon 1/AQCDThus, one does not expect medium-modifications of the hard partonic cross section itself. However, there is a class of high-px observables in e-A and p-A for which the medium-modification can be described by nuclear enhanced twist-four matrix elements whose importance is determined by the pole structure of the hard partonic cross section 43 ' 44 > 50 . This is the only case of a nuclear dependence for which higher twist factorization theorems are proven, iv) At sufficiently large center of mass energy or for sufficiently extended projectiles (A1/3 > 1), the probability of two independent hard parton interactions in the same hadronic collision becomes significant 4 9 . This is e.g. a significant background to some of the hadronic Higgs decay channels searched for in p-p collisions at the LHC 16 . (v) Finally, the presence of spatially extended matter can affect the fragmentation and hadronization of hard partons produced in nucleus-nucleus collisions 12 ' 24,3 . This effect is often referred to as "final state partonic energy loss" or "jet quenching" and is expected to affect essentially all high-px hadronic observables in

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heavy ion collisions at collider energies. The main motivation for the study of these observables is that the degree of nuclear modification may allow for a detailed characterization of the hot and dense matter produced in the collision. All the above mentioned effects should emerge from a theory of QCD processes in nuclear matter. The development of such a theory is still at the very beginning. The high rate at which new data are becoming available from RHIC, as well as the rapid development of new theoretical ideas and phenomenological interpretations make it likely that any comprehensive review will be outdated very soon. In this situation, we have decided to focus entirely on a simple partonic multiple scattering formalism which underlies at present almost all "jet quenching" calculations. This limited scope excludes many important issues. In particular, we do not discuss to what extent experimental data from RHIC provide indications of partonic energy loss (see 4 6 ) . Moreover, many phenomenologically proposals for discovering partonic energy loss, such as the measurement of back-to-back correlations 12,54,55 , the specific behavior of massive fr-quarks 18 ' 41 . 42 > 17 0 r particle ratios at high p± are not included in the discussion. Finally, we do not discuss recent alternative formulations of partonic energy loss, e.g. in the framework of nuclear enhanced higher twist matrix elements 2 3 , 5 6 . The presentation is organized as follows: in section 2, we discuss the formalism of eikonal scattering in which 5-matrix amplitudes are determined in terms of eikonal Wilson lines in the target field. Section 3 discusses the generalization of this formalism to non-eikonal trajectories which take the transverse Brownian motion of the partonic projectile into account. The medium-induced gluon energy distribution radiated off a hard parton is calculated in this generalized formalism. Section 4 discusses the properties of this medium-induced spectrum and section 5 turns to some applications.

2. Gluon Bremsstrahlung in the Eikonal Approximation At high energy, the propagation time of a parton through a target is short, partons propagate independently of each other and their transverse positions do not change during the propagation. The only effect of the propagation is that the wave function of each parton in the projectile acquires an eikonal phase due to the interaction with the target field 15,38 . To be specific, we consider a hadronic projectile wave function whose relevant degrees of

196 A. Kovner and U. A. Wiedemann

freedom are the transverse positions and color states of the partons, *in =

53

^({"i,Xi})|{ai,Xi}).

(1)

{c*i,Xi}

The color index a* can belong to the fundamental, anti fundamental or adjoint representation of the color SU{N) group, corresponding to quark, anti quark or gluon in the wave function. We choose the Lorentz frame in which the projectile moves in the negative z direction. Thus at high energy the light cone coordinate x+ of the projectile does not change during the propagation thought the target. The projectile emerges form the interaction region with the wave function Vout=SVin=

]T

i>({ai,xi})Y[W(yii)ai0i\{pi,Xi}).

{ai,Xi}

i

(2)

Here S is the 5-matrix, and the W s are Wilson lines along the (straight line) trajectories of the propagating particles W(Ki)=Vexp{iJdz-TaA+(^i,z-)}

(3)

with A+ - the gauge field in the target and Ta - the generator of SU(N) in a representation corresponding to a given parton a . The interaction in the target field changes the relative phases between components of the wave function and thus "decoheres" the initial state. As a result the final state is different from the initial one, and contains emitted gluons. The component of the outgoing wave function which belongs to the subspace orthogonal to the incoming state reads \6*) = [l-\*in)(*in\]\*mit).

(4)

From this, one can calculate for example the inelastic cross section which up to the factor of total flux is given by the probability Pinel = mf a

= 1 - |S| 2 ,

(5)

In this section we use the light cone gauge A~ = 0, in which the parton distributions of the projectile are simply expressed in terms of the parton number operators. In this gauge the color fields of the target have a large A+ component, thus the eikonal S - matrix eq.(3) is given in terms of A+. The physical observables are of course gauge invariant and can be calculated in any gauge. In Appendix A we present the same calculation in the "target light cone gauge", A+ = 0, where the only non vanishing component of the target field is A1.

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where S = (VinWaut) •

(6)

The S'-matrix element in eq.(6) has to be averaged over the distribution of the color fields in the target. Also, the total and diffractive cross sections can be calculated by projecting other components out of * 0 „ t 38 . In what follows, we apply the above framework to calculate the gluon radiation spectrum radiated off a hard quark which propagates at high energy through a nuclear target. At this point we assume the quark to be coming from outside the target rather than being produced inside the target in a hard scattering event. Thus it carries with it the fully developed wave function which contains the cloud of quasi real gluons. In the first order in perturbation theory the incoming wave function contains the Fock state \a) of the bare quark, supplemented by the coherent state of quasi real gluons which build up the Weizsacker-Williams field / ( x ) , *?„ = |a> + j dx de/(x) T*,, |/3; 6(x, 0>

(7)

fWOTOTW- b

-

2

a

.

a

| _

Here Lorentz and spin indices are suppressed. In the projectile light cone gauge A~ = 0, the gluon field of the projectile is the Weizsacker-Williams field ^(x)cx6>(ar)/i«,

/i(x)ocff^,

(8)

where x~ = 0 is the light cone coordinate of the quark in the wave function. The integration over the rapidity of the gluon in the wave function (7) goes over the gluon rapidities smaller than that of the quark. In the leading logarithmic order the wave function does not depend on rapidity and we suppress the rapidity label in the following. The interaction of the projectile (7) with the target leads to a color rotation a, —> /?, of each projectile component i, resulting in an eikonal phase W(x.i)otipi. The outgoing wave function reads *out = W£r(0) |7> + JdKf(K)T^W^(0)

WbAc(x) | 7 ; c(x)>,

(9)

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Wiedemann

where WF(0) and WA(x) are the joint representations respectively, at the transverse position xq = 0 an inelastic process, for which we I**,,) = |*? ut > - £

Wilson lines in the fundamental and adcorresponding to the propagating quark and gluon at x s = x. Gluon radiation is have to calculate the projection (4)

|*in(7))(*in(7)l*lt>

(10)

< « W, fa{*°> a

x

ap

W

a

L

pY

0/°>

Here, the index 7 in the projection operator runs over the quark color index. It appears in eq.(10) since we have to project out the components of the wave function which differ from the initial state only by a global rotation in the color space 3 8 b . The number spectrum of produced gluons is obtained from (10) by calculating the expectation value of the number operator in the state S^Q, averaged over the incoming color index a. After some color algebra, one obtains ^prod(k) = l ^ ( < 5 * Q | a + ( k ) a d ( k ) | ( 5 * Q )

?I

asCF

dx dy e t k - ( x - y )

x

• y

X2y2

i-Nrzi(0*[wAH*)wA{o)]))t

L-((Tr[WAHy)WA(0)]))t A

A

+JJ—{{TI{W Hy)W (x)}))t

(11)

Here, we have used / ( x ) / ( y ) = §£ ^ ^ 2 for the Weizsacker-Williams field of the quark projectile in configuration space and the symbol ()t denotes the averaging over the gluon fields of the target. To obtain an explicit expression, we require the target average of the product of adjoint Wilson lines. Various averaging procedures are discussed b

A n event in which the final state differs from the initial one only by a global rotation of the color index would be an elastic scattering event.

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and Parton Energy Loss

199

in the literature 61.15>S7,58,38 rp0 ^ e specific, we consider a target which consists of static scattering centers with scattering potentials a+ (q) at positions (x„, zn). In the high energy approximation, each scattering center transfers only transverse momentum to the projectile, J 4+(x,z-)

= ^y^_e^-i^a+(q)5(z--z-).

(12)

We define the target average as an average over the transverse positions of the static scattering centers. Introducing the longitudinal density of scattering centers, n{z~) = J2n^(z~ ~ ^n)> w e n n < i

({Jdz-dz-A+(X,z-)A+(y,z-)))t

= Jdtn{Z)^v{x-y),

(13)

where we have denned the dipole cross section
2

(l - e^*-*)) .

(14)

We now consider the gluon number spectrum (11) in two limiting cases. 2.1. N=l opacity limit)

approximation

(single hard

scattering

This limit is relevant for a small target with weak target fields A+ = O(g). In this case the projectile can scatter at most once. Thus the adjoint Wilson lines can be expanded to leading order in the single scattering potential a+(q). Using (13), we find A

JJ—((T,[W

^X)WA(y)]))t^l-^-Jdin(Oa(K-y).

(15)

The corresponding gluon number spectrum reads x

A r P rod(k)

8TT

= a

s

/

d£ n(0 ^x ^y e.tk-(x-y)x 2 y• 2y

. C , / d e n ( 0 / | ^ | a + ( q ) |


2

+ a{y) - a(x - y)

^ _ _ .

(16)

Differentiated with respect to the momentum transfer q, this result is the well-known Gunion-Bertsch radiation cross section 22 for the gluon radiation

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Wiedemann

cross section in quark-quark scattering

daGB dlnxdqdk

CACHS (q)l2 k 2 ( k - q ) 2 7T2 '-Viuuv-wi

(17)

Integrated over the transverse momentum exchanged with the "target" quark, this radiation spectrum can be written in terms of the dipole cross section (14), daGB d In x dk

4CA

a.f

d(j q2 2 2 (2 TT) k (k - q) 2

-TJ T ^ I ^ Q C D W

exp{iq-r}

(18)

For a scattering on a static source described by a Yukawa type potential, asM* Although perturbatively gluons are massone has c IQQCD (q)l2 oc less, the screening mass M is frequently introduced to mimic the medium effects which cut off the infrared divergence24. The factor a3 in this expression accounts for the fact that the gluon field in eqs.(13,14) is emitted by the single particle source. It is the presence of this coupling constant that justifies the expansion of the Wilson lines to leading order in a. It is interesting to compare the expression eq.(17) with the high energy limit of the Bethe-Heitler photon radiation cross section in QED, daBH d In x dq dk

c

aem x2q2 em ,| x 2 + a k2 n, (k - „^i xq)2 nT l QEDV(q)l ,.a

(19)

T h e absolute normalization of a+ in eqs.(17) and (16) is different, which accounts for the different constant factors in these equations.

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and Parton Energy Loss

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Here, k is the transverse photon momentum and x — cjy/E the fraction of the energy carried away by the photon. Again, if integrated over the transverse momentum transfer q from the target, this radiation spectrum can be written in terms of a dipole cross section daBH

. 2a

d l n ^ k = ~ ™J

f dqdr

q2

(2^k»(k-q)»ffQBD(")C

4

•

(20)

The radiation cross section (17) and (19) for QCD and QED differ only by their ^-dependence. The QCD radiation is flat in rapidity, whereas the QED radiation cross section peaks at projectile rapidity x = 1. The physical reason is that in QCD, the radiated gluon is charged. The distribution of gluons in the incoming quark wave function is flat in rapidity. Thus when a gluon scatters directly off a scattering center into the final state, the rapidity plateau is produced. It is the non-abelian diagram contributing to (17) which dominates the QCD radiation spectrum. This is also the reason why the explicit Casimir factor in (17) is the adjoint and not the fundamental one. In QED on the other hand only the electron can scatter directly off a source. This scattering is most effective for electrons with small fraction of longitudinal momentum. Such scattering events result in final state photons which carry most of the initial longitudinal momentum, and thus the photon spectrum is peaked around x = 1. In configuration space, the different z-dependence of the QCD and QED radiation spectrum is reflected in the different arguments of the dipole cross sections: the QCD dipole cross section is tested at transverse distances r but the QED one is tested at xr. For an intuitive understanding of this behavior, consider the incoming projectile electron or quark in the light cone frame as a superposition of the bare particle and its higher Fock states, |projectile> = |g) + |
(21)

If all Fock components interact with the scattering potentials with the same amplitude, then the coherence between these amplitudes is not disturbed, and no bremsstrahlung is generated. The radiation amplitude depends on the decoherence between the different Fock state elements, i.e., on the difference between the elastic scattering amplitudes of different fluctuations. This difference is characterized by the impact parameter difference between different Fock state components. It can be estimated from the transverse separation of the \q) and \q^) fluctuations. On a characteristic time scale

202

A. Kovner and U. A.

1

/

fe2

\

Wiedemann

_ 1

=

AS ( 2x(i-x)E ) se ^ ^ * n e transverse energy of the two-particle Fock state, the quark, gluon or photon move in the impact parameter plane by distances k± 1 (1 - x) x X 7 > x x „ « _ _ a n £ — , (22) (23)

^'^^O^XJEXE^I:-

Since the photon does not carry charge, the transverse distance x±^ plays no role in the decoherence of QED Fock state components, while the distance x± g plays the dominant role in the corresponding QCD process. The QED and QCD dipole cross sections thus measure the transverse separations xr

^ QED: y QCD: g

(l-x)r

^f^~~ ~~H e e q q

To leading order 0(l/x), the cross sections in (18) and (20) thus differ by a factor x in the argument of a(r). Based on the above picture, Nikolaev and Zakharov proposed 48 to write the dipole cross section for the qqgconfiguration beyond leading 0(l/x) as
+ a((l

- x) r)} - - a{x r ) ,

(24)

where the prefactors account for the color representations of quark and gluon. Due to this distinct x-dependence the QED expression cannot be obtained in the eikonal approximation. The eikonal limit where the energy loss of the electron is neglected is equivalent to the limit x = 0, where according to eq.(19) no photons are emitted. In our wave function formalism this arises since the components of the wave function with no photons and with one photon rotate with the same phase during the propagation through the target, if the energies of the electron in both components are assumed to be the same. Thus in QED the eikonal approximation leads to the vanishing inelastic cross section. In QCD this is of course not the case, since the gluon component of the wave function scatters most efficiently.

Gluon Radiation

2.2. Multiple

soft scattering

and Partem Energy Loss

limit (dipole

203

approximation)

We now consider the opposite limit, when the projectile undergoes many scatterings. Assume that these scatterings are independent in the sense that the vector potentials of the different scattering centers in eq.(12) are uncorrelated in color space. Then the dipole cross section in eq.(15) exponentiates. We also assume that the cross section is dominated by the low momentum transfer in each individual scattering. In this limit the dipole cross section
«Tr [WAHy)WA(x)]))t

« exp \-±£| d £ n ( 0 < x ( x - y) ACF (x-y)2CA 8 CF Ql

exp

(25)

Here, the saturation momentum is dq

Ql

(26)

!4(q)lV,

which in the model (12) of the target fields characterizes the average transverse momentum squared transferred from the target to the projectile. Using the average (25) to calculate the produced gluon number spectrum (11), we obtain 34 iVprod(k) =

^Ljd*dyeik^^.

X2y2

2TT

x | 1+ e

•(x-y)

ot3 CF 2

CF

WCAQI

2

\l

„2

£&Ql 2CF

dqe

C,

Ql _ e-y2

K£«2

_2

C Q

A

k2 (k - q) 2

(27)

Expression (27) contains again the Gunion-Bertsch radiation cross section (17). In the dipole approximation (25), however, the entire medium acts coherently as a single scattering center with a Gaussian momentum distribution

K(q)l 2

JCp.

ex e

2

(28)

204

A. Kovner and U. A.

Wiedemann

There is an interesting attempt to generalize (27) to the case of a dense projectile realized in nucleus-nucleus collisions 3 6 . The eikonal approximation described in this section is valid as long as the average momentum transfer from the medium is much smaller than the longitudinal momentum of the produced gluon. Only in this limit all scattering centers contribute coherently to the production amplitude. The purpose of the following is to generalize (27) to the kinematic region where this is not necessarily the case, and incoherent scatterings also play an important role. 3. Gluon Radiation beyond the Eikonal Approximation In the eikonal approximation, the Wilson lines (3) determine the S-matrix for the scattering of partonic wave functions. In this section, we derive the leading finite energy corrections 0 ( 1 /p~) to this high energy limit. We explain the form of the target averages including 0(l/p~) corrections and we give two applications: the photo absorption cross section and the medium-induced gluon radiation spectrum. 3.1. Wilson Lines for Non-Eikonal

Trajectories

We consider the AT-fold scattering amplitude of a high energy quark. In the light cone gauge A — 0, it reads F(x0,p)

= \ ~ N=O

PA

P?\

PA

PA

!

!

!

©

©

©

_ "^TT-n ( fdp+dp-dpi

x

( _ j 7 - A+(xt,Xi))

P'a

M • •

*

*

1

1

C^D ® f

AT'

Aw

/rv .

eixt-(~pi+i-pi)+ixi-{pt+i-P^eiXi
xeix°-pi v(p).

(29) +

Here, the target gauge fields A = A+Ta are path-ordered along the trajectory of the quark; we suppress color and spin indices on the quark propagators and the spinor v(p). The additional phase factor elx°'Pl in (29) is

Gluon Radiation

and Parton Energy Loss 205

useful in order to connect the multiple scattering diagram to the vertex of a more complicated amplitude by integration over xo- In the absence of scattering centers, (29) reduces to a free quark wave function elx°'p v(p). We approximate eq. (29) to leading order O ((l/p~) ) in the norm and to subleading order O ( 1 / p - ) in the phase. To this order we can neglect the dependence of the target fields on xf, as indicated in (29). This is because the incoming quark propagates in the negative z direction and thus only feels the fields at x+ = 0. The integration over xf and p^ is then trivial: all propagators have the same component p~ = p~~. To simplify (29) further, we use

1

=S.».(P)«.(P)+^1>

pz — mz

p—m

(30 )

2p

and neglect the last term to leading order in the energy l / p ~ . The spinor structure can be simplified: us(p) ( - r y • A) us,(p) ~ p~ A+ 6SS,.

(31)

With these simplifications, eq. (29) takes the form oo N

^•"-tn"(/^/*r*Srr(p? + m?)/px (iA+(xi,xr))

+ it

e^-fcr-i-Oe-to-txi-i-xirt

xeixi-p-eix-p+-ixN.pv(py

(32)

The spatial ordering of the scattering centers along the trajectory is fixed explicitly by performing the p^"-integrations. Neglecting the mass term, we find a product of 0-functions times phase factors dpj fdpt_

i 27r pf (p? + m2)/pJ 2TT •Bill

-

e*: - i

^

eipf(xi_1-xi

)

+ ie

e^r-^r-i)-

(33)

Inserting this expression into (32) and performing the transverse p j integration, we obtain the basic building block of the formalism, the free

206

A. Kowner

and U. A.

Wiedemann

light-cone Green's function n i -i - \ f dPi ^ ( x . r - i " •x-) G o t X i - i . i j . ^ X j , ^ \p ) = J — 3 2 e Pi

-»Pi-(xi_i-Xi)

\2"

ip (XJ - xi - l ) 4(i,r-xr_1) _

p

=

~

'' dfr 2 (0

(34)

- i

Here, r = dvjdz" and the boundary conditions on the free path integral are r(x^) = x,. The limit x~ — x~_x —> 0 of (34) is equivalent to the highenergy limit lim G 0 (x i _i,a;r_ 1 ;xj,a:~|p _ ) = 5 2 ( X J _ X - X j ) .

(35)

p~—>oo

The free Green's function (34) evolves the free plane wave eIX* p _ I X ; P to the position ( X J _ I , a ^ j ) . The sum of the TV-fold scattering amplitudes (29) reads oo

F(x0,p)

/ N

= e i : c o p " YlV\Yl JV=0

„ da; _ x ! J ^ iG o(xi_i,a;-_ 1 ;x i ,a;-|p-)

\ i = l ^

= eix°-p~dxG(xo,Xo;x,x-\p-)e''TX

iP

''*v(p),

(36)

where the full Green's function is G(xo,Zo;x,:r _ |p~) =

/"Pr(£)exp

^r>«>

W([r];x 0 , z ) = P exp i f

^([r];^-^-),

dC^+(r(0,0

(37) (38)

Jin

These expressions give the leading 0(l/p~) corrections to the phase of the eikonal Wilson line (3). The path integral in (37) has a very simple interpretation. It describes the quantum mechanical particle which, while propagating through the medium can move in the transverse plane. The free

Gluon Radiation

and Parton Energy Loss

207

transverse motion is governed by the kinetic term ^ - r 2 - a two-dimensional light-cone hamiltonian with "mass" given by the total energy p ~ / 2 of the projectile along the beam. During the motion the wave function of the particle also acquires a phase given by the Wilson line along the trajectory. The infinite energy limit corresponds to the classical limit, where a single straight classical trajectory dominates the sum over paths. In this limit the Green's function (37) reduces to the eikonal Wilson line in (3), lim G(-K0,XQ-,r,x^\p~) = W(r;xQ ,x~)<52(x0 - r ) .

(39)

p~ —*oo

3.1.1. Non-abelian Furry approximation (target field Ao) The multiple scattering diagram (36) contains the leading order 0(l/p~) corrections to the phase of the eikonal Wilson line, but the norm of (36) is given to O ((l/p~)°) only. For a scalar quark that would be the only important correction. For a spinor quark however this is not enough if one wants for example to discuss emission of on shell gluons. It turns out that the leading order of this propagator vanishes when connected to an emission vertex of a real gluon. To get a leading order result for this process one has to keep order 0(l/p~) corrections to the first propagator 1/(7 • p\ — m). We now discuss the non Abelian Furry approximation which addresses this issue. We also change our notations slightly in accordance with literature. The target field will not be described by a large ^4+-component but rather specified by the Gyulassy-Wang model 24 ' 53 as a collection of scattering potentials of Yukawa-type with Debye screening mass M, 00

^ W = ^ ^ ^ ( x ) T

a

^(x)=^(x-xi)<5°°i.

,

(40) (41)

Here, the i-th scattering center is located at Xj and exchanges a specific colour charge d a = a<. i n t u i t i v e l y one thinks of large A+ as being a more appropriate description of a target which itself is moving in the positive X3 direction, while large Ao being more appropriate for a static target. Technically, however there is no difference between the two sets of vector potentials as long as the projectile is very energetic and its x+ coordinate does not change during the propagation inside the target. In this sense switching to large Ao can be considered as a different gauge fixing.

208 A. Kovner and U. A. Wiedemann For this target field (40), the calculation of the multiple scattering diag r a m (29) proceeds in close analogy to section 3.1. T h e components ~f~A+ in (29) are replaced by JQAQ. T h e role of the p+-integrals (33) is played by the pj-integrals which lead to an ordering along the longitudinal direction. To simplify the spinor structure, one uses (,p)7o ^ £ ( 7 0 — 73)70 which can be iterated, ( £ ( 7 o - 73)70)" v(p) = 2 " " 1 En ( 7 o - 73)70 w(p) =± 2 " En v(p). In comparison to section 3.1, the only new feature of the present discussion is t h a t we approximate the first quark propagator to 0(1/E). In a coordinate system with p || n and for a. = 70 7, we have

( Here, the operator ^

2 \

2~E ~~ 2 ~ B ) a • ( p - p n) .

p (70 - 73)70 + la"g

—•

2

(42)

acts on wave functions of the form exp {—ipi • y } .

This structure leads to the differential operator £>i Dt =

l-i

a •V

a • (pj -

2E

npi)

(43)

2E

which carries the transverse and longitudinal spinor structure to order 0(1/E). To lowest non-trivial order 0(1/E) in the norm and phase, the multiple scattering diagram (29) takes now the form 5 8 * „ ( y ° , y , p ) = eiB*0-i*v-

=1

DF(y, PA

P,.a,

(44)

p) u ( p ) , P 4 .a 4

W

%•%

P.a

N=0

©

©

©

CED ®

Here, the differential operator D acts on the transverse wavefunctions F which for far forward longitudinal distances XL satisfies plane wave boundary conditions Foo(x-,XL,Pi) F(y

= e x p ^ -ipi

-x + i—^-XL 2 Pi

,VL,Pi) == / j d x G ( y , ? / L ; x , a : z / | p ) F 0 0 ( x , a ; / / , p ) .

(45) (46)

Gluon Radiation

and Partem Energy Loss 209

The full Green's function in this expression is given in close analogy to (37) by the sum over paths weighted with the Wilson lines. It satisfies the recursion relation involving the free non-interacting Green's function Go, G(r,z;r',z'\p)

=

G0(r,z;r\z'\p) Z

- * / d e / d p G o ( r , z; P,£|p) M{p,0 G0(p,£; r', z'\p) z XL

+~P

XL

dtli / d£,2 J

*L

dpidp2G0{r,z\px,ix\p)iAa{p\,ii)

Zl

xGo(px,tx;p2,&\p)iAo(p2,b)G(p2,&,r',z'\p),(47) G0(v,z;r',z'\p)^—-£ 2ni(z' - z)-exp ( i p2{z' ( r ~- 'z)T ) •

(48)

The path ordering V in (47) ensures that the potentials AQ are ordered along the longitudinal axis. This formula is in fact a direct non Abelian generalization of the corresponding expression in Quantum Electrodynamics. Consider the solution of the Dirac equation for an electron in a spatially extended scattering potential C/(x) d U(x) — m 70 + i a • V ^(x,p) = 0 . (49) dt In the high-energy approximation in which we neglect corrections of order 0(U/Ei) and 0(1/Ef), the solution to this Dirac equation is the Furry wave function 33 - 57 ! *F(x,

Pi)

= eiEtt-i\Pi\z £,. F

( X ) p<) y ( p i )

.

( 5 Q )

This solution has the same form as the non-abelian multiple scattering diagram in (44). The Green's function used to evolve the wave function F is now given in terms of the scattering potential £/(x), G(r,z;r',z'\p)

=

J Vr(Z) exp I j d$ |f 2 (o-it/(r(o,e)

(51)

Again, it describes the quantum mechanical evolution of the projectile particle in the plane transverse to the beam.

210

A. Kovner and U. A.

Wiedemann

3.2. Target averages for non-eikonal

Wilson

lines

In section 2, we discussed the target average of two eikonal Wilson lines. In section 3.1 we showed that the 0(l/.E)-improved generalization of eikonal Wilson lines are given in terms of the path integral (37). We can compute the target average of two such Green's functions expanding both Green's functions in powers of AQ according to (47) and then resumming the series. We assume that the different scattering centers in (40) are uncorrelated and that the projectile can interact with a given scattering center only once (elastically or inelastically). For generality we take the two Greens' function to act on initial states with different energies (1 — a)p and ap, respectively. This is a generic situation in the application to deeply inelastic scattering, where the initial quark and anti quark share the energy in the initial state. This will also be useful for us in the following, since we will need to calculate phase differences between the wave functions of an initial state parton and a final state parton with a different energy. Performing the target average as in (13), the target average for two adjoint Green's functions59 leads to the appearance of the dipole cross section e a, ((G{ruz;T[,z'{l-a)p)\G(T2,z-y2,z'\ap)))t r2(z')=r2

ri(z')=r'i

/ ri(z)=ri

Pri

/

X>r2exp

Ijf d£f (arl + U-«)*?)

r 2 (z)=r 2

+*-n(0a(r1(0-r2(£))

(52)

With the help of the transformation ro(0 = ( l - o ) n ( 0 + ara(0,

(53)

r6(0 = n ( 0 - r 2 ( 0 .

(54)

the target average can be written as {(G(Ti,z-yi,z'{l-a)p)\G(r2,z;r'2,z'\ap)))t = /Co(ra(z')> z'\ ra(z), z\p) K(rb(z'), z'; rb{z), z\pa(l - a)) e

(55)

In the rest of this article we use a to denote the cross section for scattering of dipole consisting of two adjoint charges. It differs from a defined in (13) by the factor ^A.

Gluon Radiation

and Parton Energy Loss

211

Here, the new path integrals are of the form i/j, (r' — r)

Ko(r',z';r,z\u)

2iri(z' - z) exp

(56)

2{z'-z)

Z

/C(r',z';r,z\\i) = / Vr exp i / d£ | r 2 + z l n ( O a ( r )

(57)

In the high-energy limit, this path-integral coincides with the average (26) of two eikonal (fundamental) Wilson lines, lim IC(r',z';r,z\(j,)

= exp

- 2 » K M r ) 6^(r'-r).

(58)

The path-integral (57) determines the 0(l/E)-corrections to this eikonal average. Quite generally, it is seen to compare the relative distance (54) between the trajectories of the two generalized Wilson lines as a function

of£. 3.3. The medium-induced

gluon radiation

spectrum

We consider now the medium-induced radiation amplitude for a quark of color a splitting into a quark and a gluon of colors b and c respectively. The diagram for arbitrarily many M-, L-, and TV-fold scatterings of the in- and outgoing partons is

©

©

©

"/"' a

"m

Pi

! P.m

©

»J

"2

PlJ

Pi!

©

a, ;

©

Pn

K

/ b[

Pzi

P2!

©

;

©

W

b„

b

PJJ

Pi.

! Pj

©

(59)

These multiple scattering contributions are summed up in the Green's functions (37) which include the 0(1/E) corrections to the eikonal Wilson lines.

212

A. Kovner and U. A.

Wiedemann

The corresponding radiation amplitude reads Ma^bc

= -ifdyi

x e

eiiiy< e - ^ ' l J d2y f d 2 X l d 2 x 2 d 2 x s • "1 i •pl - « 5 ^ - S l + » 2^*1

»Xl-pi-lX2-p2-tXs-k

• k2 -l2UX<

G ^ ) c ( y ; x s | W ) G ^ ( y ; x 2 | P 2 ) I (60)

^l^i\y\pi)ty^)aibl where the interaction vertex reads

v1'(pi)£>i70e • iD2v{p2)

(61)

•

Here, the differential operators £>, stem from the Furry approximations of the quark lines, see eq. (43). The gluon radiation amplitude (60) is a typical example for which the order 0(E°) term in the norm of the Green's function vanishes. As a consequence, the derivatives £- which are subleading in the Green's function itself, appear to leading order in the transition amplitude and in the radiation cross section, d3a(in)

Ols

d(lnx)dk

(2TT)2

4£2(l-a;)2

_

9 Re /

- Jh

(2TT)2 x2 E\ (1 - xY

Jdp(\Ma^bc\2) dyi

J«_ Jz

/ Jyi

dy{ iq(vi-vi)-t\yi\-z\vi\

/ dy dy dxi d5ii dxg dxg dx 2 e ~ , k ' ( x 9 _ * 9 ) . x( ^ ( x u y l p O T ^ V

^G^(y;xfl|u,)j

G ^ ( y ; x 2 | p 22)

d

G$(x2;y|p2) ^ _ G - ( x s ; y | W ) J 3 ^ G ^ y ; ^ ) ^ ) •

(62)

Graphically, this can be visualized as C:

lJ

-^ a

XL

hi a, b, yL

i—*

yL

d b XL

io Pi frr*—r-H: ib b,:a, XL

yL

i, 1—•> a

yL

XL

,ggs

As discussed earlier, the softer parton, which in this case is the gluon, rescatters much more efficiently. Keeping only the leading 0(x) terms the

Gluon

Radiation

and Parton

Energy

Loss

213

final result for this radiation cross section reads 59 J3

(in)

1

n

2

d(\nx)dk

rz+ 2

(2n) UJ

Jz

/««+ <*"«) °^

xjdue-^e-*

rz+

Jyi A . . A /C( y = o, m- u, ft|w). (64)

The e-regularization in this expression has been introduced to suppress the contributions of gluons emitted at t = ±00. With this adiabatic switching on (off) of the interaction at asymptotically early (late) times, the state of a single quark is an eigenstate of the Hamiltonian at t —» ±00, and thus the medium induced emission of an extra gluon is indeed a transition to a different eigenstate. One could in principle perform the calculation without introduction of this explicit regulator. However in this case one must take care to calculate the transition amplitude between eigenstates of the interacting Hamiltonian and thus take into account 0(as) corrections to the wave functions of the initial and final states. The initial "dressed quark" state in such a calculation will contain the admixture of a \qg) component as discussed in Section 2. To calculate the radiation spectrum one then needs to subtract from the final state wave function the component which overlaps with the "dressed quark" wave function. Introduction of the regulator makes the calculation technically much simpler. However, care is needed since the limit e —* 0 in (64) does not commute with the longitudinal integrations. In practice, one can remove the regularization e.g. by splitting up the longitudinal T/L- and j/L-hitegrations of the radiation cross sections into six parts, z + z+

0

0

0

L

0z+

L

L

L z+

II-INHHHHI-

z-

z- VL

VL

z-

0

z-

L

0 vt

0

L

z+ z+

L

™ VL

and to write the radiation cross section as a sum of the corresponding six contributions ^

^

= ^C

F

(/

1 +

/

2 +

/3 + / 4

+

/5+/a).

(66)

This allows to remove in all terms Ij the e-regularization of (64). In what follows, we are interested in the gluon energy distribution of a quark produced in the medium at position z- = 0. In this case, the contributions -fii h, h to (65) vanish. The remaining three terms denote the direct production term I4 for gluon emission inside the target in both amplitude and

214

A. Kovner and U. A.

Wiedemann

complex conjugate amplitude, the destructive interference term I5 which is the characteristic interference term between gluon emission in and beyond the target, and the hard vacuum radiation term IQ,

l4= I 5 =2Re

LO

h= (67) The contribution IQ can be associated to the medium-independent contribution to the hard radiation off a nascent quark jet propagating in the vacuum 59 . This is a consequence of the leading 0{x) approximation in which only the rescattering of the radiated gluon contributes, and the rescattering of the hard quark in (67) is neglected, 1 (68) k2 This term may be viewed as the medium-unaffected DGLAP part of the gluon radiation. In the following, we shall discuss the medium-induced modification of the gluon radiation spectrum, determined by I4 and I5. For notational convenience, we shall not denote explicitly that the mediumindependent term IQ is subtracted.

h=

4. Properties of the medium-induced gluon radiation spectrum Two approximation schemes for the radiation cross section (64) have been explored recently: the dipole approximation 3>61>6>59 and the opacity expansion 59,26 . In the dipole approximation, the medium-dependence results from the multiple soft scattering of the projectile in the spatially extended matter. The opacity expansion can be related to an expansion in the effective number of scatterings and allows for hard momentum transfers from the medium. Here, we discuss for both approximations the inclusive

Gluon Radiation

and Parton Energy Loss

215

energy distribution of gluon radiation off an in-medium produced parton. Integrating over the transverse momentum of the radiated gluon up to |k| < xw, we have 59 - 60

duj

(2TT)2LO2

Jio

7T— • -r— /

Jyi

J

J0

Drexp i / at— r — /„, - v

n(fl
(69)

»w

The radiation off hard quarks or gluons differs by the Casimir factor CR = CF or CA, respectively. The properties of the medium enter eq. (69) by the product of the density n(£) of scattering centers times the dipole cross section a(r) which determines the strength of a single elastic scattering, see (14). Note that the density can be time-dependent as is for example the case in expanding medium.

4 . 1 . Multiple

soft

scattering

Consider a medium-dependence introduced by arbitrary many soft scattering centers, rather than a few hard ones. In such a medium the projectile performs a Brownian motion in transverse plane; the transverse position of the projectile in configuration space changes rather smoothly and small relative distances in the dipole cross section are important. The integrand of (64) has its main support at small transverse distances r = |r| which allows to write the dipole cross section to logarithmic accuracy as 61 ' 64 n(0
(70)

Here, <J(£) is the transport coefficient4 which characterizes the transverse momentum squared fi2 transferred to the projectile per mean free path A. For a static medium, it is time-independent,

In the multiple soft scattering approximation, the transport coefficient q and the in-medium pathlength L are the only informations about the properties of the medium which enter the gluon radiation spectrum (69).

216 A. Kovner and U. A. Wiedemann

4.1.1. The harmonic oscillator approximation (static medium) In the so-called dipole approximation (70), the path integral (57) in the gluon radiation cross section (64) is equivalent to that of a harmonic oscillator 64 ' 60 £ o s c (y, Vi; u, m IM) - —{ e*P [iAB{y2 + u 2 ) - 2 i A y • u] , A

B

-^Ay-)'

=^ A y ) ,

(72) (73)

with complex oscillator frequency

a = L}*J±.

(74)

y/2 V 2u)

'

V

Explicit solution for parton produced inside the medium: Inserting K,osc into u>x;, the medium-induced gluon energy distribution duj becomes

^ = rfC*(/4 + /5)' dl

(75)

a

/•Z7T

/ Jo

h=

/-X"

difk I Jo

k± dk± Ij,

(76)

where the terms Ij are defined by eq. (67). Their explicit form is h = 47rRe / dzi \ Jo Jo

dz2 ( , „ . , „.._, V2 1-cosh[(l-M)Az]

4- - P-2MZ +

h = 47rRe / J0

sinh[(l+i)Az/2]/2 _ £ \

4

N2J '

K

(77)

'

dz ( —- . . .,, -—r \2 sinh[(l + i)z\ (1 + i) e~{1+i)M*

tanh[(l+t)z/2]

\

+

4 sinh[(l + i)z/2] cosh[(l + i)z/2] J ' F = z\ - 2t coth[(l +1)Ai/2] [(1 + i)(M 2 + 22) + coth[(l + t)Az/2]] , AT = i 2 sinh f ^ ( z i - za)") + ( ! - * ) cosh 0^-{z\

(?8)

(79) - z2)] • (80)

Gluon Radiation

and Partem Energy Loss

217

Here, the longitudinal distances are written in dimensionless units, rescaled by the modulus of the oscillator frequency |fi|, z = V2\il\z,

L = V2\tt\L.

(81)

The upper integration limit of the k± integral takes in dimensionless variables the from Mc = -

^

=

.

(82)

^2UJV2\U\

Totally coherent limit for parton in initial state: Assume that a parton is part of an initial state. In this case, prior to scattering on a target, the parton has already a fully developped wavefunction with additional virtual gluons. In the scattering process, these virtual gluons can be freed. As a consequence, the gluon energy distribution radiated off this parton has to include radiation vertices at arbitrary early times. One needs to include all six terms Ij of (66) in calculating eq. (75). Explicit expressions for this case are given in Ref. 6 0 . Here, we consider only the high-energy limit of the resulting expression. Since fi oc -4j, this highenergy limit of the radiation spectrum corresponds to the leading order in 0(L = y/2\il\L). It reads d3j(in) dwdk

as4nCR 2

7r

qL

f d2q

_^i

q2

J (2n I&'-*
(»)

This is the totally coherent limit of the gluon radiation spectrum for an incoming parton. It is the Gunion-Bertsch gluon radiation spectrum (28) for a target momentum transfer q which is acquired by the transverse Brownian motion of the incoming projectile, 4.1.2. Qualitative estimates ofu>^~ vs. quantitative calculations Important properties of the gluon energy distribution can be obtained from simple estimates. Their range of applicability, however, depends on phase space constraints which are not as easy to estimate. Here, we compare qualitative arguments to numerical calculations 60 ' 52 . Qualitative arguments:10 We consider a gluon in the hard parton wave function. This gluon is emitted due to multiple scattering if it picks up sufficient transverse momentum to decohere from the partonic projectile.

218

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Wiedemann

For this, the average phase 9? accumulated by the gluon should be of order one, (84) Thus, for a hard parton traversing a finite pathlenth L in the medium, the scale of the radiated energy distribution is set by the "characteristic gluon frequency" Uc = 2 9 L

(85)

•

For an estimate of the shape of the energy distribution, we consider the number iVcoh of scattering centers which add coherently in the gluon phase (84), k\ ~ JVcoh/A Based on expressions for the coherence time of the emitted gluon, tcoh - p - - A / | and Ncoh — — for the gluon energy spectrum per unit pathlength

one estimates

dl dl 1 scatt 1 a u-7—r - Tr— u - T^s ~ots\ -. (86) audz iVCoh duidz tcoh V u> This l/y'w-energy dependence of the medium-induced non-abelian gluon energy spectrum is expected for sufficiently small UJ < uc. Quantitative analysis: The gluon energy distribution (69) depends not only on uc, but also on the dimensionless "density parameter" 51 R=^qX2L3,

(87)

A finite value of R implements the kinematical constraint k± < x w m the transverse momentum phase space of the emitted gluon, see (69). The kinematical boundary is \ = 1. This constraint is neglected in the argument leading to the l/v^-energy dependence of (86). The limit R —-> 00 removes the kinematical constraint from (69) and coincides with the original result of Baier, Dokshitzer, Mueller, Peigne and Schiff4, dl lim u>— = ft-»oo

dw

2asCR

In

COS

TT

(88)

(1+i)

The limiting case for small w is as expected from the estimates (86), dl hm uj-r- ~

2a3CR J < ,

V ^ ,,,0

for for

w<wc, UJ > UJC .

9

(89)

Gluon Radiation

LU LU-2

(b)

< -6

and Parton Energy Loss

L 2 -law

,

1 — i\,C = 0.1 Irrr3 — rioC = 0.5 f m 3 — r\jc = 1.0 fnv 3 — njjc = 2.0 fnrr3

•e -10 0.5

^——^

219

1

1.5

2

2.5

L.og(U1 fm)

Fig. 1. (a) Dependence of the medium-induced radiative energy loss (AE) on the inmedium pathlength L for E = 100 GeV. The transport coefficient is defined here as | = no C. (b) Double logarithmic presentation of (a). Figure taken from 6 0 .

The average parton energy loss is the zeroth moment of this energy distribution dl asCR (AE)R. duj{ (90) lim <xL2 du 2 fi-»oo / This is the well-known independence of the average energy loss 3-4-62. The numerical calculation in Figure 1 shows deviations from the limiting case (90). [In Ref.60, an erroneous prefactor Nc was included in the definition of w ^ ; hence, the values for AE shown in Fig.l and 3 are a factor 3 too large.] In general, the strong increase of the average energy loss AE with in-medium pathlength indicates the strong sensitivity of partonic energy loss to the geometry of the nuclear collision. The average energy loss AE is the first moment of the energy distribution (69). Fig. 2 shows this distribution for finite values of the density parameter R. At sufficiently large gluon energy, the distribution approaches for any value of R the BDMPS limit (88) . Below a critical gluon energy u>, however, the finite size gluon spectrum is depleted in comparison to the BDMPS limit. This suppression results from a phase space constraint: gluons are radiated on average at a characteristic angle 52 ,

e

fci

-3/2

/wq

<1.

W

(91)

This depletes the gluon distribution for gluon energy Q below

iV /3 Ur.

\RJ

(92)

220

A. Kovner and U. A.

Wiedemann

\ BDMPS (R=oo)

w/o) c Fig. 2. The energy distribution of radiated gluons ui^ matical constraint R = wcL. Figure taken from 6 2 .

for different values of the kine-

This independent suppression is seen clearly in Fig. 2. 4.1.3. Angular Dependence of the gluon energy distribution Quantum Chromodynamics is a finite resolution theory; the energy of a single quark or gluon is not a measurable quantity. One rather measures the fraction of the hadronic decay products of a single quark or gluon which are radiated within a certain jet opening angle. Thus, the angular dependence of the medium-induced modification of gluon bremsstrahlung is important. The angular dependence of w ^ can be studied by varying the kinematic cut-off x w of the transverse momentum integration 7>60>8. For fixed characteristic gluon energy wc = \qL2 and fixed dimensionless parameter \qLz, the radiation spectrum w ^ inside a given angle 0 = arcsin(x) depends on the parameters wc and R = y^hql?. From Fig. 2 one thus concludes

Gluon Radiation

and Parton Energy Loss

221

immediately that the harder gluons are emitted more collinear. From the gluon energy distribution inside a given cone opening angle, one obtains the medium-induced additional average energy emitted outside a cone of given opening angle 0 , AE AE AE (93) ( 9 = arcsin(x)) = ~g"(x ~E~ D-^(x). As seen from Fig. 3, ^£(0) does not decrease monotonously with in—.0.03

©

*

"

m

— ••• -—

nno

001

1 *

\

1

%

gO.14

•

HI

L = 4fm L = 7lm L=10fm L=13lm

070.12 UJ <_ 0.1

n0C = 0.1 frrr3 E = 100 GeV

\

0.06 0.06 0.04 0.02 0

)

10

20

30

40

50

60

70

80

90

I

—• L = 4fm

\ \ \

--- L = 7fm - - L=10fm — L=13fm

.

r\,C = 0.5 frrr3 E = 100 GeV

:\\ ' \\

£^_ 10

20

30

40

70

50

0 in °

80

90

0 in °

S " 0.3

— — -—

go.25

<

"

0.2 0.15

A 1

0.1 0.05 0

L = 4fm L = 7fm L=10fm L=13fm

(TQC = 1.0 frrr3 E = 100 GeV

>\ \\

— — -—

L = 4fm L = 7fm L = 10fm L = 13fm

rinC = 2.0 frrr3 E = 100 GeV

*"***^%>«_ 10

20

30

40

50

60

70

I

80

90

0 in "

Fig. 3. The fraction of the total radiative energy loss AE/E emitted outside a jet cone of fixed angle 0 . Here, the transport coefficient is defined as q = 2 n n C . Figure taken from 6 0 .

creasing 9 but has a maximum at finite jet opening angle. The reason is that the radiative energy loss outside a cone angle 9 receives additional contributions from the Brownian fc_|_-broadening l •1 — (94) 2 K k (k + q) 2 ' of the hard vacuum radiation term IQ . Such contributions do not affect the total k-integrated yield ^ ( 9 = 0), since they result only in a shifting of

222

A. Kovner and U. A.

Wiedemann

the transverse momentum phase space distribution of the emitted gluon. However, this shift in transverse phase space shows up as soon as a finite cone size is chosen. We conclude from Fig. 3 that the total k-integrated radiative energy loss^P(0 = 0) is not the upper bound for the radiative energy loss outside a finite jet cone angle ^ ( 0 ) . Depending on the transport coefficient and the in-medium path length, the latter can be larger by more than a factor 2.

4.1.4. Harmonic oscillator approximation (expanding medium) In nucleus-nucleus collisions at collider energies, the produced hard partons propagate through a rapidly expanding medium. The density of scattering centers and thus the transport coefficient q(£) is expected to reach a maximal value <jd around the plasma formation time £OJ and then decreases rapidly due to the strong longitudinal and - to a lesser extent - transverse expansion,


05)

Here, a = 0 characterizes the static medium discussed above. The value a = 1 corresponds to a one-dimensional, boost-invariant longitudinal expansion and approximates the findings of hydrodynamical simulations. The formation time £o of the medium may be set by the inverse of the saturation scale pSat 19 and is « 0.2 fm/c at RHIC and « 0.1 fm/c at LHC. Since the time difference between the formation of the hard parton and the formation of the medium bulk is irrelevant for the evaluation of the radiation spectrum (69), one can replace in (69) the production time £o of the parton byO. For a dynamically evolving medium of the type (95), the path-integral (57) in (69) is the path integral of a 2-dimensional harmonic oscillator with time dependent (imaginary) frequency fi2(£) = ^ 7 a n o - mass UJThe explicit solution can be written in terms of the variables z(£) =

2iM(^0

(£) 1 / 2 , / and v = ^ :

lC(r(yi),yi;r(yi),yi\uj) 2mD(yi,yi)

exp

5

= 2iD(yi,yi)

(96) (cir(j//) 2 + c2v{yi)2 - 2r(j/j) • r(yi))

Gluon Radiation

0.5

::

0.4

:

0.2 0.1

i / /

0

•i •*

0.6

if

r./ J/ J

* ' * "»

\

1

* /* \/ * */ * A *»

V \

\

R=4000

0.5

l

*J

. ! * 1 \ '* i \

r //

0.3

0.7

' * /\ •• •; •V ; ?.

223

\ \ \ \

0.4

i.

0.3

r

0.2

«#.

3

0.6

TTTT

TJ

ill

\

-

0.7

Ill

3 "O

and Parton Energy Loss

0.1

\

0

L

10 "10 5 10 4 10 3 10 2 10 ' 1

:

JIIL

i_ • • •!•"

10 "

10

<^ I

WW

2

10

1

_

1

u/ue=c*)/€(L

Fig. 4. LHS: The medium-induced gluon energy distribution radiation for a dynamically expanding collision regions (95) with expansion parameter a = 0, 0.5, 1.0 and 1.5. The value of the transport coefficient q0 is taken at initial time £o- RHS: The same gluon radiation spectrum with parameters rescaled according to (100). Figure taken from 5 2 .

Here, the coefficients are given by modified Bessel functions Iv(z) and Kv{z) [here: z = z{yt), z = z(yL)} ci = ci(yL,yL) c2 = c2(yL,yL)

= z{z/z)v

[Iv^{z)Kv{z)

+ Kv^{z)Iv{z)]

,

(97)

v

[Kv{z)Iv^{z)

+ II/{z)KlJ_1(z)}

.

(98)

= z{z/z)

and the fluctuation determinant reads

One checks that for a = 0, v = ~, the solution (72) is regained. Using the explicit form of the path integral (96), one can calculate the medium-induced gluon energy distribution (69) for a dynamically expanding medium 5 . The result is shown in Fig.4. The radiation spectrum u>^ satisfies a simple scaling law which relates the radiation spectrum of a dynamically expanding collision region to an equivalent static scenario. The linearly weighed line integral 51 _2_ L2 /

rio+L

ho

«#(£-£<>)

9(0

(100)

224

A. Kovner and U. A.

Wiedemann

defines the transport coefficient of the equivalent static scenario. The linear weight in (100) implies that scattering centers which are further separated from the production point of the hard parton are more effective in leading to partonic energy loss. In contrast to earlier believe that parton energy loss is most sensitive to the hottest and densest initial stage of the collision, this implies for a dynamical expansion following Bjorken scaling [a = 1 in eq. (95)] that all timescales contribute equally to the average transport coefficient. This makes partonic energy loss a valuable tool for the measurement of the quark-gluon plasma lifetime.

4.1.5. Properties of the transport coefficient The only property of the medium which enters the medium-induced energy distribution (69) is the transport coefficient
2

M q ) |

2

(101)

For phenomenological estimates of the value q for cold nuclear matter, one often invokes the relation of q to the gluon structure function q — ^gf^p[xG(x,qL)}. This leads to estimates 4 qco\d < (200MeV) 2 /miThere are alternative approaches 4,x which extract qco\
Gluon Radiation

and Parton Energy Loss

225

10.0

1.0

CD 0.1

0.01 0.1

100

e (GeV/fm J )

Fig. 5. Estimate of the transport coefficient as a function of the energy density e. Figure taken from 1 0 .

The transport coefficient has been related also to the saturation momentum scale 10

Ql

qL.

(102)

The saturation scale entering here should be denned for a color octet dipole - thus, it is a factor CA/CF larger than the saturation scale determined from the qq scattering probability in DIS. The numerical value of Qs is very uncertain but Q2S < (3 GeV) 2 may be considered as an upper bound at LHC. For in-medium path lengths L up to twice the nuclear size, this corresponds to values up to R = \qL? < 40000.

4.2. Opacity Expansion

of the radiation

cross

section

In what follows, we discuss the opacity expansion of the radiation cross section (69). This is based on the expansion of the path-integral (57) in

226

A. Kovner and U. A.

Wiedemann

powers of the dipole cross section, K(r,yi;f,yi)

= /C0(r, yi\f,y{)

z'

- J d£n(t) J dri /C (r, y ; n , £ ) - cr(n) /C (r!,£; r, y ) 0

t

0

t

Z

z'

z'

dri

X

dr2ICo(r,yi;ri,£i)

1 1 2(7(ri)K(ri^i;r2,C2)2(T(r2)^o(r2,^;r,!/i).

(103)

The iV-th order in opacity is obtained by expanding the integrand of the gluon energy distribution (69) to N-th order in (n(£) cr(r))N 59 ' 26 . 25 . We start with a discussion of the first order term for which the medium acts effectively as a single hard momentum transfer positioned within path length L.

4.2.1. Expansion to order N = 0 and N = 1 For the gluon energy distribution (69) of a free incoming quark, the zeroth order in opacity vanishes. A free incoming particle does not radiate without interaction. A parton produced in some hard process, however, does radiate without further interaction in order to decrease its virtuality. This is the DGLAP parton shower whose medium modification is accessed by the medium-induced spectrum UJ4^-. To zeroth order in LUQ, this spectrum is duj ' given by d3I(N = 0)

a3 „

1

Tr„ .

®

£-

.

1

,,„„,

—-i»»~

*•

(105)

This is the characteristic gluon radiation spectrum associated to the production of a hard parton. Medium modifications to (104) appear to first order in opacity. Introducing the transverse energies Q = j^j, Qi = ' 2 ^ l J , one

Gluon Radiation

and Parton Energy Loss

227

finds d3I(N = 1) dwcflc

a3 CR

/• dqi .

-kqi

.

,|2

LQi - sin(L<5!)

(106)

»«• + g F - r - ^ — •

©

+ <8>

o

(107)

The formation time for a medium-induced gluon is tf » 1/Qi. For a finite distance ~ L between the production point of the parton and the point of its rescattering, there are thus gluons whose formation times are too long: L/tf = LQi < 1. The interference term L Q I ~ Q " ( L O I ) in (106) prevents such gluons from being radiated. It provides a logarithmic cut-off for an otherwise infrared divergent k- integrated expression, see eq. (117) below. This radiation spectrum interpolates between the totally coherent and totally incoherent limiting cases. These can be discussed in terms of the isolated contributions for the hard (vacuum) and the medium-induced GunionBertsch radiation terms: Il(ki=

Hk,+ qi>

®-

<SK^=

L-*- °° +

R(ki,qt) =

©

©

_^r (108)

In the totally coherent limit, the extension of the target is negligible, and medium-corrections are absent, lim i^o

,„ \ „ d(ln a:) dk

'- = 0 .

(109)

In the incoherent L —> oo limit, the radiation cross section (69) expanded

228

A. Kovner and U. A.

Wiedemann

up to first order in opacity, takes the form ^ d?I{m) L->oo ^-—' dwdk

as 7r2

TO=0

+

S C F n 0 L /(Si2 | a 0 ( q i ) | 2 ( / / ( k + qi) + j R ( k ' q i ) )- ( 1 1 0 )

The three terms on the r.h.s. of (110) have a simple physical meaning: the first is the hard, medium-independent radiation (104) reduced by the probability w\ that one interaction of the projectile occurs in the medium. The second term describes the hard radiation component which rescatters once in the medium. The third term is the medium-induced Gunion-Bertsch contribution associated with the rescattering. For the case N = 1, the above discussion that eq. (106) pattern interpolates between simple and physically intuitive limiting cases. In general, the N-th order term in the opacity expansion is a convolution of the radiation associated to ./V-fold scattering and a readjustment of the probabilities that rescattering occurs with less than N scattering centers 5 9 .

4.2.2. Qualitative estimates ofu)^

vs. quantitative calculations

In general, the radiation spectrum u>^|~ displays an interference pattern between medium-induced and hard (vacuum) radiation amplitudes. Formation time effects determine this interplay and allow for some qualitative estimates. In the simplest case, consider a hard partonic projectile which picks up a single transverse momentum /x by interacting with a single hard scatterer. An additional gluon of energy UJ decoheres from the projectile wave function if its typical formation time tcoh = r? is smaller than the typical distance L between the production point of the parton and the position of the scatterer. The relevant phase is 7=7^

= —-

£coh

(HI)

W

Thus one expects that the radiation of gluons is suppressed if their energy u) is larger than the characteristic gluon energy u>c = ^fi2L.

(112)

Gluon Radiation

and Parton Energy Loss

229

From this one can estimate the gluon energy spectrum per unit path length in terms of the coherence time fcoh, dIN=l as p? , , duiaz tcoh w This estimate can be compared to the full N = 1 opacity term (106) integrated over transverse momentum up to |k| < \bj. Introducing the kinematic constraint in transverse momentum phase space, R = X2"cL,

(114)

the gluon radiation spectrum takes the form dIN=1 du)

f V "-

= 2asCfi(n0L)7 7T

52

Jo

.-(

l

w2

VwI R

sin(r) r2

. . . (115) \r + 7 >/((ii/27) + r + 7) 2 - 4rR/2^ J The expression (115) is plotted in Fig. 6. It shows qualitatively the same shape as the spectrum in the multiple soft scattering case. The position of the maximum of u)—'da)= ' changes oc -4=. This is in accordance with the assumption that gluon radiation is suppressed for energies below which the characteristic angle of the gluon emission is of order one c

u>c

y/R

In the limit R —=> oo in which the phase space constraint is removed, one recovers some pocket formulas from the literature 25 . First, this limit satisfies the characteristic 1/w-energy dependence of the estimate (113) for sufficiently large gluon energies UJ > Qc, dI(N=l) _^a3CR /m r , _ f°° Jim u>——: = 2 (no L) 7 / R-*oo

diJ

7T

1

r-

sinjr)

r+7

Jo

„2^^(noL)/iog[|]

*» e c > a , di7)

y *• I f ^f for wc < w v Moreover, this expression allows to estimate the average parton energy loss for a single hard scattering. Remarkably, this seems dominated by contributions from the region w >
lim (AE)N=1 fi-»oo

= lim

f duuj^—^-

R->oo J

dw

~ ^^(n0L)Qc

\og[E/GJc] ,

2 (118)

230

A. Kovner and U. A.

Wiedemann

3 \

\GLV(R=oo) 3 o C

0.8

R=40000

Fig. 6. The medium-induced gluon energy distribution u > ^ in the N — 1 opacity expansion. Different curves are for different values of the kinematic constraint R = uic L. Figure taken from 5 2 .

which appears to be logarithmically enhanced in comparison to the region u> < u>c for which

lim

du)LJ

Jo

d!N=1 dw

_

2asCR {n0L) u>c n

(119)

However, the logarithmic enhancement of (118) does not persist upon closer inspection: All calculations so far worked for fixed coupling constant which may be justified if all momentum transfers are of the same order. In calculating (118), however, the k-integration was extended up to the total energy E. If one takes the running of the coupling into account, this logarithmic enhancement factor is reduced to a log log factor.

Gluon Radiation

and Parton Energy Loss

231

4.2.3. Parameters in the opacity expansion In the multiple scattering approximation, the gluon energy distribution (69) depends on two quantities: the transport coefficient q and the in-medium path length L (we ignore the angular dependence in the following discussion). In the opacity expansion, there are three parameters instead: the inmedium path length L, the transverse momentum squared fi2 transferred on average by a single scattering, and the inverse mean free path no = j . The multiple scattering approximation describes the average momentum transfer per unit path length while the opacity expansion specifies in addition by how many scattering centers this momentum is transferred on average. In the limit of multiple soft scattering, this additional information is redundant since fj?noL = qL,

for Brownian motion.

(120)

However, deviations from Brownian motion arise due to the high transverse momentum tails of the elastic scattering cross sections |a(q)| 2 = (fi2(2n)2) / (7r(q2 + /x 2 ) 2 ). These tails lead to a logarithmically enhancement factor in the transport coefficient ~ r

r

qL = n0L

I

J

^ ^ l l / \ l 9 - * - 2

2

t

7 " \ 2 l

—-^ \a{q)\z - q J cos