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The rho meson in dense matter and its influence on dilepton production rates
Nuclear Physics A555 ( 1993) 329-354 North-Holland
NUCLEAR PHYSICSA
The rho meson in dense matter and its influence on dilepton production rates* G. Chanfray Institut de Physique Nucliaires, 43 Avenue du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
P. Schuck Znstitut des Sciences Nuclkaires, 53 Avenue des Martyrs, 38026 Grenoble Cedex, France Received 15 June 1992 (Revised 26 November 1992)
Abstract:
Rho meson propagation in dense nuclear matter and dilepton production are considered in a phenomenological model with coupling to P+C states. Special attention is paid to constraints from gauge invariance. Taking into account the usual r-Ah coupling it is found that the rho meson in matter is slightly quenched and shifted upwards. A new interesting structure at -500 MeV appears in the rho-meson mass spectrum.
1. Introduction
Recently there has been considerable interest in dilepton production rates (DPR) originating from heavy-ion collisions or from proton-induced reactions ‘). This stems from the fact that the lepton pair couples via the photon directly to the rho meson (vector-dominance model (VDM)) and thus the dilepton invariant-mass spectrum could give important information on the rho-meson mass spectrum in dense and hot nuclear matter. Since the rho meson strongly couples to the T+T- channel the dilepton spectrum may also be very relevant for our knowledge on pion dynamics in dense matter. Preliminary studies ‘) have indeed shown that commonly accepted medium corrections to the pion propagation can strongly influence the mass distribution in the rho- and sigma-meson channels. In this work we pursue these studies and concentrate on the rho meson. We will for instance be concerned with the influence of gauge invariance on the mass spectrum. In detail the paper is organised as follows. In sects. 2 and 3 we make some comments on the dilepton production rate and the rho meson and discuss in a heuristic way medium effect and gauge invariance. In sect. 4 we present in detail our rho-meson propagator (with coupling to TT states) in free space and show that gauge invariance can be maintained even at a phenomenological level. In sect. 5 we consider the rho-meson propagator at l
Dedicated to the memory of A.B. Migdal.
03759474/93/$06.00
@ 1993 - Elsevier Science Publishers B.V. All rights reserved
330
finite
G. Chanfray,
density
and consider
gauge invariance.
again
P. Schuck / Rho meson
the corrections
In sect. 6 we give our
numerical
induced results
from the constraint and conclude
of
in sect. 7.
2. Rho-meson propagator and dilepton production As already
mentioned
in the introduction,
the dilepton
production
rate in relativis-
tic heavy-ion collision is strongly influenced by the rho-meson properties in hot and dense matter. This is a direct consequence of the vector-dominance hypothesis which states that, to a very good approximation, the hadronic electromagnetic current is a linear combination of light-vector-meson fields such as p, w or 4 for photon invariant-mass scale less than one GeV. For example it is well established that the pion electromagnetic form factor is almost completely dominated by the p-meson. It follows that the vector-dominance model (VDM) should give an accurate description of DPR for dielectron invariant masses larger than 2m,, since pion-pion annihilation is the dominant process. Moreover the universal rho-meson-hadron coupling allows one to incorporate into this VDM scheme processes involving nucleons and deltas. Since the excited nuclear system is in general produced from a heavy-ion collision the problem of dilepton production is in fact a dynamical one involving, at least in the initial phase, an out of equilibrium situation. However, assuming, as many authors did, the system to be equilibrated nuclear matter at some temperature T and some density p, one should obtain the DPR straightforwardly. Nevertheless the fact that free pions cannot survive in the medium due to their coupling to delta-hole states has to be taken into account. As a consequence only collective modes made out of pions and delta-hole (Ah) states can exist in the medium. In addition gauge invariance requires to couple the photon not only to pions but also to Ah states. All these effects can be incorporated in a consistent way through a medium modification of the rho propagator whose imaginary part is proportional to the DPR. Such an approach allows to calculate not only the in-medium renormalised electromagnetic vertex but also the in-medium corrected form factor
as required
by gauge invariance
and unitarity.
We will first clarify the connection between the DPR and the imaginary part of the rho-meson propagator in the VDM scheme. We postpone the detailed model calculation of this propagator to sect. 4 (free case) and sect. 5 (in-medium case). Let us assume that the initial macroscopic state is described by a statistical ensemble where Pi = exp (-BE,)/2 is the normalised probability to find the state Ii) with energy pi0 = Ei. Using standard Feynman rules, a straightforward calculation gives the DPR (per unit volume) for ultrarelativistic dielectrons with total momentum q = p+ +p= (q,,, q) and invariant mass M2 = qt = q“q, > 0
&= The leptonic
part which includes
~Yq)&Lv(q).
the explicit
(2.1)
factor e4 = (47~~)~ as well as the photon
331
G. Chanfray, l? Schuck / Rho meson
propagator
is given by
The physics is contained in the hadronic piece H,.(q)=;?
F
H,,(q)
(ilJ”(o)l~~lJ”(o)li>(2~)36’4’(q+Pf-Pi)
9
(2.3)
where J@(O) is the hadronic electromagnetic current and If) the final hadronic states. Using current conservation the DPR can be written as dR h4X
ff2 -H(q) = 37r2M2
(2.4)
with (2.5)
H(q) = -g”“%(q). Let us now relate the hadronic tensor r..
Jp’“(q; T)=-i
Using translational Jc”“(q;
d4xe’qxCL
to the retarded propagator
H&‘“(q)
p-w
J
(2.6)
invariance one easily finds T)
=
I-
eepEi
i,f
z
(~lJ”(~)lf)(flJ”(~)li> c2nJ36t3jcq+p. _p ) 1
qo+Ei-Ef+iv
/
~2T~3s(3)~q+p _p.j --eepEi W”(WWlJ”(O)l~) I * f Z
qo-J$+Ef+i~
(2.7)
Taking the imaginary part of Jp” and exchanging i and f indices in the first term one obtains the very general result -i
Im J’““(q; T) = (epqo- l)Hp”(q).
(2.8)
In the VDM the electromagnetic current is related to the third isospin component of the rho-meson field by the field current identity Jfi = (mt/g)pT where mp and g are the rho-meson mass and the universal VDM coupling constant appearing in the effective VDM lagrangian respectively. Let us define the retarded rho-meson propagator: Gp”(q; T) = -i
d4x eiqXCi 9
(il[~f(x),~XO)llW(xo).
(2.9)
G. Chanfray, P. Schuck/ Rho meson
332 Thus
using the field current H(q)
In the following
=
identity
and eq. (2.8) one obtains Im G““(q;
(ePqa- l)-‘(m~/7rg*)g,,
we will consider
case, from gauge invariance,
only back-to-back
-1 Q2
where the rho propagator
piece
T) .
(2.10)
kinematics,
only the space components
H(q,,q=0)=(e”4~-1))1m4P
for the hadronic
i.e. q = 0. In that
contribute:
i ImG”(q,,q=O,T) 7Ti=i
,
(2.11)
>
has the form 6” G”(qo;
T) = q;-m;-~(qO;
The detailed
derivation
of the rho-meson
(2.12)
T)’
propagator
and the scalar part of the rho
self-energy E(q,; T) is given in sect. 4 in the free case (see eqs. (4.4), (4.16) and (4.17)) and in sects. 5 and 6 for the in-medium case (see eqs. (5.1)-(5.7), (5.17), (5.18) and (6.10)) in the zero-temperature limit. The hadronic piece takes the form H(q,,
T) = (epqo- I)-‘lF,(qO;
Here F, is the (in-medium
modified)
IF,&,;
T)12( --$
Im z(qO;
pion electromagnetic
T)) .
form factor 7X1’.
T)I*= Im:l(&-$--WI,;
(2.13)
(2.14)
Before coming to the influence of the nuclear matter density which is the main purpose of this paper, let us first briefly discuss the finite-temperature effects. Assuming a free dispersion relation for the pion and ignoring all possible finitedensity effects the only mechanism for DPR is rr annihilation. The corresponding piece in 2 (q,, , T) (i.e. that piece having a non-vanishing imaginary part) contributing to H(q,; T) of eq. (2.13) is given by: 4g2 S(q0;
d3k
T)=y
(2T)3
v’(
k)k2G2,(q,;
T) +
purely real terms .
(2.15)
i
Here v(k) is a phenomenological form factor which defines the scale of the effective theory (i.e. -1 GeV). Its precise form fitted to the 7rm phase shifts was given in ref. ‘) and all the details concerning this effective VDM theory will be presented in sect. 4. The retarded two-pion propagator at finite T is given by3) G2,(q,,;
k;
T)=l wk
1+2h(T)
q:-4wz+
is ’
(2.16)
where fk( T) = (e@“*- 1))‘. The reader may notice that the quantity G,, which depends on the internal pion momentum k is not strictly speaking a propagator, since the “conventional” propagator is obtained after integration over k. However,
G. Chanfray, P. Schuck / Rho meson
333
for simplicity, it has become usual in our context to call G2_ also a propagator. We immediately deduce
H(qo; 7-1= krh;
T)l*j-
$
(5)@qo-24
~2W-2W
(2.17)
which is the result quoted in a number of places [see e.g. ref. “)I. Sometimes the Bose distribution is modified by the introduction of a chemical potential to account for non-equilibrium effects. If the temperature is not too high one can neglect its effect in the rescattering processes (i.e. in 1); in this case T appears only in an overall multiplying factor H(qO)=e-PqOIF,(q,;
T=O)/‘(---$*mT(q,;
T=O))
(2.18)
which is essentially the approximation made in all the papers devoted to DPR. In what follows we will implicitly use this approximation and concentrate on density effects and gauge invariance constraints at T = 0. For that reason the connection to DPR is still tenuous. The unified treatment of density and temperature effects is under investigation and will be the subject of a future publication. 3. Medium effects and gauge invariance: a heuristic approach It has been argued “) that the modification of the pion-dispersion relation may provoke a drastic enhancement of the DPR near the two-pion threshold. The origin of the effect can be seen very easily by simply replacing in formula (2.18) the free pion energy wk by the energy R,(k) of the quasi-pion in the medium [see ref. ‘) for notations]. Integrating over k one obtains:
With the possibility of d&(k)/dk becoming very small at threshold for finite densities this can give rise to a spectacular enhancement of DPR near 2m,. To see the magnitude of the effect one can assume (this is a quite good approximation) that 0;(k) = m’,+ */k2 with y a decreasing function of the density “). For instance at equilibrium density y is of the order of 0.3-0.4 depending essentially on the value of g’ (see below). We thus obtain
H(h) = +
e-Bq4~A90)12 & 66 - 4J3/2
(3.2)
which gives an enhancement factor of ten at normal nuclear-matter density. At even higher densities y may come close to zero (Kisslinger catastrophe). However, as already pointed out by Korpa and Pratt “) the above considerations are too naive, since gauge invariance is not respected. Restoration of gauge invariance essentially
G. Chanfray,
334
suppresses process
the enhancement.
is the annihilation
P. Schuck / Rho meson
Let us explain
this point in a heuristic
of a pion r+ with four momentum
(fig. 1). The corresponding current between a two-pion
amplitude is the matrix element state and the vacuum, i.e.
jP(q;
ki, M = K(q)(k,
where F,(q) is the (possibly medium we will assume it is given by VDM,
k’l_with a pion 6(
kc)
of the electromagnetic
- k2) p,
modified) rr form factor. that is there is a universal
coupling to baryons through -ye conversion (only the isovector here). Current conservation tests gauge invariance. We find qPjP(q;
way. The basic
(3.3) In the following electromagnetic part is considered
kl, k2)=FAq)(k:-k:)
(3.4)
which vanishes for on-shell free pions ( k2 = mt). However, if the process occurs in the medium, the quasi-pion corresponds to the pole of the in-medium pion propagator D,(k) = ( k2 - rn’, - S(k))-’ where the pion self-energy S(k) = k2fio( k,, k2) originates from the p-wave coupling of the pion to Ah states (ph states are practically negligible beyond the pion threshold) corrected by a screening effect (g’) due to short-range correlations. Ignoring for the moment complications linked to the imaginary part of S(k) (width of the delta and 2p2h states), “on shell” quasi-pions _ are characterised by a modified dispersion relation OJ: = rni + k2 + k*l7,,( k). It follows that qP.&(q;
k,, W = FAq)(k%(kA
-k%(k2))
(3.5)
which does not vanish in general. This clearly demonstrates that all the calculations of dilepton production including modification of the pion-dispersion relation violate gauge invariance, since the r.h.s. of eq. (3.5) does not vanish. The reason for this failure is quite obvious and is due to the fact that the photon only couples to the pions while via gauge invariance it also should couple to Ah excitations. In particular one has to take into account processes such as those depicted in fig. 2 where the dashed
box represents
the inverse
(virtual)
pion-photoproduction
amplitude
A’+*-)
of a r(+*-) with pion-pole term subtracted. In a model with pseudovector coupling this amplitude involves a seagull term and can be calculated to leading order in the inverse nucleon mass; one finds (A, is the photon field) (3.6)
x+(k,b.
“..
“‘b----nA
,.*.’.” y (q=k,+k$ n-(k2) ,.:.’ Fig. 1. Basic P+T-
annihilation
graph in free space.
G. Chanfray, J? Schuck / Rho
335
meson
rc+.. ..
.a..
.$Qjp .. .x” R_....”
Fig. 2. In medium gauge correction graphs for n+rr- annihilation.
where Pm = 2fm and S, T are the standard spin and isospin transition operators. In other words within this static approximation the amplitude reduces to a one-body spin-isospin operator. Thus the diagrams of fig. 3 are proportional to a,,(k) once short-range correlations are included. Therefore in the context of this simplified model the current matrix element becomes: j”(9; k,, kz)=F,(q)
(
84; ki, k2) = FAq)((I
+k+k,
40
+&(k,))k,
* k,(fio(k,)-fio(k,))
-(I +fio(k2))Q
> .
,
(3.7a) (3.7b)
Taking its divergence one obtains 4% =F,(q){(k:-mZ,-k:~o(k,))-(k:-m2,-k:~o(k,))},
(3.8)
which vanishes for “on-shell quasi-pions”. More generally we see that the divergence of the current involves the difference of the two inverse pion propagators and has the form of a generalized Ward identity. In the two-level model used in our previous papers ‘) one has two kinds of quasi-pions with energy O,(k), L?,(k) and carrying strength Z,(k), Z,(k) with Zi + Z, = 1. Let us call cu,(k) and al(k) the irreducible pion self-energy along those branches namely: a&k)
= ii&=
%2(k), k) .
Near the two-pion threshold only the lower branch 0, contributes modified by these gauge corrections becomes
-$
H(e) = e-Bq0)F,(qo)12(Z,(k)(l
(3.9) and the DPR
+a,(k)) $J2eF)-‘12*,.;,~ (3.10)
G. ChQnfrQy, I? Schuck / Rho meson
336
(30)
(3b)
(3c)
(31)
(se)
(St)
(39)
C3R)
,,._~
Z? Fig. 3. The various contributions
(39
--4-J-to the rho-meson
(3i) self-energy;
only (3a) and (3e) survive in free space.
It is now very easy to show that the modified hadronic current Z,(k) (l+ ~,(~))~/~,(~) is just equal to the group velocity d&$/d/c in the static approximation (i.e. a&/& very small which is true in practice). Consequently H(qo) is proportional to dO1/dk = yk/LJ,. This feature leads Korpa and Pratt “) to the conclusion that the DPR exhibits a dip in place of enhancement near q. = 2m,. However at a given invariant mass q. the DPR is also proportional to k3 = ($q$- M:)/ Y”~. More precisely one finds that (3.11)
G. Chanfray, l? Schuck / Rho meson
331
The net effect of gauge invariance
is to replace the factor y-“’ in eq. (3.2) by a factor Y-“~. This slight enhancement effect will be confirmed by more detailed numerical calculation (see sect. 6). However, in conjunction with the finite lifetime of the pion in a nuclear medium this is so small that it is practically invisible on the scale of our graph (figs. 5,6). 4. The rho-meson propagator in free space In this section we will be concerned with the rho-meson propagator the 7~1~channel in free space. The bare rho-meson propagator has the well-known form
coupled to
(4.1) where nr, is the rho-meson mass appearing in the effective VDM lagrangian. The coupling of the rho meson to the two pions can be described by the introduction of an appropriate mass operator. The dressed rho-meson propagator is then the solution of the corresponding Dyson equation. G““(q) = G?‘(q)+ According to gauge invariance energy has to be of the form
GY’(&&)GQY(q).
(q@E,J
q) =
0) and Lorentz covariance
(4.2) the self-
(4.3) The solution of the Dyson equation is straightforward (4.4a) with G(q’)=[q’-m;-iE(q’)]-‘.
The pion electromagnetic
(4.4b)
form factor in VDM is given by F,(q2)=
mz “;+H(q2)-q2
(4.5)
and the condition F,,(O) = 1 imposes that X(O) =o.
(4.6)
We now proceed to evaluate the self-energy .Z within our effective VDM lagrangian model. There are two contributions: 2 p” (fig. 3a) corresponds to the diagram with
G. Chanfray, l? Schuck / Rho meson
338
two pions Seagull
in the intermediate
states and Z-2”’ (fig. 3e) accounts
for the purely
real
contribution: v”(t)4k~kYoo,(k+~q)~~~(k-fq)
-
-
(4.7)
u*(r)UMk+tq)+%(k-fq)),
where Do,, is the free pion propagator. The effective hadronic theory is completed form factor depending on
through
the use of a Lorentz
invariant
t=f[(w(k+fq)+w(k-fq))2-4mZ,-q2]1’2 which is nothing The coupling factor
but the relative
constant
momentum
(4.8)
in the two-pion
g = 5 and the parameter
center-of-mass
qd = 3.3m,
appearing
frame.
in the form
(4.9) are fixed by fitting the pion electromagnetic form factor and the I = J = 1 PT phase shifts [for details see our previous publication ‘)I. The form factor has a flat behavior up to C= qd and strongly decreases beyond this value. The invariant mass scale defining the range of validity of the effective theory is thus A =2(q!j+m~)“*-1 In the rest frame of the rho meson
(qg = @,
P”(qo*,
which is a necessary of the self-energy
condition tensor
GeV. q = 0) it is easy to check that
0) =o
(4.10)
for gauge invariance.
The non-vanishing
components
are -
#.v(q,*,o)=6”~(q*)
i, j = 1,2,3’
t* q*-4t02(t)+iseg2
(4.11a)
, d3t m
v*(t) w(t).
(4-11b)
We immediately see that this calculation yields a non-vanishing value of x(O) and thus F,(O) is different from one. We now show that the problem is related to a of the model. To see violation of gauge invariance in this “naive” interpretation what is going on we repeat the calculation in the laboratory frame and, taking the 3-axis along q, we find q&p”(q)
=o
foruf3,
(4.12a)
G. Chanfray, k? Schuck / Rho meson
339
which does not vanish due to the (k - q)2 dependence of the form factor. This integral can be calculated in boosting k into the two-pion c.m. frame with a jacobian J
=
4k+fq) o(t)
(
4’ t
1-(4w2(t)+q2)1/2
(4.13)
>*
The result is
qp~r3(9) = -W_(O)
(4.14)
showing that it is precisely the violation of gauge invariance which entails F,(O) # 1. In addition it is possible, using the previous change of variables, to derive the explicit form of S:“(q): (4.15)
_w(q)=-(g’.-yp(q2)-_s(o))+P(o)
showing that both gauge invariance and Lorentz covariance are violated. The obvious solution to all these problems is simply to reinterpret the physical self-energy tensor as @(q2)-X(O))
*
(4.16)
The rho-meson self-energy appearing in the rho-meson propagator is then given by
=$ i (~~{(q~,O)-~~i(O)) i=l
1 q2 1 n2(‘) -4~*(t) -w(r) q2-4w2(f)+ie
’
(4.17)
In other words the real part of the self-energy is obtained from the imaginary part by a once subtracted (at q = 0) dispersion relation. This is essentially the procedure which we will adopt later on also at finite density. The agreement of the model with the pion form factor and the (I = J = ~)PCT phase shifts S(E)=arctg(ImE(E)/(E2-mz-ReE(E))) is excellent. Numerical calculations show that Re .Z(mE) almost exactly vanishes and thus the physical rho-meson mass is practically equal to the lagrangian mass. Notice that the mass parameter MP appearing in our previous work 2, is related to mP (we take mp =773 MeV) by M$ = mi-x(0).
G. Chanfray, P. Schuck / Rho meson
340
5. The rho-meson propagator at finite density As demonstrated
in the second
the basic PTW vertex first piece
section,
are renormalised
of the rho-meson
xf”(q)
EY(s;
P) = g2
4% (2r)4
I
the basic
YGTTT~~ vertex and consequently
in the medium self-energy
tl’(t)j’l(q;
according
to eq. (3.7). The
k)D,(k+;q)D,(k-+q)
(5.1)
becomes
k)j”(q;
with D,(k) the full pion propagator in the medium (as before, we only consider p-wave pion-nucleon interaction) D,(k) = (kz - w2(k) - k2fio( ko, k))-’ and j’(q;
k)=jp(q;
k,=$q+k,
k,=;q-k)
is given in eq. (3.7). The diagrammatic content of 2, is displayed in fig. (3a-3d) where the bubbles represent the irreducible pion self-energy (dominated by Ah states) already introduced in sect. 3. The second piece is the purely real seagull contribution (fig. 3e). XfY(
q; p)
= -g2g
WY
$+2(f)(Dm(k+fq)+D,(k-~q)). I
(5.2)
The third contribution involves in the intermediate states a one-quasipion line (D,) together with a delta-hole state (I?“) as shown in fig. 3f. To leading order in l/M,., the part Er” of the mass operator is then given by EY(q;
k) = g2
I
id4k -[v*(t)O;“(q; t2T14
k)l?“(k-;q)D,(k+;q)+(q+-q)]
(5.3) with 1 =--++:q)*, 40
Oy(q;
k)
Oyi(q;
k) = O;‘(q;
Of(q;
k) = 8”.
k) = (k’+$q’)/q,,
The next contribution is diagrammatically analog to the one of fig. 3d with the pion line connecting the two Ah bubbles replaced by a rho-meson line. One obtains (see fig. 3g) 2:”
= g2
id’k[v2(t)O:v(q; (2T)4
k)D,(k+$q)(fi°CpD$o)(k-;q)+(q+-q)] (5.4)
with O,“‘(q;
k) = O;‘(q;
k) = (q’k-
k’q)
O:“(q; #(q;
k) = ik x qi2/q:, k)=S”lk-~q12-(ki-4qi)(kj-~qj).
. (k-;q)/q,,
G. Chanfray, P. Schuck / Rho meson
341
In eq. (5.4) DP( k) = [ ki - wz( k) - C,k*l7’( ko, k)]-’ is the full rho propagator. C, is the usual parameter entering the PNN and pNA couplings “); we take in the following C, = 2.2 (strong rho coupling). One may argue that the rho propagator appearing in the self-energy should be consistent with the time-like rho propagator we are searching for. However, such a self-consistent calculation is unnecessary for our purpose, since this internal rho involves a completely different kinematical range (i.e. high-momentum range of the order of a few m,c). It is also important to notice that this contribution to the self-energy is separately gauge invariant. Grouping together figs. 3d, 3f and 3g allows us to reconstruct diagrammatically in the intermediate state the full longitudinal and transverse spin-isospin response functions as is shown in fig. 4. In a similar way the fifth contribution can be obtained from fig. 3c by simply replacing a pion line by a rho-meson line as shown in fig. 3h where the right-hand side bubble represents the spin-dependent part of the rho-meson photoproduction amplitude. To leading order in l/MN the dominant contribution relevant for our purpose connects the time component of the incident rho meson (photon) p’(q) to the space component of the outgoing rho p’(k); the corresponding amplitude is Jl”(k;
q)=i~pi(k)(Sxk)‘TpO(q)/qo.
(5.5)
As in the case of the pion-photoproduction amplitude (eq. (3.6)) it reduces to a spin-isospin one-body operator. The contribution to the rho-meson self-energy does not have pure space components (which are relevant at q = 0) but this is necessary to ensure gauge invariance. For completeness one should also add a graph deducible from Xry by simply exchanging the role of the pion and the rho; to leading order in l/M, this last graph contributes only to the pure time component 2:” of the self-energy tensor
25” = 8 +
id’k[02(t)(O;:(q; (2T)4
W’(q;
k)(iiOD,)(k+~q)(fioCpDp)(k-;q)
k)(ii’D~‘)(k+~q)C,D,(k-Iq))+(q~ -411
(5.6)
a W+@f)+W=
+a
Fig. 4. Reconstruction
.,./,,.............
of the full longitudinal and transverse intermediate state.
‘..,
spin-isospin
response
function
in the
G. Chanfray, P. Schuck/ Rho meson
342
with 0:: = -21k
x q/*/q;,
O;:=Og=-(qik-k’q).(k-$q)/qO, oya CO 3 0:: = ikx o;;=
qb’qi,
@=
O&z0
We finally add a graph which corresponds graph shown in fig. 3i which is given by ZY = -2
I
to a medium correction
k)[(fi”D,)(k+~q)+(jioD,)(k-$q)]
id4k (2p)4 o’(t)O:“(q;
to the seagull
(5.7)
with 0%;
k) = (k+$q)‘/q;,
O:‘(q; k)=O;*(q; O!(q;
k)=(k’++q’)/qo,
k)=O.
This contribution which is entirely real does not have a pure space component but again this is necessary to ensure gauge invariance. The last contribution Z; (fig. 3j) is just the usual irreducible rho-meson self-energy due to its coupling to ph and Ah bubbles. This gauge invariant piece vanishes at q = 0. Since our calculation will in practice be limited to q = 0, we will not consider it here. Let us now come to the problem of gauge invariance at finite density. The occurrence of explicit breaking of Lorentz covariance due for instance to the treatment of short-range correlations (g’ parameter) certainly forbids a structure for ~Fy(q) such as (4.3). As a consequence it is in general very difficult to maintain a vanishing value for the divergence of the self-energy tensor, (especially at finite momentum of the rho meson). However we will see that the subtraction of Z’“(0) in the definition of the rho-meson self-energy will allow us to maintain gauge invariance in the limit of small 141.To achieve that we will follow a procedure very similar to what we did in the free case. For the rho meson at rest (i.e. q = 0) one can easily show that PO{ go* =
J41,O)
= Y(
go*, 0) = 0
(5.8)
This means that gauge invariance is minimally satisfied in that case which corresponds to the condition of the Bevalac experiment ‘). The non-vanishing components of the self-energy tensor (i.e. x1, .&, ifs, .&) have the form 8@(qo*,O)
= S’jTz(q)
)
s(q)=+;
Fi(q$,O), i-1
(5.9)
G. Chanfray, F? Schuck / Rho meson
343
Explicitly this gives ~(qo,O) = g2
id4k @+ u’(~k~)[$k’(l+~(fi”(k+fq)
I
+fi”(k-+q)))2D,(k+$q)D,(k-;q) +2D,(k)+2ij0(k-$q)D,(k+fq)
+~(~°CpD~o)(k-4q)Drr(k+fq)lq=(qo,0)
(5.10)
which again does not vanish in general at qo= 0 and thus the medium-corrected form factor is not 1 at q = 0. As before this problem is actually related to gaugeinvariance violation in this naive interpretation of the model. To see this point let us calculate the divergence of the self-energy tensor at finite 141.An explicit calculation (using eq. (3.8)) shows that 9$%Io,
4) = -2g2
I
$$
v2(t)[(ki+fqi)(l+iio(k+;q))Dm(k+fq)
-(k’-+q’)(l+fi’(k-$q))D,,(k-fq)] id4k
-~‘(t)(k’(;q’+k+
+2g2 I (2?T)4
q)-qi(k2++q.
k))
x (fi°Cp@“)(k+~q)D,(k-$q) .
(5.11)
In the first piece of the above integral the time arguments h*iq, can be shifted to b by a change of variable. One can also remark that to leading order in q one has R(ko,
k+&q) = Wko,
k)+q*
k(l+fi’(b,
k))DZ,(ko,
k) ,
(5.12)
where we have used the fact that in the static approximation VJTO(k,, k) =o. After some angular integration one easily finds that to leading order in q qpx %lo, 4) = G(O) + 0(q2) - ji (O)+O(q’) =+qjE =+qpPi(0)+O(q2).
(5.13)
In the same way one can also show that 4$P0(40,
4) = O(q2) *
(5.14)
Thus to order q2 the two results of above can be summarised by 4JS
Yqo, 4) - 2
“WI = Wq2) .
(5.15)
Hence the redefinition oft; as e(q) -x(O) allows us to maintain gauge invariance in the limit of small (q(, independently of the precise form of v(t).
344
G. Chanfray, P. Schuck / Rho meson
Let us now come back to a rho meson at rest (q = 0). The the rho propagator is then:
Dyson equation for
G”(q~,O)=6”(q~-m~+i~)+Sik(q~-m~+i~)~Ck’(q,,O)GU(qo,O)
(5.16)
with ;I;“(qo,o)=6”(~(qo,o)-~(o))=~s”Ic(qo,o)
(5.17)
The solution of eq. (5.16) is G@(q,,O)=SU[q$-m;-iY(q,,O)]-’
(5.18)
To close this section let us make a final remark on the subtraction procedure which is again equivalent to a once subtracted dispersion relation for the real part of the self-energy. The substracted quantity is a priori different from &JO). However, the numerical calculations performed in the two-level model (see next section) show that the subtraction of &JO) in place of %(O, p) gives a negligible effect: in particular the deviation of the in-medium pion form factor F,(O) from one is of the order of 0.01 at 1.5~~ once transverse Ah states appearing in & and E’, are explicitly taken into account. 6. Results The numerical calculations are performed in the framework of the two-level model
already introduced in our previous work “). The main approximation of this model is to neglect the Fermi motion of the nucleon in the Ah bubble. In addition we first also neglect the width of the delta resonance and the 2p2h contribution in the polarisation propagator. This last approximation will be relaxed below. In the previous version of the model the Ah bubble has the form (using standard notation)
(6.1) with Egk= (Mi + k’) “2 - MA ,
(6.2a)
&,A(k)p.
(6.2b)
The polarisation bubble appearing in the rho and pion propagator short-range screening
iiO(k&,k)
= JT”,,/(l -gV2Q
=
is corrected by
C(k) k$-&+irf
’
(6.3)
where the effect of g’ (here g’ = 0.5) is to push the Ah energy upwards, i.e. .&,k= [& “t-C(k)g’]“‘.
(6.4)
G. Chanfray, I? Schuck 1 Rho meson
345
notice that we neglect the ph bubble but this has a negligible effect at the (high) energies of interest. The pion propagator in this model becomes
Also
D,(k,,~)=[~~-~2(W~2fio(%,~)l-‘=ii,
k2_;;(;))+irl
0
(6.5)
3
I
where the energies L?i of the collective pion and Ah modes and their respective strengths Zi (with Z, + Z2 = 1) are given in ref. ‘). In an analogous way the rho-meson propagator can be expressed in terms of two weakly collective transverse modes with energies Si( k) and strengths Yi(k):
The full transverse response function which will explicitly appear is given by Wko,
1- Y,(k)
k) = C(k) i
i=l ki-8f(k)+iv
*
(6.7)
The rho-meson self-energy can now entirely be expressed in terms of the various longitudinal modes L?,(k) and transverse modes 69i(k), i.e.
(6.8)
~(q0,0)=~LL(qO,O)+&-(qO,O) with 2 4: u2(k)k2
$~2(fIi(k)+f2j(k))Oi(k)LJj(k)
xZi(k)q(k)(l+f(ai(k)+aj(k)))2 q~-(Ri(k)+Oj(k))2+i~
(6.9)
’
2
4;
C i,j=12(f&(k)+
%j(k))fli(k)Z9j(k)
Z,(k)(l- y(k)) x
2
qo-(Ri(k)+%j(k))2+iT
’
(6.10)
The result of the calculation for the strength function S(q) = -1m G is
Im z(4) S(q)= - Iq2-m~-ReE(q)(2+IIm2(q)12
(6.11)
and is shown in fig. 5. We see that the rho meson itself is only mildly modified with a quenching and a slight upward shift. On the other hand, a spectacular structure at q = m, + iakxo shows up. This structure actually corresponds to the opening of the longitudinal-transverse (&) channel and completely disappears when &-r is arbitrarily put to zero (fig. 6). A priori one could imagine that this peak is related
G. Chanfray, P. Schuck / Rho meson
346
0.05
-
iNVAR MASS (Me’/)
Fig.
5.
Imaginary part of the rho-meson propagator at various values of p/p0 when the A-width is neglected (pure two-level model).
0.175
0
-&I INVAR MASS (M&f
Fig.
6.
Same as fig. 5 but with the longitudinal-transverse
contribution (.&) to the rho self-energy omitted.
G. Chanfray, P. Shuck
/ Rho meson
341
to a peak of Im ZLT but this is not the main origin, since the pion form factor, not directly proportional to Im 1, also exhibits this structure (fig. 7) (6.12) ‘The effect is actually mainly due to RPA correlations and is related to a strong peak ofReILT at q=m,,+E"ak=o. It should not be confused with the enhancement of the dilepton production rate originally put forward in ref. ‘). A very similar structure has been found recently in a work by Herrmann et al. lo) [see also ref. ‘)I. It is extremely important to check whether this narrow structure will survive when the widths of the states L!i and 8i are taken into account, i.e. when the imaginary part of the polarisation propagator is considered [see also ref. ‘)I. A simple extension of our two-level model is obtained, if the propagators of the mesons are generalised in the following way 1
1
(6.13)
k,-Ri(k)+i~i(k)-k,+Ri(k)-i~(k)
and an analogous expression for the rho-meson propagator. In other words the real part of the pole in the complex plane is taken using the above two-level model and the width of these states is obtained by explicitly taking into account the width of
L
INVAR MASS (MeVJ
Fig. 7. F’ion electromagnetic
form factor at various values of p/p,, when the A-width two-level model).
is omitted (pure
348
G. Chanfray,
the delta resonance
P. Schuck / Rho meson
and the coupling Pi(k) =
-I
to 2p2h states,
i.e.
.
2fli (k)
k2 Im fi”( o = fli( k), k)
(6.14)
with ii’(w,
k) =l7’(w,
k)/(l
fl’(w,
k) = Ei,,(w,
-g’Lro(w,
k)),
k) + fl;pa,b,
(6.15)
k) .
The Ah piece is for w > 0
I;T;,Jw,k)
1 =;‘$pv2(k)
w-Edk+$T(w,
?i
1
-k)
w+cAk
> ’
(6.16a)
The delta width T(w, k) receives two contributions. For the first one, r,, corresponding to the usual TN decay channel (but suppressed by Pauli blocking) we use the explicit form given by Oset and Salcedo [formula A.1 of ref. ‘)I. The second piece rh [$b is noted -1m Ed in ref. ‘)] which essentially gives a 2p2h contribution to the optical potential k217’ is also taken from ref. ‘). We actually take its value along the free pion line which in turn is very close to its value along the photon line. This approximation is certainly good, since the pion branch line is located in between these two branches. We parametrise r; according to
Ci(w; k) 2
=-ImE,
=24p
l+
w(MeV) - 140 280 Y(W)
PO
w(MeV) -450 20
)I 9 -’
(6.16b)
where the cut-off function y(w) allows to reproduce the drop of Im 24 obtained by Oset and Salcedo beyond o = 380 MeV. However, it should be emphasised that the precise form of this cut-off function which governs the high-energy behavior of rh has practically no effect on our numerical result for the rho-meson spectrum. Furthermore, there are other 2p2h contributions, not reducible to a delta width, which are especially important if one aims to reproduce the absorbtive part of the pion-nucleus
potential.
We take the usual form [see subsect. 7.2.3 and 7.2.4 of ref. “)I.
Z7ip2,,(~; k) = -iO.25
0
i
'ri,,(k)y(w)
,
(6.17)
where we explicitly introduce, as in n”,,, the TNA form factor T,,,(k) and the cut-off function y(w) although this has again practically no effect in our numerical results. What really matters in practice for our purpose is the low-energy behavior (o = m,) of the whole 2p2h contribution (which has a p* dependence). As a check, taking the imaginary part of the II0 = n”,, + L!!,,, , one can extract the net absorbtive contribution lTg2 proportional to p2 at w = m, and one finds ~EI(u
= m,) = -i4r
Im Cop2
(6.18)
G. Chanfray, P. Schuck / Rho meson
349
with Im C,,=0.13m,6 in agreement with usual phenomenology of pion-nucleus optical potential and rr-mesic atoms (see for instance table 7.1 of ref. “) where the value 0.12mi6 is quoted for Im C,. The most naive way of incorporating the width of various intermediate states in the rho-meson self-energy is to make a replacement of the type
1
1
(6.19)
4O-(fii+~j)+i(pi+@j)-qO+(0i+0j)-i(/3i+~j)
>
and similar modifications for contributions with transverse Ah states. However, this procedure is not correct. In particular it could give an important unphysical strength (i.e. a non-vanishing Im 1) at qo= 0. The correct way to calculate the “twoquasipion” propagator is to make use of the spectral representation where the only input is the one-quasipion strength function Im 0, which is known either from a microscopic model or from phenomenology (or both). In this framework the imaginary part of the “two-quasipion propagator” is for k. > 0, given by % dE Im D,(E, k) Im D(k,,-E, k), Im G(k,, k) = -i (6.20) I0 where 0, is given in eq. (6.13). This formula was already used in ref. ‘I) to incorporate the finite lifetime of the pion in nuclear matter into the dilepton production rate. An analytic evaluation yields Pi+Pj
Im G(k,, k) = -1 zi5
j 4L?iflj ( f(ni”j;E)(E-~i-n,)2+(~i+p,)2
i,
Pi+Pj
(6.21)
-f(niyn,;-E)(E+f&+R,)2+(pi+A)2
>
with f(Oi,Oj;
E)=-!-&(arctg:-arctgy) I
I
I
+‘~(arctg;-arctg~)~ 7r
Pi+Pj
(6.22) I
Pj
We see that f(LJi, flj ; E = 0) = 0 and the unphysical strength disappears at E = 0. At the position of the pole one has f(L?i,L!j;E=Oi+Oj)=~P~$B,(arctg~+arctg~) I
I
I
+1(,rctg~+arCtg~). (6.23) m Pi+Pj J As we can see from fig. 8 the effect of the width is to strongly wash out the narrow structure seen in fig. 6. However, around normal nuclear matter densities a shoulder
G. Chanfray,
350
P. Schuck / Rho meson
0.15
0.125
0.1
0.075
0.05
0.025
0 INVAR MASS (WV)
Fig. 8. Imaginary part of the rho-meson propagator at various values of p/p0 with imaginary parts from 2p2h states and A-resonance included (for details see text).
survives and at p = 2p, even a small resonance can still clearly be seen. On the other hand the quenching of the rho meson is considerably reduced as well as its upward shift. To calculate the full dilepton collision rate we have to integrate the DPR given in eq. (2.1) over space-time. Clearly such a calculation requires a dynamical theory. However, to have a first estimate, we proceed in the following way. We assume the system to be equilibrated at some temperature T. As shown in sect. 2, the DPR is, in the limit of not too high temperature given by Im G,,(q,, q=O).
(6.24)
Ignoring medium effects in the rho-meson propagator, we fix the space-time volume and the temperature 73by adjusting our result to the one obtained by Wolf er al, 12> using a VUU code where also all medium effects on the pion have been omitted. Once the space-time volume and the temperature are in this way fixed, we include the medium effects and compare our results to the existing data ‘) (Ca+Ca at 1 and 2 GeV/nucleon). In the 1 GeV case (fig. 9a) where the temperature is T = 65 MeV we show the results at normal and two times normal nuclear matter density (pO). In the 2 GeV case (fig. 9b) the results are shown for p = 2p, (with T = 95 MeV). We see that the dilepton spectrum is considerably modified once in medium effects of
G. Chanfray, P. Schuck / Rho meson
351
the rho meson are taken into account. For instance the rho-meson peak is quite quenched. However, the interesting feature now is that the lower peak (at -500 MeV) seen on fig. 8 which was due to the XLT piece of the mass operator (6.10) gives a relatively pronounced structure. This reinforcement is due to the fact that the thermal factor in eq. (6.24) is much less effective at lower energies than at higher ones. If one is optimistic, one could interpret the data in such a way as to support our loo
0.1
0.1
0.2
0.2
0.3
0.3
0.5 M (GeV)
0.4
0.4
0.5
0.6 M
0.6
0.7
0.7
0.6
0.8
0.9
0.9
1.0
(Gd)
Fig. 9. (a) Dilepton production rate for %a + @‘Ca at 1 GeV per nucleon calculated for p/p0 = 1 and p/p0 = 2 according to eq. (6.24) 65 MeV (for details see text). The dashed curve corresponds to the rr+rrannihilation contribution obtained by Wolf et al. lo) used for the normalisation of our cross section (full line). The data are from ref. r). (b) Same as (a) but at 2 GeV per nuelcon and for p/po=2 only (T=95 MeV).
G. Chanjiiay, P. Schuck / Rho meson
352
shoulder
at -500
MeV. However,
on the one hand,
much
more precise
data are
necessary for firm conclusions and on the other hand one must add to our values the dilepton pairs from bremsstrahlung, Dalitz decay etc. These processes deform the spectrum
appreciably
figs. 9a, b should
with increasing
be interpreted
importance
to lower
only in a more or less qualitative
energies
‘*). Thus
way.
7. Conclusion In this work we continued our earlier studies **3)on the rho meson in dense matter in connection with dilepton production. We for instance are concerned here with the constraints coming from gauge invariance. First we show that even within our phenomenological model we can maintain gauge invariance at least for small total dilepton momenta. Since the dileptons are measured back to back our theory is adapted to the experimental situation. The way we achieve that goes essentially via once subtracted dispersion relations. Gauge invariance introduces vertex renormalisations which suppress formation of rho-meson strength around the 2m, threshold. On the other hand gauge invariance introduces many new contribution to the rho-meson self-energy. For instance graphs with pion plus rho meson and pion plus delta-hole now appear as intermediate doorway states. The latter ones give rise to mass spectrum. Such a feature a narrow structure at -500 MeV in the rho-meson is completely absent in the free space. Inclusion of the A-width and a width coming from couplings to 2p2h states washes out this unexpected structure to a large extent. However, at densities twice the normal one it clearly survives and due to its low energy it is actually favored over the rho meson in the dilepton production rates. Looking at figs. 9a, b one may deduce from the data something like a shoulder at the expected energy but certainly more precise data are needed before any firm conclusion can be drawn. Concerning our results for the rho meson proper we see that it is relatively little affected by medium effects. Its position, depending on the density
is slightly
shifted
upwards
and its strength
somewhat
quenched.
However,
one must realize that other effects, not taken into account here, may go into the opposite direction 13) and thus the in-medium position of the rho meson is theoretically certainly not settled. The future experimental determination of this quantity will be most interesting.
One of us (G.C.) would like to thank J. Delorme and M. Ericson for numerous discussions. We have also benefitted from discussions with W. Niirenberg, M. Hermann and B. Friman at the early stage of this work.
Note added in proof: After completion ref. 14) which also confirms our structure
of the present work we became aware of (Figs. 5, 8) in the rho-meson form factor.
G. Chanfray, P. Schuck / Rho meson
353
References 1) G. Roche et al., Phys. Rev. Lett. 61 (1988) 1069; Phys. Lett. B226 (1989) 228 2) G. Chanfray and P. Schuck, hoc. Int. workshop on pions and nuclei, Peniscola, Spain, June 1991; G. Chanfray, Z. Aouissat, P. Schuck and W. Niirenberg, Phys. Lett. B256 (1991) 325; G. Chanfray and P. Schuck, Nucl. Phys. A545 (1992) 271~ 3) Z. Aouissat, G. Chanfray, P. Schuck and G. Welke, Z. Phys. A340 (1991) 347 4) C. Gale and J. Kapusta, Phys. Rev. C35 (1987) 2107 5) P. Schuck, W. Niirenberg and G. Chanfray, Z. Phys. A330 (1988) 119 6) Cl. Korpa and S. Pratt, Phys. Rev. Lett. 69 (1990) 1502 7) E. Oset and L.L. Salcedo, Nucl. Phys. A468 (1987) 631 8) T.E.O. Ericson and W. Weise, Pions and nuclei (Oxford Science Publications, 1989) 9) L.H. Xia, CM. Ko, L. Xing and L.Q. Wu, Nucl. Phys. A485 (1988) 721 10) M. Herrmann, B. L. Friman and W. Niirenberg, Nucl. Phys. A545 (1992) 267~ 11) C.M. Ko, L.H. Xia and P.J. Siemens, Phys. Lett. B231 (1989) 16 12) G. Wolf, G. Batko, W. Cassing, U. Mossel, K. Niita and M. Schafer, Nucl. Phys. A517 (1990) 615 13) P.Y. Bertin and P.A.M. Guichon, Phys. Lett. B274 (1992) 133 14) M. Asakawa, CM. Ko, P. Levai and X.J. Qiu, Phys. Rev. C46 (1992) R1159
NUCLEAR PHYSICSA
The rho meson in dense matter and its influence on dilepton production rates* G. Chanfray Institut de Physique Nucliaires, 43 Avenue du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
P. Schuck Znstitut des Sciences Nuclkaires, 53 Avenue des Martyrs, 38026 Grenoble Cedex, France Received 15 June 1992 (Revised 26 November 1992)
Abstract:
Rho meson propagation in dense nuclear matter and dilepton production are considered in a phenomenological model with coupling to P+C states. Special attention is paid to constraints from gauge invariance. Taking into account the usual r-Ah coupling it is found that the rho meson in matter is slightly quenched and shifted upwards. A new interesting structure at -500 MeV appears in the rho-meson mass spectrum.
1. Introduction
Recently there has been considerable interest in dilepton production rates (DPR) originating from heavy-ion collisions or from proton-induced reactions ‘). This stems from the fact that the lepton pair couples via the photon directly to the rho meson (vector-dominance model (VDM)) and thus the dilepton invariant-mass spectrum could give important information on the rho-meson mass spectrum in dense and hot nuclear matter. Since the rho meson strongly couples to the T+T- channel the dilepton spectrum may also be very relevant for our knowledge on pion dynamics in dense matter. Preliminary studies ‘) have indeed shown that commonly accepted medium corrections to the pion propagation can strongly influence the mass distribution in the rho- and sigma-meson channels. In this work we pursue these studies and concentrate on the rho meson. We will for instance be concerned with the influence of gauge invariance on the mass spectrum. In detail the paper is organised as follows. In sects. 2 and 3 we make some comments on the dilepton production rate and the rho meson and discuss in a heuristic way medium effect and gauge invariance. In sect. 4 we present in detail our rho-meson propagator (with coupling to TT states) in free space and show that gauge invariance can be maintained even at a phenomenological level. In sect. 5 we consider the rho-meson propagator at l
Dedicated to the memory of A.B. Migdal.
03759474/93/$06.00
@ 1993 - Elsevier Science Publishers B.V. All rights reserved
330
finite
G. Chanfray,
density
and consider
gauge invariance.
again
P. Schuck / Rho meson
the corrections
In sect. 6 we give our
numerical
induced results
from the constraint and conclude
of
in sect. 7.
2. Rho-meson propagator and dilepton production As already
mentioned
in the introduction,
the dilepton
production
rate in relativis-
tic heavy-ion collision is strongly influenced by the rho-meson properties in hot and dense matter. This is a direct consequence of the vector-dominance hypothesis which states that, to a very good approximation, the hadronic electromagnetic current is a linear combination of light-vector-meson fields such as p, w or 4 for photon invariant-mass scale less than one GeV. For example it is well established that the pion electromagnetic form factor is almost completely dominated by the p-meson. It follows that the vector-dominance model (VDM) should give an accurate description of DPR for dielectron invariant masses larger than 2m,, since pion-pion annihilation is the dominant process. Moreover the universal rho-meson-hadron coupling allows one to incorporate into this VDM scheme processes involving nucleons and deltas. Since the excited nuclear system is in general produced from a heavy-ion collision the problem of dilepton production is in fact a dynamical one involving, at least in the initial phase, an out of equilibrium situation. However, assuming, as many authors did, the system to be equilibrated nuclear matter at some temperature T and some density p, one should obtain the DPR straightforwardly. Nevertheless the fact that free pions cannot survive in the medium due to their coupling to delta-hole states has to be taken into account. As a consequence only collective modes made out of pions and delta-hole (Ah) states can exist in the medium. In addition gauge invariance requires to couple the photon not only to pions but also to Ah states. All these effects can be incorporated in a consistent way through a medium modification of the rho propagator whose imaginary part is proportional to the DPR. Such an approach allows to calculate not only the in-medium renormalised electromagnetic vertex but also the in-medium corrected form factor
as required
by gauge invariance
and unitarity.
We will first clarify the connection between the DPR and the imaginary part of the rho-meson propagator in the VDM scheme. We postpone the detailed model calculation of this propagator to sect. 4 (free case) and sect. 5 (in-medium case). Let us assume that the initial macroscopic state is described by a statistical ensemble where Pi = exp (-BE,)/2 is the normalised probability to find the state Ii) with energy pi0 = Ei. Using standard Feynman rules, a straightforward calculation gives the DPR (per unit volume) for ultrarelativistic dielectrons with total momentum q = p+ +p= (q,,, q) and invariant mass M2 = qt = q“q, > 0
&= The leptonic
part which includes
~Yq)&Lv(q).
the explicit
(2.1)
factor e4 = (47~~)~ as well as the photon
331
G. Chanfray, l? Schuck / Rho meson
propagator
is given by
The physics is contained in the hadronic piece H,.(q)=;?
F
H,,(q)
(ilJ”(o)l~~lJ”(o)li>(2~)36’4’(q+Pf-Pi)
9
(2.3)
where J@(O) is the hadronic electromagnetic current and If) the final hadronic states. Using current conservation the DPR can be written as dR h4X
ff2 -H(q) = 37r2M2
(2.4)
with (2.5)
H(q) = -g”“%(q). Let us now relate the hadronic tensor r..
Jp’“(q; T)=-i
Using translational Jc”“(q;
d4xe’qxCL
to the retarded propagator
H&‘“(q)
p-w
J
(2.6)
invariance one easily finds T)
=
I-
eepEi
i,f
z
(~lJ”(~)lf)(flJ”(~)li> c2nJ36t3jcq+p. _p ) 1
qo+Ei-Ef+iv
/
~2T~3s(3)~q+p _p.j --eepEi W”(WWlJ”(O)l~) I * f Z
qo-J$+Ef+i~
(2.7)
Taking the imaginary part of Jp” and exchanging i and f indices in the first term one obtains the very general result -i
Im J’““(q; T) = (epqo- l)Hp”(q).
(2.8)
In the VDM the electromagnetic current is related to the third isospin component of the rho-meson field by the field current identity Jfi = (mt/g)pT where mp and g are the rho-meson mass and the universal VDM coupling constant appearing in the effective VDM lagrangian respectively. Let us define the retarded rho-meson propagator: Gp”(q; T) = -i
d4x eiqXCi 9
(il[~f(x),~XO)llW(xo).
(2.9)
G. Chanfray, P. Schuck/ Rho meson
332 Thus
using the field current H(q)
In the following
=
identity
and eq. (2.8) one obtains Im G““(q;
(ePqa- l)-‘(m~/7rg*)g,,
we will consider
case, from gauge invariance,
only back-to-back
-1 Q2
where the rho propagator
piece
T) .
(2.10)
kinematics,
only the space components
H(q,,q=0)=(e”4~-1))1m4P
for the hadronic
i.e. q = 0. In that
contribute:
i ImG”(q,,q=O,T) 7Ti=i
,
(2.11)
>
has the form 6” G”(qo;
T) = q;-m;-~(qO;
The detailed
derivation
of the rho-meson
(2.12)
T)’
propagator
and the scalar part of the rho
self-energy E(q,; T) is given in sect. 4 in the free case (see eqs. (4.4), (4.16) and (4.17)) and in sects. 5 and 6 for the in-medium case (see eqs. (5.1)-(5.7), (5.17), (5.18) and (6.10)) in the zero-temperature limit. The hadronic piece takes the form H(q,,
T) = (epqo- I)-‘lF,(qO;
Here F, is the (in-medium
modified)
IF,&,;
T)12( --$
Im z(qO;
pion electromagnetic
T)) .
form factor 7X1’.
T)I*= Im:l(&-$--WI,;
(2.13)
(2.14)
Before coming to the influence of the nuclear matter density which is the main purpose of this paper, let us first briefly discuss the finite-temperature effects. Assuming a free dispersion relation for the pion and ignoring all possible finitedensity effects the only mechanism for DPR is rr annihilation. The corresponding piece in 2 (q,, , T) (i.e. that piece having a non-vanishing imaginary part) contributing to H(q,; T) of eq. (2.13) is given by: 4g2 S(q0;
d3k
T)=y
(2T)3
v’(
k)k2G2,(q,;
T) +
purely real terms .
(2.15)
i
Here v(k) is a phenomenological form factor which defines the scale of the effective theory (i.e. -1 GeV). Its precise form fitted to the 7rm phase shifts was given in ref. ‘) and all the details concerning this effective VDM theory will be presented in sect. 4. The retarded two-pion propagator at finite T is given by3) G2,(q,,;
k;
T)=l wk
1+2h(T)
q:-4wz+
is ’
(2.16)
where fk( T) = (e@“*- 1))‘. The reader may notice that the quantity G,, which depends on the internal pion momentum k is not strictly speaking a propagator, since the “conventional” propagator is obtained after integration over k. However,
G. Chanfray, P. Schuck / Rho meson
333
for simplicity, it has become usual in our context to call G2_ also a propagator. We immediately deduce
H(qo; 7-1= krh;
T)l*j-
$
(5)@qo-24
~2W-2W
(2.17)
which is the result quoted in a number of places [see e.g. ref. “)I. Sometimes the Bose distribution is modified by the introduction of a chemical potential to account for non-equilibrium effects. If the temperature is not too high one can neglect its effect in the rescattering processes (i.e. in 1); in this case T appears only in an overall multiplying factor H(qO)=e-PqOIF,(q,;
T=O)/‘(---$*mT(q,;
T=O))
(2.18)
which is essentially the approximation made in all the papers devoted to DPR. In what follows we will implicitly use this approximation and concentrate on density effects and gauge invariance constraints at T = 0. For that reason the connection to DPR is still tenuous. The unified treatment of density and temperature effects is under investigation and will be the subject of a future publication. 3. Medium effects and gauge invariance: a heuristic approach It has been argued “) that the modification of the pion-dispersion relation may provoke a drastic enhancement of the DPR near the two-pion threshold. The origin of the effect can be seen very easily by simply replacing in formula (2.18) the free pion energy wk by the energy R,(k) of the quasi-pion in the medium [see ref. ‘) for notations]. Integrating over k one obtains:
With the possibility of d&(k)/dk becoming very small at threshold for finite densities this can give rise to a spectacular enhancement of DPR near 2m,. To see the magnitude of the effect one can assume (this is a quite good approximation) that 0;(k) = m’,+ */k2 with y a decreasing function of the density “). For instance at equilibrium density y is of the order of 0.3-0.4 depending essentially on the value of g’ (see below). We thus obtain
H(h) = +
e-Bq4~A90)12 & 66 - 4J3/2
(3.2)
which gives an enhancement factor of ten at normal nuclear-matter density. At even higher densities y may come close to zero (Kisslinger catastrophe). However, as already pointed out by Korpa and Pratt “) the above considerations are too naive, since gauge invariance is not respected. Restoration of gauge invariance essentially
G. Chanfray,
334
suppresses process
the enhancement.
is the annihilation
P. Schuck / Rho meson
Let us explain
this point in a heuristic
of a pion r+ with four momentum
(fig. 1). The corresponding current between a two-pion
amplitude is the matrix element state and the vacuum, i.e.
jP(q;
ki, M = K(q)(k,
where F,(q) is the (possibly medium we will assume it is given by VDM,
k’l_with a pion 6(
kc)
of the electromagnetic
- k2) p,
modified) rr form factor. that is there is a universal
coupling to baryons through -ye conversion (only the isovector here). Current conservation tests gauge invariance. We find qPjP(q;
way. The basic
(3.3) In the following electromagnetic part is considered
kl, k2)=FAq)(k:-k:)
(3.4)
which vanishes for on-shell free pions ( k2 = mt). However, if the process occurs in the medium, the quasi-pion corresponds to the pole of the in-medium pion propagator D,(k) = ( k2 - rn’, - S(k))-’ where the pion self-energy S(k) = k2fio( k,, k2) originates from the p-wave coupling of the pion to Ah states (ph states are practically negligible beyond the pion threshold) corrected by a screening effect (g’) due to short-range correlations. Ignoring for the moment complications linked to the imaginary part of S(k) (width of the delta and 2p2h states), “on shell” quasi-pions _ are characterised by a modified dispersion relation OJ: = rni + k2 + k*l7,,( k). It follows that qP.&(q;
k,, W = FAq)(k%(kA
-k%(k2))
(3.5)
which does not vanish in general. This clearly demonstrates that all the calculations of dilepton production including modification of the pion-dispersion relation violate gauge invariance, since the r.h.s. of eq. (3.5) does not vanish. The reason for this failure is quite obvious and is due to the fact that the photon only couples to the pions while via gauge invariance it also should couple to Ah excitations. In particular one has to take into account processes such as those depicted in fig. 2 where the dashed
box represents
the inverse
(virtual)
pion-photoproduction
amplitude
A’+*-)
of a r(+*-) with pion-pole term subtracted. In a model with pseudovector coupling this amplitude involves a seagull term and can be calculated to leading order in the inverse nucleon mass; one finds (A, is the photon field) (3.6)
x+(k,b.
“..
“‘b----nA
,.*.’.” y (q=k,+k$ n-(k2) ,.:.’ Fig. 1. Basic P+T-
annihilation
graph in free space.
G. Chanfray, J? Schuck / Rho
335
meson
rc+.. ..
.a..
.$Qjp .. .x” R_....”
Fig. 2. In medium gauge correction graphs for n+rr- annihilation.
where Pm = 2fm and S, T are the standard spin and isospin transition operators. In other words within this static approximation the amplitude reduces to a one-body spin-isospin operator. Thus the diagrams of fig. 3 are proportional to a,,(k) once short-range correlations are included. Therefore in the context of this simplified model the current matrix element becomes: j”(9; k,, kz)=F,(q)
(
84; ki, k2) = FAq)((I
+k+k,
40
+&(k,))k,
* k,(fio(k,)-fio(k,))
-(I +fio(k2))Q
> .
,
(3.7a) (3.7b)
Taking its divergence one obtains 4% =F,(q){(k:-mZ,-k:~o(k,))-(k:-m2,-k:~o(k,))},
(3.8)
which vanishes for “on-shell quasi-pions”. More generally we see that the divergence of the current involves the difference of the two inverse pion propagators and has the form of a generalized Ward identity. In the two-level model used in our previous papers ‘) one has two kinds of quasi-pions with energy O,(k), L?,(k) and carrying strength Z,(k), Z,(k) with Zi + Z, = 1. Let us call cu,(k) and al(k) the irreducible pion self-energy along those branches namely: a&k)
= ii&=
%2(k), k) .
Near the two-pion threshold only the lower branch 0, contributes modified by these gauge corrections becomes
-$
H(e) = e-Bq0)F,(qo)12(Z,(k)(l
(3.9) and the DPR
+a,(k)) $J2eF)-‘12*,.;,~ (3.10)
G. ChQnfrQy, I? Schuck / Rho meson
336
(30)
(3b)
(3c)
(31)
(se)
(St)
(39)
C3R)
,,._~
Z? Fig. 3. The various contributions
(39
--4-J-to the rho-meson
(3i) self-energy;
only (3a) and (3e) survive in free space.
It is now very easy to show that the modified hadronic current Z,(k) (l+ ~,(~))~/~,(~) is just equal to the group velocity d&$/d/c in the static approximation (i.e. a&/& very small which is true in practice). Consequently H(qo) is proportional to dO1/dk = yk/LJ,. This feature leads Korpa and Pratt “) to the conclusion that the DPR exhibits a dip in place of enhancement near q. = 2m,. However at a given invariant mass q. the DPR is also proportional to k3 = ($q$- M:)/ Y”~. More precisely one finds that (3.11)
G. Chanfray, l? Schuck / Rho meson
331
The net effect of gauge invariance
is to replace the factor y-“’ in eq. (3.2) by a factor Y-“~. This slight enhancement effect will be confirmed by more detailed numerical calculation (see sect. 6). However, in conjunction with the finite lifetime of the pion in a nuclear medium this is so small that it is practically invisible on the scale of our graph (figs. 5,6). 4. The rho-meson propagator in free space In this section we will be concerned with the rho-meson propagator the 7~1~channel in free space. The bare rho-meson propagator has the well-known form
coupled to
(4.1) where nr, is the rho-meson mass appearing in the effective VDM lagrangian. The coupling of the rho meson to the two pions can be described by the introduction of an appropriate mass operator. The dressed rho-meson propagator is then the solution of the corresponding Dyson equation. G““(q) = G?‘(q)+ According to gauge invariance energy has to be of the form
GY’(&&)GQY(q).
(q@E,J
q) =
0) and Lorentz covariance
(4.2) the self-
(4.3) The solution of the Dyson equation is straightforward (4.4a) with G(q’)=[q’-m;-iE(q’)]-‘.
The pion electromagnetic
(4.4b)
form factor in VDM is given by F,(q2)=
mz “;+H(q2)-q2
(4.5)
and the condition F,,(O) = 1 imposes that X(O) =o.
(4.6)
We now proceed to evaluate the self-energy .Z within our effective VDM lagrangian model. There are two contributions: 2 p” (fig. 3a) corresponds to the diagram with
G. Chanfray, l? Schuck / Rho meson
338
two pions Seagull
in the intermediate
states and Z-2”’ (fig. 3e) accounts
for the purely
real
contribution: v”(t)4k~kYoo,(k+~q)~~~(k-fq)
-
-
(4.7)
u*(r)UMk+tq)+%(k-fq)),
where Do,, is the free pion propagator. The effective hadronic theory is completed form factor depending on
through
the use of a Lorentz
invariant
t=f[(w(k+fq)+w(k-fq))2-4mZ,-q2]1’2 which is nothing The coupling factor
but the relative
constant
momentum
(4.8)
in the two-pion
g = 5 and the parameter
center-of-mass
qd = 3.3m,
appearing
frame.
in the form
(4.9) are fixed by fitting the pion electromagnetic form factor and the I = J = 1 PT phase shifts [for details see our previous publication ‘)I. The form factor has a flat behavior up to C= qd and strongly decreases beyond this value. The invariant mass scale defining the range of validity of the effective theory is thus A =2(q!j+m~)“*-1 In the rest frame of the rho meson
(qg = @,
P”(qo*,
which is a necessary of the self-energy
condition tensor
GeV. q = 0) it is easy to check that
0) =o
(4.10)
for gauge invariance.
The non-vanishing
components
are -
#.v(q,*,o)=6”~(q*)
i, j = 1,2,3’
t* q*-4t02(t)+iseg2
(4.11a)
, d3t m
v*(t) w(t).
(4-11b)
We immediately see that this calculation yields a non-vanishing value of x(O) and thus F,(O) is different from one. We now show that the problem is related to a of the model. To see violation of gauge invariance in this “naive” interpretation what is going on we repeat the calculation in the laboratory frame and, taking the 3-axis along q, we find q&p”(q)
=o
foruf3,
(4.12a)
G. Chanfray, k? Schuck / Rho meson
339
which does not vanish due to the (k - q)2 dependence of the form factor. This integral can be calculated in boosting k into the two-pion c.m. frame with a jacobian J
=
4k+fq) o(t)
(
4’ t
1-(4w2(t)+q2)1/2
(4.13)
>*
The result is
qp~r3(9) = -W_(O)
(4.14)
showing that it is precisely the violation of gauge invariance which entails F,(O) # 1. In addition it is possible, using the previous change of variables, to derive the explicit form of S:“(q): (4.15)
_w(q)=-(g’.-yp(q2)-_s(o))+P(o)
showing that both gauge invariance and Lorentz covariance are violated. The obvious solution to all these problems is simply to reinterpret the physical self-energy tensor as @(q2)-X(O))
*
(4.16)
The rho-meson self-energy appearing in the rho-meson propagator is then given by
=$ i (~~{(q~,O)-~~i(O)) i=l
1 q2 1 n2(‘) -4~*(t) -w(r) q2-4w2(f)+ie
’
(4.17)
In other words the real part of the self-energy is obtained from the imaginary part by a once subtracted (at q = 0) dispersion relation. This is essentially the procedure which we will adopt later on also at finite density. The agreement of the model with the pion form factor and the (I = J = ~)PCT phase shifts S(E)=arctg(ImE(E)/(E2-mz-ReE(E))) is excellent. Numerical calculations show that Re .Z(mE) almost exactly vanishes and thus the physical rho-meson mass is practically equal to the lagrangian mass. Notice that the mass parameter MP appearing in our previous work 2, is related to mP (we take mp =773 MeV) by M$ = mi-x(0).
G. Chanfray, P. Schuck / Rho meson
340
5. The rho-meson propagator at finite density As demonstrated
in the second
the basic PTW vertex first piece
section,
are renormalised
of the rho-meson
xf”(q)
EY(s;
P) = g2
4% (2r)4
I
the basic
YGTTT~~ vertex and consequently
in the medium self-energy
tl’(t)j’l(q;
according
to eq. (3.7). The
k)D,(k+;q)D,(k-+q)
(5.1)
becomes
k)j”(q;
with D,(k) the full pion propagator in the medium (as before, we only consider p-wave pion-nucleon interaction) D,(k) = (kz - w2(k) - k2fio( ko, k))-’ and j’(q;
k)=jp(q;
k,=$q+k,
k,=;q-k)
is given in eq. (3.7). The diagrammatic content of 2, is displayed in fig. (3a-3d) where the bubbles represent the irreducible pion self-energy (dominated by Ah states) already introduced in sect. 3. The second piece is the purely real seagull contribution (fig. 3e). XfY(
q; p)
= -g2g
WY
$+2(f)(Dm(k+fq)+D,(k-~q)). I
(5.2)
The third contribution involves in the intermediate states a one-quasipion line (D,) together with a delta-hole state (I?“) as shown in fig. 3f. To leading order in l/M,., the part Er” of the mass operator is then given by EY(q;
k) = g2
I
id4k -[v*(t)O;“(q; t2T14
k)l?“(k-;q)D,(k+;q)+(q+-q)]
(5.3) with 1 =--++:q)*, 40
Oy(q;
k)
Oyi(q;
k) = O;‘(q;
Of(q;
k) = 8”.
k) = (k’+$q’)/q,,
The next contribution is diagrammatically analog to the one of fig. 3d with the pion line connecting the two Ah bubbles replaced by a rho-meson line. One obtains (see fig. 3g) 2:”
= g2
id’k[v2(t)O:v(q; (2T)4
k)D,(k+$q)(fi°CpD$o)(k-;q)+(q+-q)] (5.4)
with O,“‘(q;
k) = O;‘(q;
k) = (q’k-
k’q)
O:“(q; #(q;
k) = ik x qi2/q:, k)=S”lk-~q12-(ki-4qi)(kj-~qj).
. (k-;q)/q,,
G. Chanfray, P. Schuck / Rho meson
341
In eq. (5.4) DP( k) = [ ki - wz( k) - C,k*l7’( ko, k)]-’ is the full rho propagator. C, is the usual parameter entering the PNN and pNA couplings “); we take in the following C, = 2.2 (strong rho coupling). One may argue that the rho propagator appearing in the self-energy should be consistent with the time-like rho propagator we are searching for. However, such a self-consistent calculation is unnecessary for our purpose, since this internal rho involves a completely different kinematical range (i.e. high-momentum range of the order of a few m,c). It is also important to notice that this contribution to the self-energy is separately gauge invariant. Grouping together figs. 3d, 3f and 3g allows us to reconstruct diagrammatically in the intermediate state the full longitudinal and transverse spin-isospin response functions as is shown in fig. 4. In a similar way the fifth contribution can be obtained from fig. 3c by simply replacing a pion line by a rho-meson line as shown in fig. 3h where the right-hand side bubble represents the spin-dependent part of the rho-meson photoproduction amplitude. To leading order in l/MN the dominant contribution relevant for our purpose connects the time component of the incident rho meson (photon) p’(q) to the space component of the outgoing rho p’(k); the corresponding amplitude is Jl”(k;
q)=i~pi(k)(Sxk)‘TpO(q)/qo.
(5.5)
As in the case of the pion-photoproduction amplitude (eq. (3.6)) it reduces to a spin-isospin one-body operator. The contribution to the rho-meson self-energy does not have pure space components (which are relevant at q = 0) but this is necessary to ensure gauge invariance. For completeness one should also add a graph deducible from Xry by simply exchanging the role of the pion and the rho; to leading order in l/M, this last graph contributes only to the pure time component 2:” of the self-energy tensor
25” = 8 +
id’k[02(t)(O;:(q; (2T)4
W’(q;
k)(iiOD,)(k+~q)(fioCpDp)(k-;q)
k)(ii’D~‘)(k+~q)C,D,(k-Iq))+(q~ -411
(5.6)
a W+@f)+W=
+a
Fig. 4. Reconstruction
.,./,,.............
of the full longitudinal and transverse intermediate state.
‘..,
spin-isospin
response
function
in the
G. Chanfray, P. Schuck/ Rho meson
342
with 0:: = -21k
x q/*/q;,
O;:=Og=-(qik-k’q).(k-$q)/qO, oya CO 3 0:: = ikx o;;=
qb’qi,
@=
O&z0
We finally add a graph which corresponds graph shown in fig. 3i which is given by ZY = -2
I
to a medium correction
k)[(fi”D,)(k+~q)+(jioD,)(k-$q)]
id4k (2p)4 o’(t)O:“(q;
to the seagull
(5.7)
with 0%;
k) = (k+$q)‘/q;,
O:‘(q; k)=O;*(q; O!(q;
k)=(k’++q’)/qo,
k)=O.
This contribution which is entirely real does not have a pure space component but again this is necessary to ensure gauge invariance. The last contribution Z; (fig. 3j) is just the usual irreducible rho-meson self-energy due to its coupling to ph and Ah bubbles. This gauge invariant piece vanishes at q = 0. Since our calculation will in practice be limited to q = 0, we will not consider it here. Let us now come to the problem of gauge invariance at finite density. The occurrence of explicit breaking of Lorentz covariance due for instance to the treatment of short-range correlations (g’ parameter) certainly forbids a structure for ~Fy(q) such as (4.3). As a consequence it is in general very difficult to maintain a vanishing value for the divergence of the self-energy tensor, (especially at finite momentum of the rho meson). However we will see that the subtraction of Z’“(0) in the definition of the rho-meson self-energy will allow us to maintain gauge invariance in the limit of small 141.To achieve that we will follow a procedure very similar to what we did in the free case. For the rho meson at rest (i.e. q = 0) one can easily show that PO{ go* =
J41,O)
= Y(
go*, 0) = 0
(5.8)
This means that gauge invariance is minimally satisfied in that case which corresponds to the condition of the Bevalac experiment ‘). The non-vanishing components of the self-energy tensor (i.e. x1, .&, ifs, .&) have the form 8@(qo*,O)
= S’jTz(q)
)
s(q)=+;
Fi(q$,O), i-1
(5.9)
G. Chanfray, F? Schuck / Rho meson
343
Explicitly this gives ~(qo,O) = g2
id4k @+ u’(~k~)[$k’(l+~(fi”(k+fq)
I
+fi”(k-+q)))2D,(k+$q)D,(k-;q) +2D,(k)+2ij0(k-$q)D,(k+fq)
+~(~°CpD~o)(k-4q)Drr(k+fq)lq=(qo,0)
(5.10)
which again does not vanish in general at qo= 0 and thus the medium-corrected form factor is not 1 at q = 0. As before this problem is actually related to gaugeinvariance violation in this naive interpretation of the model. To see this point let us calculate the divergence of the self-energy tensor at finite 141.An explicit calculation (using eq. (3.8)) shows that 9$%Io,
4) = -2g2
I
$$
v2(t)[(ki+fqi)(l+iio(k+;q))Dm(k+fq)
-(k’-+q’)(l+fi’(k-$q))D,,(k-fq)] id4k
-~‘(t)(k’(;q’+k+
+2g2 I (2?T)4
q)-qi(k2++q.
k))
x (fi°Cp@“)(k+~q)D,(k-$q) .
(5.11)
In the first piece of the above integral the time arguments h*iq, can be shifted to b by a change of variable. One can also remark that to leading order in q one has R(ko,
k+&q) = Wko,
k)+q*
k(l+fi’(b,
k))DZ,(ko,
k) ,
(5.12)
where we have used the fact that in the static approximation VJTO(k,, k) =o. After some angular integration one easily finds that to leading order in q qpx %lo, 4) = G(O) + 0(q2) - ji (O)+O(q’) =+qjE =+qpPi(0)+O(q2).
(5.13)
In the same way one can also show that 4$P0(40,
4) = O(q2) *
(5.14)
Thus to order q2 the two results of above can be summarised by 4JS
Yqo, 4) - 2
“WI = Wq2) .
(5.15)
Hence the redefinition oft; as e(q) -x(O) allows us to maintain gauge invariance in the limit of small (q(, independently of the precise form of v(t).
344
G. Chanfray, P. Schuck / Rho meson
Let us now come back to a rho meson at rest (q = 0). The the rho propagator is then:
Dyson equation for
G”(q~,O)=6”(q~-m~+i~)+Sik(q~-m~+i~)~Ck’(q,,O)GU(qo,O)
(5.16)
with ;I;“(qo,o)=6”(~(qo,o)-~(o))=~s”Ic(qo,o)
(5.17)
The solution of eq. (5.16) is G@(q,,O)=SU[q$-m;-iY(q,,O)]-’
(5.18)
To close this section let us make a final remark on the subtraction procedure which is again equivalent to a once subtracted dispersion relation for the real part of the self-energy. The substracted quantity is a priori different from &JO). However, the numerical calculations performed in the two-level model (see next section) show that the subtraction of &JO) in place of %(O, p) gives a negligible effect: in particular the deviation of the in-medium pion form factor F,(O) from one is of the order of 0.01 at 1.5~~ once transverse Ah states appearing in & and E’, are explicitly taken into account. 6. Results The numerical calculations are performed in the framework of the two-level model
already introduced in our previous work “). The main approximation of this model is to neglect the Fermi motion of the nucleon in the Ah bubble. In addition we first also neglect the width of the delta resonance and the 2p2h contribution in the polarisation propagator. This last approximation will be relaxed below. In the previous version of the model the Ah bubble has the form (using standard notation)
(6.1) with Egk= (Mi + k’) “2 - MA ,
(6.2a)
&,A(k)p.
(6.2b)
The polarisation bubble appearing in the rho and pion propagator short-range screening
iiO(k&,k)
= JT”,,/(l -gV2Q
=
is corrected by
C(k) k$-&+irf
’
(6.3)
where the effect of g’ (here g’ = 0.5) is to push the Ah energy upwards, i.e. .&,k= [& “t-C(k)g’]“‘.
(6.4)
G. Chanfray, I? Schuck 1 Rho meson
345
notice that we neglect the ph bubble but this has a negligible effect at the (high) energies of interest. The pion propagator in this model becomes
Also
D,(k,,~)=[~~-~2(W~2fio(%,~)l-‘=ii,
k2_;;(;))+irl
0
(6.5)
3
I
where the energies L?i of the collective pion and Ah modes and their respective strengths Zi (with Z, + Z2 = 1) are given in ref. ‘). In an analogous way the rho-meson propagator can be expressed in terms of two weakly collective transverse modes with energies Si( k) and strengths Yi(k):
The full transverse response function which will explicitly appear is given by Wko,
1- Y,(k)
k) = C(k) i
i=l ki-8f(k)+iv
*
(6.7)
The rho-meson self-energy can now entirely be expressed in terms of the various longitudinal modes L?,(k) and transverse modes 69i(k), i.e.
(6.8)
~(q0,0)=~LL(qO,O)+&-(qO,O) with 2 4: u2(k)k2
$~2(fIi(k)+f2j(k))Oi(k)LJj(k)
xZi(k)q(k)(l+f(ai(k)+aj(k)))2 q~-(Ri(k)+Oj(k))2+i~
(6.9)
’
2
4;
C i,j=12(f&(k)+
%j(k))fli(k)Z9j(k)
Z,(k)(l- y(k)) x
2
qo-(Ri(k)+%j(k))2+iT
’
(6.10)
The result of the calculation for the strength function S(q) = -1m G is
Im z(4) S(q)= - Iq2-m~-ReE(q)(2+IIm2(q)12
(6.11)
and is shown in fig. 5. We see that the rho meson itself is only mildly modified with a quenching and a slight upward shift. On the other hand, a spectacular structure at q = m, + iakxo shows up. This structure actually corresponds to the opening of the longitudinal-transverse (&) channel and completely disappears when &-r is arbitrarily put to zero (fig. 6). A priori one could imagine that this peak is related
G. Chanfray, P. Schuck / Rho meson
346
0.05
-
iNVAR MASS (Me’/)
Fig.
5.
Imaginary part of the rho-meson propagator at various values of p/p0 when the A-width is neglected (pure two-level model).
0.175
0
-&I INVAR MASS (M&f
Fig.
6.
Same as fig. 5 but with the longitudinal-transverse
contribution (.&) to the rho self-energy omitted.
G. Chanfray, P. Shuck
/ Rho meson
341
to a peak of Im ZLT but this is not the main origin, since the pion form factor, not directly proportional to Im 1, also exhibits this structure (fig. 7) (6.12) ‘The effect is actually mainly due to RPA correlations and is related to a strong peak ofReILT at q=m,,+E"ak=o. It should not be confused with the enhancement of the dilepton production rate originally put forward in ref. ‘). A very similar structure has been found recently in a work by Herrmann et al. lo) [see also ref. ‘)I. It is extremely important to check whether this narrow structure will survive when the widths of the states L!i and 8i are taken into account, i.e. when the imaginary part of the polarisation propagator is considered [see also ref. ‘)I. A simple extension of our two-level model is obtained, if the propagators of the mesons are generalised in the following way 1
1
(6.13)
k,-Ri(k)+i~i(k)-k,+Ri(k)-i~(k)
and an analogous expression for the rho-meson propagator. In other words the real part of the pole in the complex plane is taken using the above two-level model and the width of these states is obtained by explicitly taking into account the width of
L
INVAR MASS (MeVJ
Fig. 7. F’ion electromagnetic
form factor at various values of p/p,, when the A-width two-level model).
is omitted (pure
348
G. Chanfray,
the delta resonance
P. Schuck / Rho meson
and the coupling Pi(k) =
-I
to 2p2h states,
i.e.
.
2fli (k)
k2 Im fi”( o = fli( k), k)
(6.14)
with ii’(w,
k) =l7’(w,
k)/(l
fl’(w,
k) = Ei,,(w,
-g’Lro(w,
k)),
k) + fl;pa,b,
(6.15)
k) .
The Ah piece is for w > 0
I;T;,Jw,k)
1 =;‘$pv2(k)
w-Edk+$T(w,
?i
1
-k)
w+cAk
> ’
(6.16a)
The delta width T(w, k) receives two contributions. For the first one, r,, corresponding to the usual TN decay channel (but suppressed by Pauli blocking) we use the explicit form given by Oset and Salcedo [formula A.1 of ref. ‘)I. The second piece rh [$b is noted -1m Ed in ref. ‘)] which essentially gives a 2p2h contribution to the optical potential k217’ is also taken from ref. ‘). We actually take its value along the free pion line which in turn is very close to its value along the photon line. This approximation is certainly good, since the pion branch line is located in between these two branches. We parametrise r; according to
Ci(w; k) 2
=-ImE,
=24p
l+
w(MeV) - 140 280 Y(W)
PO
w(MeV) -450 20
)I 9 -’
(6.16b)
where the cut-off function y(w) allows to reproduce the drop of Im 24 obtained by Oset and Salcedo beyond o = 380 MeV. However, it should be emphasised that the precise form of this cut-off function which governs the high-energy behavior of rh has practically no effect on our numerical result for the rho-meson spectrum. Furthermore, there are other 2p2h contributions, not reducible to a delta width, which are especially important if one aims to reproduce the absorbtive part of the pion-nucleus
potential.
We take the usual form [see subsect. 7.2.3 and 7.2.4 of ref. “)I.
Z7ip2,,(~; k) = -iO.25
0
i
'ri,,(k)y(w)
,
(6.17)
where we explicitly introduce, as in n”,,, the TNA form factor T,,,(k) and the cut-off function y(w) although this has again practically no effect in our numerical results. What really matters in practice for our purpose is the low-energy behavior (o = m,) of the whole 2p2h contribution (which has a p* dependence). As a check, taking the imaginary part of the II0 = n”,, + L!!,,, , one can extract the net absorbtive contribution lTg2 proportional to p2 at w = m, and one finds ~EI(u
= m,) = -i4r
Im Cop2
(6.18)
G. Chanfray, P. Schuck / Rho meson
349
with Im C,,=0.13m,6 in agreement with usual phenomenology of pion-nucleus optical potential and rr-mesic atoms (see for instance table 7.1 of ref. “) where the value 0.12mi6 is quoted for Im C,. The most naive way of incorporating the width of various intermediate states in the rho-meson self-energy is to make a replacement of the type
1
1
(6.19)
4O-(fii+~j)+i(pi+@j)-qO+(0i+0j)-i(/3i+~j)
>
and similar modifications for contributions with transverse Ah states. However, this procedure is not correct. In particular it could give an important unphysical strength (i.e. a non-vanishing Im 1) at qo= 0. The correct way to calculate the “twoquasipion” propagator is to make use of the spectral representation where the only input is the one-quasipion strength function Im 0, which is known either from a microscopic model or from phenomenology (or both). In this framework the imaginary part of the “two-quasipion propagator” is for k. > 0, given by % dE Im D,(E, k) Im D(k,,-E, k), Im G(k,, k) = -i (6.20) I0 where 0, is given in eq. (6.13). This formula was already used in ref. ‘I) to incorporate the finite lifetime of the pion in nuclear matter into the dilepton production rate. An analytic evaluation yields Pi+Pj
Im G(k,, k) = -1 zi5
j 4L?iflj ( f(ni”j;E)(E-~i-n,)2+(~i+p,)2
i,
Pi+Pj
(6.21)
-f(niyn,;-E)(E+f&+R,)2+(pi+A)2
>
with f(Oi,Oj;
E)=-!-&(arctg:-arctgy) I
I
I
+‘~(arctg;-arctg~)~ 7r
Pi+Pj
(6.22) I
Pj
We see that f(LJi, flj ; E = 0) = 0 and the unphysical strength disappears at E = 0. At the position of the pole one has f(L?i,L!j;E=Oi+Oj)=~P~$B,(arctg~+arctg~) I
I
I
+1(,rctg~+arCtg~). (6.23) m Pi+Pj J As we can see from fig. 8 the effect of the width is to strongly wash out the narrow structure seen in fig. 6. However, around normal nuclear matter densities a shoulder
G. Chanfray,
350
P. Schuck / Rho meson
0.15
0.125
0.1
0.075
0.05
0.025
0 INVAR MASS (WV)
Fig. 8. Imaginary part of the rho-meson propagator at various values of p/p0 with imaginary parts from 2p2h states and A-resonance included (for details see text).
survives and at p = 2p, even a small resonance can still clearly be seen. On the other hand the quenching of the rho meson is considerably reduced as well as its upward shift. To calculate the full dilepton collision rate we have to integrate the DPR given in eq. (2.1) over space-time. Clearly such a calculation requires a dynamical theory. However, to have a first estimate, we proceed in the following way. We assume the system to be equilibrated at some temperature T. As shown in sect. 2, the DPR is, in the limit of not too high temperature given by Im G,,(q,, q=O).
(6.24)
Ignoring medium effects in the rho-meson propagator, we fix the space-time volume and the temperature 73by adjusting our result to the one obtained by Wolf er al, 12> using a VUU code where also all medium effects on the pion have been omitted. Once the space-time volume and the temperature are in this way fixed, we include the medium effects and compare our results to the existing data ‘) (Ca+Ca at 1 and 2 GeV/nucleon). In the 1 GeV case (fig. 9a) where the temperature is T = 65 MeV we show the results at normal and two times normal nuclear matter density (pO). In the 2 GeV case (fig. 9b) the results are shown for p = 2p, (with T = 95 MeV). We see that the dilepton spectrum is considerably modified once in medium effects of
G. Chanfray, P. Schuck / Rho meson
351
the rho meson are taken into account. For instance the rho-meson peak is quite quenched. However, the interesting feature now is that the lower peak (at -500 MeV) seen on fig. 8 which was due to the XLT piece of the mass operator (6.10) gives a relatively pronounced structure. This reinforcement is due to the fact that the thermal factor in eq. (6.24) is much less effective at lower energies than at higher ones. If one is optimistic, one could interpret the data in such a way as to support our loo
0.1
0.1
0.2
0.2
0.3
0.3
0.5 M (GeV)
0.4
0.4
0.5
0.6 M
0.6
0.7
0.7
0.6
0.8
0.9
0.9
1.0
(Gd)
Fig. 9. (a) Dilepton production rate for %a + @‘Ca at 1 GeV per nucleon calculated for p/p0 = 1 and p/p0 = 2 according to eq. (6.24) 65 MeV (for details see text). The dashed curve corresponds to the rr+rrannihilation contribution obtained by Wolf et al. lo) used for the normalisation of our cross section (full line). The data are from ref. r). (b) Same as (a) but at 2 GeV per nuelcon and for p/po=2 only (T=95 MeV).
G. Chanjiiay, P. Schuck / Rho meson
352
shoulder
at -500
MeV. However,
on the one hand,
much
more precise
data are
necessary for firm conclusions and on the other hand one must add to our values the dilepton pairs from bremsstrahlung, Dalitz decay etc. These processes deform the spectrum
appreciably
figs. 9a, b should
with increasing
be interpreted
importance
to lower
only in a more or less qualitative
energies
‘*). Thus
way.
7. Conclusion In this work we continued our earlier studies **3)on the rho meson in dense matter in connection with dilepton production. We for instance are concerned here with the constraints coming from gauge invariance. First we show that even within our phenomenological model we can maintain gauge invariance at least for small total dilepton momenta. Since the dileptons are measured back to back our theory is adapted to the experimental situation. The way we achieve that goes essentially via once subtracted dispersion relations. Gauge invariance introduces vertex renormalisations which suppress formation of rho-meson strength around the 2m, threshold. On the other hand gauge invariance introduces many new contribution to the rho-meson self-energy. For instance graphs with pion plus rho meson and pion plus delta-hole now appear as intermediate doorway states. The latter ones give rise to mass spectrum. Such a feature a narrow structure at -500 MeV in the rho-meson is completely absent in the free space. Inclusion of the A-width and a width coming from couplings to 2p2h states washes out this unexpected structure to a large extent. However, at densities twice the normal one it clearly survives and due to its low energy it is actually favored over the rho meson in the dilepton production rates. Looking at figs. 9a, b one may deduce from the data something like a shoulder at the expected energy but certainly more precise data are needed before any firm conclusion can be drawn. Concerning our results for the rho meson proper we see that it is relatively little affected by medium effects. Its position, depending on the density
is slightly
shifted
upwards
and its strength
somewhat
quenched.
However,
one must realize that other effects, not taken into account here, may go into the opposite direction 13) and thus the in-medium position of the rho meson is theoretically certainly not settled. The future experimental determination of this quantity will be most interesting.
One of us (G.C.) would like to thank J. Delorme and M. Ericson for numerous discussions. We have also benefitted from discussions with W. Niirenberg, M. Hermann and B. Friman at the early stage of this work.
Note added in proof: After completion ref. 14) which also confirms our structure
of the present work we became aware of (Figs. 5, 8) in the rho-meson form factor.
G. Chanfray, P. Schuck / Rho meson
353
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