27 July 1995 Physics Letters B 355 (1995) 288-294 PHYSICS LETTERS B Effective chiral lagrangian from QCD at nonzero chemical potential R.F. Alvarez-Estrada, A. G6mez Nicola Departamento de Fisica Tedrica, Universidad Complutense, 28040, Madrid, Spain Received 23 May 1995 Editor: R. Gatto Abstract We start from the euclidean QCD action for gluons and massless quarks with NC colours at finite baryon chemical potential ,!& and zero temperature. For ,u~ of the order of external momenta o(p) we derive an euclidean effective real chiral lagrangian at finite /.LB, up to and including o(p4), in terms of Goldstone Bosons (GB), in the large NC limit, including gluon contributions. Our effective action generalizes non-trivially the one obtained for ,& = 0 by previous authors, and it includes new ,uB-dependent terms. In particular, a topological term ,UBNB is found, NB being the baryon number in terms of GB fields with the correct normalization factor. Physical implications of the remaining PB-dependent terms are discussed briefly. In recent years, the low-energy dynamics of QCD has been extensively studied in the framework of chi- ral effective lagrangians. In the first works [ 11, in which chiral perturbation theory (,@T) was applied successfully, the most general effective lagrangian dis- playing all the symmetries required, and containing only Goldstone Bosons (GB) and external sources, was derived up to order o(p4) in external momenta and quark masses. Another approach also aimed at obtaining the effective lagrangian starts directly from QCD [2-51 and assumes that the number of colours (NC) is large. We will make use of this method below. The effective lagrangian and xF’T programs have also been extended so as to include finite temperature effects. Thermodynamical properties of a hadronic medium in thermal equilibrium have been studied at zero density in [ 6,7]. In this paper, we shall extend this program for the case of a medium at finite den- sity by inciuding the baryon chemical potential in the QCD lagrangian. Previous studies of chiral matter at finite temperature and density can be found in [8- lo]. We will perform our analysis for the T = 0 case. Our starting point will be the QCD partition function for massless quarks (chiral limit) at nonzero chemi- cal potential, which is given in euclidean space-time [ll-131 by Zbbl = I [d”Gl [dql [&I x exp- J [ d4n C - 2jeZ,,q NC I where q are the unrenormalized quark fields repre- sented by a column of Nf Dirac spinors, Nf being the number of flavours. The gluonic measure [ &] incor- porates the Faddeev-Popov determinant and the Yang- Mills piece exp ( -i GzVG&) _ We have L = qY,A,q with A, = 8, + $AaGE, Gt being the unrenormalized 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved ,SSDI0370-2693(95)00734-2 R.F. Aluarez-Estrada, A. G&nez Nicola /Physics Letters B 355 (1995) 288-294 gluon fields, and GE,, = a,G; - a,G; - gf”bcG;G; 289 incorporate the corresponding sources ucL (x) , aEL (x) into the QCD lagrangian when building up the gener- ating functional. These u = (v,), a = (a,) fields can also be considered as gauge fields, thereby gauging the symmetry SUL (Nf) x SU, (Nf) . Specifically, the sources up(x) and up(x) are traceless antihermitian matrices of SU( Nf) . In the non-anomalous effective action this will imply that ordinary derivatives acting on U fields will be replaced by covariant ones. Thus, by consistency, the U, a fields are o(p) quantities, as it happens with ordinary derivatives acting on GB fields. A term depending on U(x) and some mass parameter M will also be included [4,5]. This is the parame- ter that sets the scale of the theory and allows us to perform the derivative expansion. This M-dependent term can be regarded as a constituent quark mass term [ 161, or simply as the linear sigma model coupling between quarks and GB fields. The effective action I to be built up in this way will be a finite density gen- eralization of the one obtained in [ 1,4] . Then, we will study the following euclidean effective action: with A“ the generators of SU( NC) and pbc its s&uc- ture constants. We shall use euclidean gamma matrices yp and ys, which are hermitian, with {yp, yp} = 2S,, and {yp, ys} = 0. Overall normalization constants in path integrals will be included in the gluonic measure and will never be written explicitly. In Eq. ( 1) we have restricted ourselves to the case of a chemical potential term %I, attributing the same chemical potential to each f&our, which reflects baryon number conserva- tion in QCD. Thus, I is the unit Nf x Nf matrix and /cLg # 0 is the baryon chemical potential. Physically, this corresponds to a medium at finite baryon number density. We are interested in deriving an effective lagrangian at finite density in terms of only GB fields, starting from ( 1). This lagrangian would then lead to an ef- fective theory, valid at low energies. To be consistent, we will treat ,LLB as a quantity of order o(p), that is, of the same order as the external momentum appear- ing in the theory. Our effective lagrangian will then be valid in the low density regime, with respect to some energy scale. We will be more precise about the order of magnitude of this scale below. The partition function in Eq. ( 1) could also be written as a path in- tegral (not over quark and gluon fields but over GB fields) which, in turn, would contain an effective ac- tion depending also on GB fields. This program can be implemented up to a given momentum order, us- ing xPT, in terms of chiral loops, as it was done for low temperatures and ,_&g = 0 in [ 71. In order to ob- tain the effective lagrangian from QCD we shall need an important approximation, namely, we shall work in the large N, limit in which chiral loops are down by powers of l/N,. This is the approach followed in [ 2-41 for deriving the non-anomalous part of the ef- fective lagrangian. In [3] and [5], this program has been extended to the anomalous part, that is, the Wess- Zumino-Witten (WZW) action [ 14,151. To be pre- cise, we parametrize the GB field in the usual way as U(X) = exp(iti&/f) with U(x) E SU( Nf). For instance, for Nf = 2, fl and f are the pion fields and the pion decay constant fV, respectively. In the large NC limit the U(X) field can be regarded as a classical field, since chiral loops are neglected. Fur- thermore, in order to derive the Green functions with vector and axial-vector current insertions, we should = [I~“Gl~~l~~ql~~p--S~C~,q,~,~,~,U,CLBl J = J [d”Gl exp-rq[Gu,a,U,,u~l SE = J d4x[C +4yp(u,(.x) + Ysa,(x) - && + ~G&WX>clL +w+mR)] (2) whereCisthesameasin(l),b, = (,%g/Nc,O),and, as usual, qL = 9 q. As in [4], the following change of basis inRthe quark fields will yield a more suitable form for SE: QL =&L; QR=@~R (3) with t2( x) = U(x) . In this basis the euclidean action in Eq. (2) becomes: SE = J d4xDVQ 290 R.F. Alvarez-Estrada, A. Gbmez Nicola/Physics Letters B 355 (1995) 288-294 where vP = $ ( rP + I+) and uP = i ( yP - ZP) . Notice that Ifi and t/L are both antihermitian. We see that in this basis A4 gives rise to a mass term. The change of basis (3) induces a jacobian in the path integral mea- sure that should be taken into account in the effective action. We shall now ignore this jacobian but we will study it later. We are only interested in the real part Ir of the effective action I4 defined in Eq. (2). The imaginary part gives rise to the WZW action [ 51, and will not be dealt with here. We have: IT = ReI’,[~B,u,5,U] = -$_logdetD+D (6) The determinant of Z)+D can be calculated using the J-function regularization method [ 17,l X] : x / d4xTr(x] exp( --7D+D) IX) (7) where Y is an arbitrary energy scale, the trace Tr is taken over Dirac, flavour and colour indices and l?(s) is the Euler Gamma function. From Eq. (4) we find: fi = --apv [V,, V,l - 2W,ascT,u + 2My,b, (8) where tag,, = [ yP. yV] 14. Notice that IP, 5, and g depend on x and that they are not differential operators. The operator D+D in Eq. (8) has the generic form of an elliptic second order differential operator. The following expansion: =g ” H,(x) n=O (4*& (9) is called the Heat Kernel (HK) one (in the coinci- dence limit). The Seeley-de Witt coefficients H, (x) are explicitly known up to and including H6 [ 181. Here, D is the euclidean space-time dimension. Notice that the expansion in powers of the momentum does not correspond directly with the HK expansion. We will treat first the case in which gluon fields are dis- regarded. Recall that in the ,UB = 0 case [4] : i) This approximation leads to the o(p4) effective lagrangian and to specific values for the coefficients of the var- ious terms (with encouraging numerical agreement when compared to experimental data). ii) The o(p2) lagrangian is the non-linear sigma model, which, in turn, is used to relate the scale M to f,, and to derive the equations of motion for the U field. iii) The gluon dependence can then be taken into account in the large NC limit, at a later stage, and it turns out to give rise to corrections to the various coefficients of the model. The same features are going to be met in our case, as we will see. We now turn to calculate the effective action for rug # 0, first without gluonic corrections. Then I?, in Eq. (2) coincides with I (as s[d-G] = 1, by a suit- able choice of the normalization constant embodied in [ d”G] ) _ When the coupling of gluon fields to quarks are neglected, the chiral power counting is easier. The relevant coefficients in the HK expansion are given by [18]: Ho(x) = 1; Hi(X) = -B H2(x) = ;A2 - i[V,, [$,,8]] + $?P&, H3(x) = -ik3 + A({& 16,, [VP, 81 I} + ([~,m([qLA) t-Q3 ff4(x) = &I? + o(p6> where: (10) = rPLu +r55pv + [S,,Svl +‘W+[5,a,5vpl rpv =a,r, -4x,+ r,,r,l S/w = d,.dv - &Sp ; dp z 3, + D’,, -1 4111 If the gluon fields are neglected, k,, is o(p2), h is o(p), and then only the terms displayed in Eq. (10) contribute to the o(p4) lagrangian. Now, from (6), (7), (8) and (9) we have: R.F. Alvarez-Estrada, A. GbmezNicola/Physics Letters B 3.55 (1995) 288-294 291 ’ d u2s m&rS-‘e-7[ii,2-2b2) G,. = --- 2dvv) I x /d4xli.i O” H,(X)? (497-r p/2 1~ iii FGjn) (12) n=a Notice that the condition & < N:M2/2 appears naturally here and can be interpreted as what char- acterizes the maximum range of applicability of our approximation. The low density regime may then be defined as ,ug < M2. We obtain up to o (p4): NC G(O) = - r 1679 s d4x V2 (M2 - 2b2) Gt2) = i- &log v2 (M2-22b2) /d4x x tr[ - $((GpLY + Lt,,6vl)2+S$,) -8M2C2 + 8M2b2 - Sb4] NC M2 Gc3) = - r 32~~ M2 - 2b2 s d4x NC Gc4) = - r M4 d4x 32~~ (M2 - 2b2)2 J x tr [; (2E252 - SaSpSaSp ) + +pzx&&&5~ - 16b212 + ybaba&& + ;b4] (13) where the trace tr runs only over flavour indices. In Eq. ( 13 ), terms that are total d, derivatives have been omitted. The final step will lead from Eq. (13 ) to the ef- fective action in terms of GB fields U(x) and Ffv = a,.J, - &l, + [I,, I, I, F;L”y = apry - &rp + [rp, r, I. This can be done from the definitions in Eqs. (5 ) and (1 1 ), by using properties given in [4]. In addition, since we treat U (X ) as a classical field, it satisfies the equations of motion in the chiral limit: (D21J)Ut - U(D2Ut) = 0 (14) where the covariant derivatives acting over U fields are given by: D,U = i$JJ i- r,U - Ul, D,Ut = apUt - Utrp + l,Ut (15) Eqs. (14) follow from the o (p2) effective la- grangian, which is the non-linear sigma model. Then, it is consistent to use them in the o(p4) lagrangian. These equations remain as they stand when pug 3 0 is included, because there are no terms at o (p2 ) de- pending on the U field and PB simultaneously. To be consistent, as PB iS of order 0 (p ) and 0 (p6 ) terms have been neglected, we have to expand the constant factors appearing in Eq. (13) up to o (~2 ) in Gj”) and up to o (pi ) in the second and third terms in G (2). In the remaining terms, those factors have to be t&en at PB = 0. Regarding the various ,Uug dependent terms appear- ing in Bq. (13 ) we notice that the terms b212 and b,bpsa[p would give rise in the effective action to &D,UD,U~ (which, in turn, would imply a renor- malization of fr, as it will appear in (16) ) and to p%DoUDoUt, respectively. One of the most surpris- ing features of our computation is the fact that the coefficients multiplying both terms vanish exactly, and independently, which can be seen directly from Eq. (13 ) by expanding into powers of ,Ug . In addi- tion, the term (daep ),fa vanishes identically in the chiral limit. Thus, in the absence of gluon fields we finally arrive to the following finite density euclidean effective action at 0 (p4): G, [U,PB,F~,F~I = Gf [,%I + s [ d4xtr $ ~;D,uD,u~ + L2 (D,UtD,UD,UtD,U) + HI (F;vF,$ -t F;,,F,& I + L9 (F;vD,UD,Ut -I- Ffi,D,UtD,U) + LloUtFR UFL PY w - $$Eijk (FkUtDkU f F:DklJUt ) - 5 (DiUUt Fi - Ut DilJc$ ) + $EijtUt DilJU’ DjUUt DklJ 1 (16) 292 R.F. Alvarez-Estrada, A. G&nez Nicola /Physics Letters B 355 (1995) 288-294 The term I?; f ,..&B] (which would represent a shift in the free energy density) is: r:bBl = & /d**Lr[$ (i +log$) +2&P (,,,,$ - 1) >I (17) The ,u~g = 0 part in Eq. (16) is the o(p2) plus o(p4> lagrangian already obtained in [ 11. The var- ious coefficients appearing in it coincide with those obtained in previous pug = 0 calculations 12-41, as they should. For instance, fr is related to M through fz = sM2 log v2/M2. A typical value for M giv- ing good numerical agreement for the coefficients is M 21 320 MeV [4]. Regarding the terms appearing in.the complete o(p4) lagrangian in [l] that are not displayed in Eq. ( 16)) notice that the corresponding coefficients vanish in the large NC limit. We shall now treat the jacobian of the change of basis given in Eq. (3). It can be computed follow- ing the methods used by Fujikawa [ 191 in the cal- culation of anomalies. Since the ;D operator is not hermitian, we choose to regularize the jacobian as it was done in the second reference of [ 191. With this method, the Q fields are expanded in terms of eigen- functions of H s @D and the c ones in terms of those of 1? E 2X@. Notice that H(b,, rp,sp, M) = ?7( -b,, rp, -lp, -M) . Then, the HK expansion is used with the coefficients given in Eq. ( 10) and only H2 survives now in the regularization procedure. We have checked that the jacobian regulated in this way is indeed independent of ps, which does agree with pre- vious analysis [ 201. Furthermore, the jacobian itself gives contributions only to the WZW action [ 3,5,17]. Consequently, it is irrelevant for our purposes and will not be taken into account here. We now turn to comment about the different ,%B- dependent terms obtained in the effective lagrangian displayed in Eq. ( 16). Based only upon symmetry arguments, one could have asked what would be the most general set of operators that have all the sym- metries of the original QCD lagrangian, in the actual ,u~ # 0 case. Notice that this point of view tells noth- ing about the precise values of the coefficients of the different operators. Obviously, if the symmetries of the pug f 0 lagrangian were the same as those for the ,%B = 0 one, the only possible rug dependencies would appear through renormalizations of the various coefficients of the operators in the original lagrangian. Then, up to the order o(p4) considered here, the only dependencies would have been in the renormalization of fr previously mentioned. However, the chemical potential term in Eq. ( 1) does not respect two proper- ties that were valid for rug = 0, namely, charge conju- gation (C) symmetry and Euclidean covariance. The breaking of the latter covariance is due to the inclu- sion of the constant four-vector bp, Since the only Eu- clidean four-vector appearing in the ,u~ = 0 lagrangian is D,, we could get different operators by replacing D, vectors by b,, in suitable combinations so as to preserve both chiral SUL (Nf) x SUR (Nf ) symmetry and parity (P) invariance. This is what happens with the second PB-dependent term in Eq. ( 16)) which can be obtained in this way from the & term in the orig- inal ps = 0 lagrangian. Another operator with these features would be &D&D&, which comes from the L2 piece (and which does not appear in ( 16)) as explained above). Other types of operators would be those violating C but preserving P and chiral invariance. Two operators of this type are the first and the third l_~g dependent ones in Eq. ( 16), The first one depends on the exter- nal sources (and on U), while the third one depends only on the U field. Notice that these two terms, al- though real, are anomalous, that is, they are odd under the spatial parity operation PO: (t, x) --) (t, -x) for any four-vector (t, x). Of course, they are also odd under the simultaneous operation U c--t Uf , L t-f R. The parity operation P is the product of PO and the latter operation, so these terms are P invariant, as they should. However, as they are anomalous, when U is expanded in terms of 7p fields, only odd powers of ti will appear. Then, they could in principle contribute to anomalous processes (in xPT), like no -+ ‘yy at pug # 0 (a process studied previously in [ 211 at fi- nite temperature but for pug = 0). From the point of view of building up PB # 0 operators commented be- fore, one can regard these anomalous terms as coming from the gauged WZW term [ 14,151 when the exter- nal fields are replaced by bp . We remark that when ex- ternal fields are not considered (so that 1, = rP = 0)) the last term in Eq. ( 16) turns out to be the baryon R.F. Alvarez-Estrada. A. Gbmez Nicola/Physics Letters B 3.55 (199.5) 288-294 293 number, written now in terms of U fields. Recall that, as discussed in [ 151, when the WZW anomalous ac- tion is considered one can cast the baryon number current as: 1 J;=-E 2&2 WC@ uQgJub,uub,u (18) Then we conclude that in our effective action in Eq. ( 16)) a term of the form ~BNB has been derived, NB being the baryon number with the correct normal- ization factor. This term is indeed a topological one, because NB turns out to be the winding number of mappings U(x) from S3 into SU(Nf) [ 15,171, Next, we will take into account the coupling of quarks to gluon fields. In the large N, limit, the lead- ing approximation in the cumulant expansion [ 41: (exp -X) c-’ exp -(X) ; (X) = J [&IX (19) appears to be correct for ,%B # 0 as it certainly was for ,.&B = 0 [ 41. That is, we replace the gluonic in- variants appearing in the calculation by their QCD expectation values. The first possible gluonic correc- tion will be then proportional to the gluon condensate (cr/rrG;,G&,) (with CY = g2/4n-) , which is known to have a non-perturbative part [ 221. Its phenomeno- logical value has many uncertainties. The value for M = 320 MeV is consistent with (a/rG$,,Gt,) 21 [ 410 MeV] 4 [ 4,231. Here, we will calculate the first gluonic corrections to the ,%B dependent terms which would vanish exactly if gluon fields were neglected (see above). We will consider only the corrections proportional to the gluon condensate. It is not easy to estimate the order of magnitude of the terms we are neglecting. However, recall that when only gluon con- densate contributions (for p.B = 0) were considered in [4], the numerical agreement for the coefficients be- came rather remarkable, the results being always bet- ter than the ones obtained without gluon fields. Fur- thermore, regarding the gluonic corrections to fr, the next ones, proportional to the triple gluon condensate (GGG), have been computed for ,Y,B = 0 in [4] and found to be subdominant. UDJJt - ?%&,UD,,Ut c (20) The first term in the above equation gives the first ghOniC correction to f: for ,Ug = 0, calculated in [ 41, If U is again expanded in terms of 7p fields, the ,Ug de- pendent contributions in Eq. (20) give rise, at the low- est order o(d), to a PB-dependent modification of the propagator for the ti fields. Thus, this effect would ap- pear, via xP, in chiral loops, now at nonzero pug. This correction, in momentum space (P = (pa, p) ) would modify the pi and the p2 dependencies of the propaga- tor by different amounts, thereby reflecting the loss of covariance previously mentioned. The ,%Ug dependent terms obtained in Eq. (20) can give rise, via PCAC, to density corrections to the physical value of fr (which for rug = 0 and to lowest order in rr fields, coincides with the f?r appearing in the lagrangian [ 171) . The absence of these terms when gluon corrections are not considered (see above) seems to agree with the results in [ 91, in which fw scales with density due to the trace anomaly of QCD, proportional to G&G$, in the chi- ral limit. In fact, it can be seen that the ,!_&B-dependent terms in ( 16) do not contribute to the physical value of f7r. When gluonic corrections are included as indicated To summarize: 1) We have obtained the real part of above, the calculation becomes very much involved, the effective action at nonzero ,.&, up to o(p4> in the even if we concentrate only in the terms we are inter- ested in. Recall that i?,, defined in Eq. ( 11) gets mod- large NC limit. 2) Precise values for the coefficients multiplying every f_@-dependent term have been ob- ified as 8,, -+ i?,, + G,, , with G,, = gAaG$,. The tained. 3) A piece ,UBNB arises, NB being the topo- new term is o( 1) in the chiral power counting. The same happens with ,!? -+ i? - apvGIUY. Consequently, we have to treat H4 in order to find a term proportional to GGt2 (the non-linear sigma model piece), which is o(p2). This term has been analyzed for j_&Lg = 0 in [4]. When trying to calculate the ,!,&B corrections commented above, we have to look for contributions proportional to &GGt2. Since these terms are o(p4) we have now to go beyond Ha. In fact, for ,UB # 0 we find the searched contributions in the H5 and H6 coefficients [ 181. Finally we arrive at the following result. When gluonic corrections are included, the real part of the effective action I in Eq. (2) is given by Re I = Ir + To, Ir being given in Eq. (16) and: 1 (+G;,G;J rG= - M2 32 M4 TD,uD,U1 294 R.F. Alvarez-Esfrada, A. G6mez Nicofa / Physics Letters B 3.55 (19951288-294 logical baryon number, which physically corresponds to baryon number conservation. 4) A term contribut- ing to anomalous processes at finite ,~a has also been obtained. 5) When gluonic corrections proportional to the gluon condensate are included, two terms quadratic in PUS and in derivatives of the U field are also found, with their corresponding coefficients. The lagrangian obtained here could be an starting point for the study of density effects on: i) physical observables, in the low density regime, using for instance xPT, ii) Thermo- dynamical properties of hadrons near the chiral phase transition. The analysis of i) and ii) lies beyond the scope of this work. We thank Prof. A. Dobado for useful informa- tion. 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