This paper is devoted to the interplay between p-wave, s-wave pion-nucleon/
nucleus interaction and in-medium pion-pion interaction with special emphasis
on the role of the nuclear pionic scalar density driving a large amount of
chiral symmetry restoration. In particular we show that the $\pi NN$ coupling
constant and the Goldberger-Treiman relation are preserved in the nuclear
medium within certain conditions. We also discuss the related problem of the
in-medium pion-pion strength function.Comment: 12 pages, Latex2e, 2 figures included as eps file

Chanfray, Guy

Davesne, Dany

Nuclear Physics A

English

arXiv

This paper is devoted to the interplay between p-wave, s-wave pion-nucleon/ nucleus interaction and in-medium pion-pion interaction with special emphasis on the role of the nuclear pionic scalar density driving a large amount of chiral symmetry restoration. In particular we show that the $\pi NN$ coupling constant and the Goldberger-Treiman relation are preserved in the nuclear medium within certain conditions. We also discuss the related problem of the in-medium pion-pion strength function

Chanfray, G.

Davesne, D.

HAL-IN2P3

Chiral Pion interaction in nuclear matter and chiralsymmetry restorationG. Chanfray, D. DavesneTo cite this version:G. Chanfray, D. Davesne. Chiral Pion interaction in nuclear matter and chiral symmetryrestoration. Nuclear Physics A, Elsevier, 1999, 647, pp.125-131. <in2p3-00009565>HAL Id: in2p3-00009565http://hal.in2p3.fr/in2p3-00009565Submitted on 26 Jan 2001HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.1Chiral symmetry restoration of QCD and the Gross-Neveu modelT. ReiszaaInstitut fu¨r Theoretische Physik,Universita¨t Heidelberg,Philosophenweg 16,D-69120 Heidelberg, GermanyTwo flavour massless QCD has a second order chiral transition which has been argued to belong to the uni-versality class of the 3d O(4) spin model. The arguments have been questioned recently, and the transition wasclaimed to be mean eld behaved. We discuss this issue at the example of the 3d Gross-Neveu model. A solutionis obtained by applying various well established analytical methods.1. CHIRAL QCDMassless QCD has a flavour singlet, chiral sym-metry that is broken spontaneously, but becomesrestored at sucently high temperature. Fortwo flavours the symmetry restoring transition iswidely accepted to be of second order. Accordingto arguments rst given by Pisarski and Wilczek,the universality class belongs to that of the O(4)-spin model in 3d. Beyond others, two main as-sumptions made are It is the symmetry pattern that determinesthe universality class. Dimensional reduction is complete at thephase transition, that is, quantum fluctu-ations along the temperature torus do notrenormalize the classical 3d IR behaviour.In particular, the second assumption impliesthat fermions both do not interact as dynami-cal degrees of freedom in the IR, and do not evenrenormalize the (purely bosonic) eective theoryin such a way that the universality class of thechiral transition changes.Recently, the O(4) scenario has been ques-tioned, and the chiral transition was proposed tobe mean eld behaved.Supported by a Heisenberg fellowship, e-mail addresst.reisz@thphys.uni-heidelberg.de2. THE GROSS-NEVEU MODELThe reliability of the arguments against or infavour of the classical reduction picture is resolvedanalytically in the 3d Gross-Neveu (GN) model atnite temperature [1]. Here we outline the meth-ods and the results.The GN model we discuss is a four-point in-teracting model of N two-component masslessfermions, with partition function and actionZ =ZD D exp(−Sgn( ; ); (1)Sgn( ; ) =Zx γ  @ −2N(  2(x);whereRxR T−10 dx0RR2 d2~x. We choose hermi-tian γi in the representation with γ0γ1γ2 = i12.The model (1) serves as a good example forstudying a \chiral" transition because it revealsconsiderable similarities to massless QCD. In 3dthe role of chirality is replaced by parity, corre-sponding to the transformation (x)!  (−x);  (x)! − (−x): (2)The model is invariant under (2). The symmetrywould be broken by a fermionic mass term m  .For suciently large coupling , parity symmetryis broken spontaneously at T = 0, and it becomesrestored at high temperature by a second orderphase transition. The order parameter is given2by1N<  (x) (x) >; (3)which now represents the parity condensate.For large N it is convenient to rewrite themodel as a Yukawa-type interacting model, usingexp2N(  2=N42 12Z 1−1d exp−12N222     :Doing the quadratic fermionic integration, we ob-tainZ = cZD exp (−N SY ());SY () =Zz(z)242− tr s logK(z; z); (4)with K = γ@+1, and the trace is taken over thetwo spinor indices. We impose periodic boundaryconditions on the auxiliary eld  in temperaturedirection.In the form (4) the GN model is studied bymeans of the saddle point expansion for large N .Whereas the expansion in powers of  is non-renormalizable, the model becomes renormaliz-able in the 1=N -expansion. Renormalizabilityprovides a large freedom in the choice of the UVcuto. For simplicity, we choose a simple momen-tum cuto, which respects parity symmetry.3. PHASE TRANSITIONThe minimizing congurations (x)min   ofthe eective action (4) are obtained as the stablesolutions of the gap equation142− J1(2; T−1)= 0; (5)withJ1(x; T−1) = TXq02FZd2~q(2)21q2 + x;where F = fTj 2 Z oddg. (T ) is equal tothe parity condensate (3) to leading order in 1=N .3.1. N =1At zero temperature, the broken phase is real-ized for suciently large coupling , and we setthe T = 0 mass scale M byJ1(M2;1) 142< J1(0;1):As function of T , J1 is monotonically decreasing.The parity condensate (T ) approaches zero con-tinuously at Tc = M=(2 ln 2). It is straighforwardto show that the critical exponents dened by theparity condensate and its correlation functions asT ! Tc at m = 0 (and m! 0 at T = Tc) becomeγ = 1;  = 1=2;  = 0;  = 0;  = 1=2 (and  = 3).These are the critical exponents as predicted bymean eld analysis.3.2. FINITE NWe solve the question of the universality classby applying the following two methods. Dimensional reduction. For large N thephase transition is described by a 2d scalareld theory. There is more than one uni-versality class with the same symmetry. Bycomputing the couplings of the eectivemodel, we identify that part of the phasediagram that corresponds to the GN model. Linked cluster expansion. High tempera-ture series expansions convergent up to thecritical points are applied to the correla-tion functions of the eective model withthe prescribed values of the coupling con-stants. The high order behaviour allows fora precise measurement of the critical expo-nents.Dimensional reduction. This well estab-lished technique applies to local, renormalizablequantum eld theories in D  3 dimensions [2].For suciently high temperature T  T0, andfor spatial momenta ~k suciently small w.r.t. T ,the correlation functions are given by those of aD − 1 dimensional eld theory at zero tempera-ture, which itself is local and renormalizable.< e(~k1); : : : ; e(~kn) > (6)=< e(~k1); : : : ; e(~kn) >eff +O(~k1T; : : : ;~knT):3The most general action is obtained by applyinga zero-momentum expansion to the non-static ef-fective action,Sloceff (st) = Tn;m;f(n;m)T (3−D)=2;~k=T[− lnZDns exp (−S(T1=2st + ns))];(this provides the loophole in the no-go theoremof Landsman [3]), where T n;m;fx;y denotes the Tay-lor expansion w.r.t. x and y of order n and m,respectively, constrained byf(n;m)  (D − 1)− (D − 32n+m)  0:For the GN model, T0(N = 1) = 0. For suf-ciently large N , dimensional reduction worksdown to the phase transition temperature Tc. Weget with the large N expansionSeff () = c(T )Z~xm202(~x)2 +12(@)(~x)2+T 2c4N(~x)4 −c6N2(~x)6 +c8N3(~x)8:where we have truncated terms of order N−4. Allcouplings are positive. They are known to leadingorders of the 1=N expansion.Linked cluster expansion (LCE). Accord-ing to renormalizability, we are allowed to putthe eective model on the lattice. In a form wellknown for lattice spin models, the action becomesin terms of the lattice eldsbS0(0) = X~x22−(2)X=1;20(~x)0(~x+ b)+[20 +4Xi=2i(20 − 1)i](~x):LCE provides convergent series expansions ofconnected correlations and susceptibilities inpowers of . For xed i we have  ’ T−1=2.Critical exponents are obtained from the high or-der behaviour of the coecients. For instance,if2 =< e0(0); e0(0) >= XL0aL (2)L;Table 1The exponents γ and  of the GN model.N γ(N) 2(N)46.1 1.751(2) 2.001(2)24.3 1.752(3) 2.001(4)4.68 1.749(2) 1.999(2)a critical behaviour as 2 ’ (c−)−γ is obtainedfromaLaL−1=12c1 +γ − 1L+ o(L−1):The coecients are computed to the 20th order.Some results on the exponents for various N aresummarized in Table 1. As a result, the parityrestoring transition of the GN model belongs tothe universality class of the 2d Ising model.4. SUMMARYFor the Gross-Neveu model with large numberN of flavours, dimensional reduction works downto the phase transition. The eective 2d modelhas a rich phase structure, including both Gaus-sian and Ising model behaviour. However the val-ues of the coupling constants are determined bythe reduction technique. As a result of the com-putation, the parity transition of the Gross-Neveumodel corresponds to the Ising part of the phasediagram.For QCD we know that dimensional reductionworks at least down to 2Tc ([4,5] and referencestherein). Whether it works down to Tc or not isan open question. There are additional conver-gence factors 1=T compared to the Gross-Neveumodel because of the higher dimension, so N isno more required to be large. However, the renor-malized gauge coupling constant becomes largeby approaching Tc, so that perturbation theorynormally used in computing the eective actionbecomes worse. Other computational techniquesare required to realize the reduction below 2Tc.REFERENCES1. T. Reisz, "The Gross-Neveu Model and QCDsChiral Phase Transition", in "Field Theoret-ical Tools for Polymer and Particle Physics",4Lecture Notes in Physics, Vol. 508 (1998) 192H. Meyer-Ortmanns and A. Klu¨mper, eds.(Springer Verlag, Heidelberg), and e-printarchive hep-lat 9712017.2. T. Reisz, Z. Phys. C53(1992)1693. N. P. Landsman, Nucl. Phys. B322(1989)4984. P. Lacock and T. Reisz, Nucl. Phys. (Proc.Suppl.) B30(1993)3075. S. Dutta and S. Gupta, hep-ph/9806034

Chiral Symmetry Restoration and Pion Interaction in Nuclear Matter

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