T h ere are several ra th e r general conclusions to be draw n from all these detailed calculations. T h e salient featu re is th a t th e re is a ro b u st p h a se tra n sitio n as th e density is increased from a sy stem of isolated skyrm ions w ith no strongly preferred sym m etry, to a reg u lar lattice of h alf skyrm ions. T h e tra n sitio n is in general second o rd er an d th e condensed sy stem has lowest energy w ith fee sym m etry. T h e ph ase tra n sitio n an d condensed p h ase look rem ark ab ly sim ilar to M a n to n ’s so lu tio n [5] on a 3-sphere. T h e energy m inim um occurs a t a density of 0 .2 1 7 F m -3 a n d th e phase tra n sitio n a t 0 .0 6 8 F m - 3 . T hese sh ould n o t be co m p ared d irectly to nuclear m a tte r density 0.17jFra-3 . T h e Klebanov[6] choice of p a ra m e te rs sets energy an d length scales d eterm in ed by / whi ch is set a t th e u nrealistic value of 64.5M ev. Also skyrm ionic m a tte r co n tain s b o th nucleons an d A an d th e la tte r have n o t been p rojected o u t. A t hig h den sity th is m ay be a reaso n ab le ap p ro x im atio n , b u t n o t a t low densities. F u rth erm o re , th e present calcu latio n s co n tain only p o te n tia l term s an d no kinetic energy co n trib utio n s. T h e effects of including these to one loop ord er can be crudely e stim ate d from
th e w ork of Zahed e t al. [22] for a single skyrm ion on R 3 a n d G eneralis an d W illiam s [23] for S3. O ne loop c o n trib u tio n s a p p e a r to lower th e sky rm io n m ass by a b o u t 20% in th e d ilu te p h ase or n e ar th e p h ase tra n s itio n , w ith m o st of th e c o n trib u tio n occurring effectively in th e e4 te rm ra th e r th a n in e2. E ven larger corrections m ay occur n e ar th e m inim um energy of th e condensed p h ase. Such changes in th e relativ e s tre n g th of e4 an d e2 w ould increase th e above densities by a significant facto r, to b rin g th e m m ore in line w ith p red ictio n s for nuclear m a tte r.
In m o st of th e above calcu latio n s we have c o n ce n tra te d on th e zero pio n m ass case, using a finite pion m ass only to explore th e energy surface as a fun ctio n of < a > . T h is seem s a good p ro ced u re for several reasons. O u r m ain in terest is to explore th e effects of th e tra n s itio n to h a lf sk yrm io n sy m m e try an d th e pion te rm destroys this sy m m etry explicitly. T h e pion m ass is im p o rta n t for well s e p a ra te d skyrm ions, because over large regions of space (far from th e cen ter of th e sk y rm io ns), it is th e d o m in an t te rm in th e action. B u t in o u r dense system this occurs now here an d so th e pion an d a d istrib u tio n s will n o t be a ltered by m uch. T h ere is also th e q u estion of w h e th er it is correct to use th e free field values of th e pion m ass a n d /*- in such dense system s. A ccording to Forkel e t a l.[24], th e h alf-skyrm ion tra n s itio n co rresp o n ds to chiral sy m m e try re sto ra tio n a n d they arg ue one sh ould consider th a t th e /*- in eq u atio n (3.5.6) really co rresp o n d s to < cr > a n d w ould th u s give no c o n trib u tio n in th e con d en sed phase. T h e m ass te rm in (3.5.6) ensures th a t skyrm ions will prefer to c o n ce n tra te a ro u n d points
w ith cr = — 1 ra th e r th a n a = + 1 . T he d o m in a n t effect of th e pion m ass on th e energy is to in tro d u ce a te rm p ro p o rtio n al to th e volum e occupied by th e th e skyrm ion; th is shifts th e m inim um , decreasing Lo by 2.2% a n d increasing its energy by 42M eV; it also slightly increases th e energy difference betw een th e bcc an d fee m inim a. T h e pion m ass te rm always leads to a non-zero < cr > . For L < L c, < cr > is sm all a n d d o m in ated by th e balan ce betw een th e < cr > te rm in E m an d th e < cr > 2 te rm in (3.5.4), it is p ro p o rtio n a l to m 2/g (L ) and so gets larger as L approaches L c. A t th e m in im u m < cr > = 0.12. N ear L c th e q u a rtic te rm in < cr > in (3.5.4) becom es im p o rta n t a n d this has th e effect of sm o o th in g th e a b ru p t increase of < cr > , w hich occurs a t th e p h ase tra n s itio n as seen in (3.5.1). Above L c, again th e shift in < cr > is d o m in an tly p ro p o rtio n al to m 2, b u t involves b o th q(L) and v (L) . Essentially, in view of th e sm allness of th e pion m ass, it m erely leads to a sm o o th tra n s itio n fro m th e low to th e high d ensity phase. O f course, < a > will never be rigorously equal to zero and in a s tric t sense th e re will be no ph ase tra n sitio n for non-zero pion m ass. Sim ilar conclusions hold on 5 3[25].
Finally, we n o te th a t K ugler et al [12] proposed th a t for values of L > L0, c ry stallin e skyrm ionic m a tte r undergoes a p h ase s e p a ra tio n ra th e r th a n ex p an d ing to fill th e increased size of th e lattice. T h u s, for L > Lo, skyrm ionic m a tte r w ould be com posed of a region in w hich th e bary o n density was th a t of th e la ttic e a t L = L 0 an d a region of triv ial vacuum w ith zero b ary o n density and zero energy density. T hu s, beyond L 0 th e energy of skyrm ionic m a tte r w ould be
c o n sta n t an d equal to th e m in im u m value.
However, since no kinetic effects have so far been considered, it seem s u n reaso n ab le a t th is stag e to ignore th e b ran ch of th e curve, w hich a t low baryon densities will corresp o n d to a well se p a ra te d a rra y of B — 1 hedgehogs. It w ould seem reaso n ab le th a t th is b ran ch of th e curve m ay rep resen t som e tra n sie n t form of skyrm ionic m a tte r.
R e fe r e n c e s
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