The equation of state of dense matter.pdf

7/30/2019 The equation of state of dense matter.pdf

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PHYSICS RE PO RTS (Rev iew Sec tion o f Phys ics Le t te r s) 163 , Nos . 1 -3 (1988 ) 167 -204 . Nor th -Ho l land , Am ste rdam

T H E E Q U A T I O N O F S T A T E O F D E N S E M A T T E R *G . E . B R O W N * *

Physics Department , State Universi ty of New York at Stony Brook, Stony Brook, N Y 11794, USA

Abstract:The ories of dense m atter are d iscussed. I t is pointe d out that th ere is considerable confluence betw een the be st nonrela t iv is t ic and re la t iv is t iccalcula t ions. In part icular, the b ehavio r of the effective mass m* w ith density is very s imilar . This behav ior is essentia l for the eq uation of s ta te ; th ema in dens i ty depende nce seems to be de te rm ined by tha t o f m* .

Momentum dependence of the effective in teraction is d iscussed in connection with the re la t iv is t ic heavy ion experiments , which involve h ighnuc leon momen ta .

Consis tency w ith the la ter trans it ion to the quark /gluon phase is d iscussed, especia l ly the restric t ions which th is p laces on the behav ior of themass of th e scalar ~r-partic le with density . Kao n cond ensatio n is a lso considered.

Ge neral consis tency of nonrelat iv is t ic and re la t ivis t ic ca lcula t ions is found. I t is pointed ou t tha t inclusion o f effects which m ake t he i t-massdec rease wi th dens i ty wi ll so f ten the equa t ion o f s ta te .

In th i s a r ti c le we wi l l o f ten ta lk abou t "c o ld" an d " ho t" equa t ions o f s ta te . By "c o ld" equa t ion o f s ta te we mean th e EO S fo r the s tat ic sy s tem.By "h o t" equa t ion o f s ta te we wi l l mean the behav io r o f the sy s tem fo rmed in the co ll is ion o f heavy ions wi th l arge re lat ive mom en ta . W e hope tha tthe se ma t te r s - w h ich fo rm the m a in pa r t o f th i s a rt ic le - w i ll b e m ade c lea r in the deve lopmen t .

"Das sind richt ige Miinner, die wacker a uf die rauh en Berge steigen."1 . I n trod u ct i on

Th e b ehavior of nuc lear and neu t ron ma t te r a t den s i t ies h igher than nu c lear mat te r dens i ty , Prim, i s ofcons iderable in te res t . Th e reg ion Pnm < P < 4Pnm is encou ntered in the co l lapse of la rge s ta rs , l eading tosup ernov a explosions, a nd has b ee n discussed in relativist ic hea vy ion coll is ions with E / A u p t o - 2 G e Vo r e c m/ A up to -4 00 M eV, a l though w e sha l l show tha t i t is un l ike ly tha t den s i t ies much h ighe r than2pnm are a chieved in the la t te r case .At h igher dens i t ies and h igher tempera tures than those encountered here , i t i s expec ted tha t nuc learma t t e r m akes a phase t r ans it ion i n to a qua rk /g luo n p l a sma . I n t e rms o f t empe ra tu r e fo r ho t m a t t e r , t h isi s predic ted a t T = 150-200 M eV. In te rms of dens i ty for co ld ma t te r , w e es t imate u = P/Pnm ~ 10. Thisla t te r es t imate has been made both for a t rans i t ion to s t range mat te r [1] and for a phase t rans i t ionwi th in the nuc leo n [2] , wh ere , wi th increas ing dens i ty , the m eson c loud is squeezed out an d the qu arksin a chiral bag model f i l l a l l of space.

Th e reg ion 1 < u < 4- 5 , T -< 100 MeV seems to be an in te rm edia te reg ion of cons iderable in te res t , inwhich a descr ip t ion of the equa t ion of s ta te (EOS) in te rms of nuc leons (and mesons) may s t i l l beappropr ia te . On the o ther hand, there may be phase t rans i t ions in th i s reg ion . Cer ta in ly one has thel iquid-ga s t rans i t ion a t qu i te a low tem pera ture , 10 MeV-< T-< 20 MeV . M igdal [3] , Saw yer [4] , and* Suppo r ted in pa r t by the D epa r tmen t o f Ene rgy , Con t rac t No . DE-AC02-76ER13001 .** Th is pape r cove rs jo in t work wi th T .L . Ainswor th , M . Prakash , W. W eise and H . W ol te r a t S tony Brook . Th is work wi ll be pub l i shed la te r ,

af ter deta i led examples are w orked throug h. Th e pres ent author is so lely responsible for the way i t is d iscussed her e .

0 370-1573/88/$12.95 Else vier Science Publishers B.V. (N orth -H ollan d Physics Publishing Division)


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168 Theory of supernovae

Scalapino [5] proposed p ion condensa t ion , a l though the dens i ty for tha t i s probably put of f un t i ldens i t ies above th is reg ion by the s t rong p-meson tensor coupl ing [6] . M ore rec ent ly , kaon co ndensa t ionhas been prop osed [7] , and those ag encies which h inder p ion conden sa t ion he lp , i f they do an yth ing ,kaon condensat ion. As we shal l discuss, kaon condensat ion is a very real possibi l i ty (probabil i ty) atp - 3 P nm. We wil l put off further discussion of kaon condensat ion unti l sect ion 5.In sect ion 2 we shal l discuss nonrelat ivist ic calculat ions of nuclear matter , especial ly the region1 < u < 4 . We rev iew the w ork of Fr ied ma n and Pan dhar ipa nde [8] and of Fan toni e t a l. [9] , wh oexten ded th is w ork by com put ing the secon d-order cor re la ted basis func t ion cor rec t ion to thes ingle-par t ic le energy , thus in t roducing w-dependence of the e f fec t ive in te rac t ion . Fantoni e t a l .ca lcu la te the opt ica l mod el , which g ives an exce l len t fi t to the empir ica l energy depe nde nce of nuc leo nscat ter ing off nuclei .In sec t ion 3 we rev iew re lat iv is t ic nuc lear ma t te r ca lcu la t ions. Inc lus ion of e f fec ts f rom vi r tua l pa i rshe lps grea t ly in sa tura t ing nuc lear mat te r a t the cor rec t dens i ty . We poin t ou t tha t in the mean f ie ldcalculat ions, the effect ive mass drop s to o sharply w ith densi ty, resul t ing in a cold EO S wh ich is too st iff .Th i s ha s been r emed ied i n D i r ac -B rueck ne r c a l cu la t ions by Horowi t z and Se ro t [ 10 ] , who inc ludedis tor t ion of the nega t ive energy sea by the sca la r mean f ie ld ( the nuc leon loop te rms) . These h ighlyrepuls ive te rms, propor t iona l to the f if th and h igher powers of the sca la r f ie ld in the w ay Horow i tz andSerot renorm al ize , ke e p - when the se lf -cons is ten t equa t ion for the e f fec tive mass m* i s so lv ed - thesca la r f ie ld re la t ive ly small and m* la rge , com pared w i th the m ean f ie ld ca lcu la t ion . The ne t resu l t is anEO S c lose to , bu t s l igh tly sof te r a t h igh dens it ies than tha t of Fr iedm an and P andh ar ipand e . T he m * / min the two theories behaves very similarly with densi ty.

Many recent a r t ic les in the l i t e ra ture s ta te tha t re la t iv i s t ic heavy ion reac t ions show the EOS to bequite s ti ff in th e region of d ensi t ies Pnm < P < 4Pnm" W e co nsider this q uest ion in sect ion 4. I t is show ntha t there i s subs tan t ia l momentum dependence in the mean f ie ld potent ia l and , consequent ly , in thebinding energy per par t ic le E / A . Thi s momen tum dependence r e su l t s f rom two sou rce s : ( i ) t hewel l -known ef fec tive mass e f fec t, ( i i ) the decoupl ing of the sca la r f ie ld beca use of the decrease in thesca la r dens i ty Ps , as compared wi th p , wi th la rge momentum. In ana lys ing the exper iments , i t i s mosts imp ly exp re s sed i n t e rms o f t he known m om en tum depend ence o f t he op t ic a l mod e l po t en t ia l , wh ichinc ludes bo th o f t he se e ff ec ts . The mo men tum d ependen ce is o f t he gene ra l magn i tude r equ i r ed t oexpla in the d i f fe rences be tween the E /A found in t he heavy i on expe r imen t s and t hose f rom theFr i edm an-Pan dha r ipan de o r H orow i t z -Se ro t o r poss ib ly even so f te r EOS ' s . Our fo rmu la t ion is madein te rms of the mean f ie ld , so the momentum dependence d iscussed should be s t ra ight forward toinco rpo ra te i n t he V la sov , Uh l ing , Uh lenbe ck (m ore p rope r ly V la sov -N ordhe im ) ca lcu l at ions .

In sec t ion 5 we cons ider p ion and kaon condensa t ion .In appendix A we re turn to the ques t ion of phase t rans i t ions . We poin t ou t tha t i f dense mat te r i sgo ing to m erge smoo th ly in to qua rk m a t t e r - phenomeno log i ca l ly , t he Lee -W ick t r ans i ti on [11 ] - t henthe mass of the sca la r t r -par t ic le must decrease wi th increas ing dens i ty . S ince in the Horowi tz -Sero ttheory m~ effect ively increases with densi ty, such a decrease, i f i t sets in the region of densi t iescons idered , wi ll lower the i r EOS . W e show in de ta i l in the appendix a m odel for how m , can decreasewith increasing densi ty.I t i s po in ted out tha t pu t t ing condi t ions on the theory cons is ten t wi th nuc lear mat te r making aqua rk /g lu on t rans i tion a t a h igher dens i ty , v iz. , requi r ing the or -mass to go to zero a t a h igh dens i typ -> 10pnm, has e f fec ts a l ready in the range of dens i t ies cons idered h ere and wi ll most l ike ly sof ten theH orow i tz -Se ro t EO S. Ka on cond ensa t ion would have a simi la r e f fect . Our ten ta t ive conclus ion i s tha t


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G.E . Brown, The equation of state of dense matter 1 6 9

t h e E O S a t h i g h d e n s i t i e s i s r e l a t i v e l y s o f t , p r o b a b l y i n t h e r a n g e o f t h e B C K e q u a t i o n s [ 1 2 ] , n e c e s s a r yt o m a k e a s u p e r n o v a b l o w o n t h e f i r s t t r y .

I n t h i s r e v ie w w e c o v e r a l o t o f w o r k , n o n e o f it in d e p t h . O u r o b j e c t i s t o e x h i b it r e l a ti o n s b e t w e e nw o r k t o d a t e a n d t o i n d i c a t e e f f e c t s w h i c h s h o u l d b e i n c o r p o r a t e d i n f u r t h e r c a l c u l a t i o n s .

2 . N o n r e l a t i v i s t i c c a l c u l a t i o n o f n u c l e a r m a t t e rT h e m o s t c a r e f u l n o n r e l a t i v i s t i c c a l c u l a t i o n i s t h a t o f F r i e d m a n a n d P a n d h a r i p a n d e [ 8 ] . T h e y

d e v e l o p e d a r e a l i s t i c n u c l e a r h a m i l t o n i a n w h i c h f i t t e d t h e a v a i l a b l e s c a t t e r i n g d a t a . T h e s e c a l c u l a t i o n sh a v e b e e n e x t e n d e d b y D a y a n d W i r in g a [ 1 3] . S i n c e t h e m o m e n t u m d e p e n d e n c e o f t h e e ff e c ti v ei n t e r a c t i o n i s o f p a r a m o u n t i m p o r t a n c e , i t i s o f p a r t i c u l a r a d v a n t a g e t h a t F a n t o n i e t a l . [ 9 ] l a t e re x t e n d e d t h e F r i e d m a n - P a n d h a r i p a n d e w o r k b y a d d i n g t h e s e c o n d - o r d e r c o r r e l a t e d b a s i s f u n c t i o nc o r r e c t i o n , f i g . 1 , t o t h e s i n g l e - p a r t i c l e e n e r g y ( a n d o p t i c a l m o d e l p o t e n t i a l ) . T h i s i n c o r p o r a t e s t h et o - d e p e n d e n c e , s o i m p o r t a n t f o r t h e b e h a v i o r o f t h e o p t ic a l m o d e l p o t e n t i a l in t h e n e i g h b o r h o o d o f th eF e r m i s u r f a c e [ 1 4 ]. F o r p = P n m , F P h a d a n e f f e c t i v e m a s s m * / m - - 0 . 6 ; i nc l us i on o f t he c or re c t i on , f i g .1 , i n c r e a s e d i t t o 0 . 8 2 a t t h e F e r m i s u r f a c e . E x c e l l e n t f i t s t o n u c l e o n - n u c l e u s s c a t t e r i n g d a t a u p t on u c l e o n e n e r g i e s - 1 0 0 M e V w e r e t h e n o b t a i n e d .

I t i s w e l l k n o w n t h a t w i t h i n t e r a c t i o n s w h i c h f i t t h e t w o - b o d y s c a t t e r i n g d a t a , n u c l e a r m a t t e rs a t u r a t e s a t p - 2 Pn m - - m 3 ,,. F r i e d m a n a n d P a n d h a r i p a n d e e n f o r c e s a t u r a t i o n a t Pnm b y i n t r o d u c t i o n o f ar e p u l s i v e t h r e e - b o d y i n t e r a c t i o n ( a t l e a s t it i s t h r e e - b o d y a t n o t t o o l a r g e d e n s i t i e s w h e r e e - vlp -- -1 - ~ l P ) ,

8v = v , ( e -~1 - 1 ) , (2 . 1 )w h e r e 71 = 0 . 1 5 f m 3. H e r e v 1 is t h e a t t r a c ti v e t w o - b o d y i n t e r a c t i o n w h i c h c o n t r i b u t e s - 1 5 0 M e V / p a r t i c l eb i n d i n g a t n u c l e a r m a t t e r d e n s i t y . F o r p - P . m , i t i s s u f f ic i e n t t o e x p a n d

e -~1p - 1 - Y lP , (2 .2 )t h e c o r r e c t i o n t e r m t h e n g i v i n g a c o r r e c t i o n t o t h e e n e r g y p e r p a r t i c l e o f

8 ( E / A ) = - v 1 3 '1 P = 3 . 6 (P / Pnm) 2 MeW, ( 2 . 3 )

F ig . 1 . S eco n d - o r d e r co r r e l a t ed b as i s f u n c t io n co r r ec t io n to th e s in gle -p a r ti c le en e r g y .


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17 0 Theory of supernovae

Fig. 2 . Con tr ibution of v ir tual pairs to th e s ingle-part ic le energy. This is the low est-order d iagram in perturba tion theory .

w i t h o 1 - - 1 5 0 M e V / p a r t i c le a t n u c l e a r m a t t e r d e n s i ty . F r i e d m a n a n d P a n d h a r i p a n d e t ie t h is r e p u ls i v et h r e e - b o d y i n t e r a c t i o n i n w i t h p r o p e r t i e s o f t h e t h r e e - a n d f o u r - b o d y s y s t e m s .

A i n s w o r t h e t a l. [ 1 5 ] s h o w t h a t t h e c h i e f d i f f e r e n c e , f o r d e n s i t i e s p -< P , m , b e t w e e n r e l a ti v i st i c a n dn o n r e l a t i v i s ti c n u c l e a r m a t t e r t h e o r i e s c o m e s f r o m t h e c o n t r i b u t i o n s o f v i r tu a l p a i r s , f ig . 2. T h e y * )e s t i m a t e

(E /A)~ , = 2 . 4 ( p / p . m ) 8/3 M e V . ( 2 . 4 )N o t e t h a t t h e d e r i v a t i v e s o f ( 2 . 3 ) a n d ( 2 . 4 ) a r e v e r y c l o s e a t p - fln m " T h e e f f e c t s ( 2 . 3 ) a n d ( 2 . 4 ) h a v er o u g h l y t h e s a m e e f f ec t in l o w e r i n g t h e s a t u r a t io n d e n s i ty . F r o m t h e d e r i v a t io n o f A B B C P ( E /A ) r ~a r i s e s f r o m a v e r a g i n g a m o m e n t u m - d e p e n d e n t t e r m o v e r t h e g r o u n d s t a t e ; i . e . ,

= p / O ' 6 p f o ) ( P / P n m ) ,E/A)reI 2 .4( 2 2 2 ( 2 . 5 )w h e r e Pf0 i s t h e F e r m i m o m e n t u m f o r p = P ,m , a n d t h e a n g u l a r b r a c k e t s i m p l y t h e a v e r a g e . I t w il l b ei m p o r t a n t i n o u r l a t e r d i s c u s s i o n o f r e l a t i v i s t i c h e a v y i o n c o l l i s i o n s t o k e e p t h e m o m e n t u m a n d d e n s i t yd e p e n d e n c e s e p a r a t e . W e b e l ie v e t h e a b o v e d i s c u ss io n t o i n d i c a te t h a t o n e s h o u l d r e a l ly k e e p t r a c k o ft h e r e l a t i v i s t i c t r a n s f o r m a t i o n p r o p e r t i e s o f t h e i n t e r a c t i o n s , a l t h o u g h t h e e f f e c t s a r e n o t l a r g e f o r t h eg r o u n d s t at e o f n u c l e a r m a t t e r a t n u c l e a r m a t t e r d e n s i ti e s. T h e t e r m s ( 2 .4 ) a n d ( 2 . 5 ) a r e e x p r e s s i o n s ,e q u i v a l e n t i n p e r t u r b a t i o n t h e o r y t o t h e e f f e ct s f r o m l e t ti n g m - -- > m * i n t h e r e l a ti v i st i c t h e o r y . T h e yd e s c r i b e t h e e f f e c t o f d e c o u p l i n g o f t h e s c a l a r f i e l d a s t h e s c a l a r d e n s i t y p ~ , t o b e d e f i n e d l a t e r , d r o p sb e l o w p . L e t t i n g m - -- > m * i s e q u i v a l e n t to g o i n g o v e r t o a F u r r y r e p r e s e n t a t i o n [ 16 ].

3 . R e l a t i v i s ti c c a l c u l a t i o n o f n u c l e a r m a t t e r

T h e c o r r e s p o n d e n c e s b e t w e e n r e la t iv i s ti c m e a n f ie ld c a lc u l a ti o n s a n d n o n r e l a ti v is t ic o n e s w e r e d r a w ni n A B B C P [ 1 5 ] . T h e c h i e f d i f f e r e n c e s , f o r l o w d e n s i t i e s , w a s t h a t t h e r e l a t i v i s t i c t h e o r y c o n t a i n e d t h ee f f e c t s f r o m v i r t u a l p a i r s , (E /A)re~ o f e q s. ( 2 . 4 ) a n d ( 2 . 5 ) . O n e m i g h t s u s p e c t t h a t i t i s i m p o r t a n t t ok e e p t r a ck o f t h e L o r e n t z t r a n s f o r m a t i o n p r o p e r t i e s o f t h e i n t e r a c t io n s i n g o i n g to h i g h e r d e n s i t ie s , a n de s p e c i a l l y t o t h e h i g h e r m o m e n t a i n v o l v e d i n t h e r e l a t i v i s t i c h e a v y i o n c o l l i s i o n s .

* ) We re fe r to th i s pape r a s AB BCP in the fu ture .


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G.E. Br ow n, The equation of state of dense matter 171

We sha ll ba se ou r d i s cuss ion he re o n two r a the r comple t e i nvest iga ti ons o f D i r ac -B rueck ne r t heo ry :( i ) D eta i led ca lcu la t ions by te r Ha ar a nd M alf liet [17] . ( i i) Ex tend ed wo rk by Horow i tz and Sero t [10] .In both works shor t - range cor re la t ions and an t i symmetry a re inc luded . Horowi tz and Sero t inc ludema ny-bod y ef fects f rom the nega t ive-energy s ta tes (nuc leon loops) as we sha l l di scuss .

What was somewhat surpr i s ing was tha t the te r Haar and Malf l ie t ca lcu la t ions , wi th inc lus ion ofef fec ts of sho r t - range cor re la t ions and an t i symm etry , gave resu lt s very s imi la r no t on ly q ua l i ta t ive ly , bu teven quan t i ta tive ly , to the or ig ina l Sero t -W aleck a [18] m ean f ie ld theory . In nonre la t iv is t ic Bru eckn ertheory , shor t - range cor re la t ions of ten chan ge the co nt r ibut ions of the repuls ive vec tor in te rac t ion to theG-m atr ix by a fac tor ->2 . W e sha ll see la te r tha t the good a greem ent i s som ewh at of a co inc idence ,a l though the wo rk of Celenz a and S hakin [19] had indica ted tha t the d i f fe rences we re somew hat lessthan fac tors of 2 .

Relat ivist ic mean f ield theories, a lso the work of ter Haar and Malfl iet , give effect ive massesm*/m ~ 0.6 for p - Pnm" The se a re too smal l , cor responding to a s ingle-par tic le leve l dens i ty substan-t ial ly grea te r tha n the empir ica l one . T o some extent processes like tha t show n in f ig . 1 wo uld fix theseup, as they did the FP effect ive masses.

A mo re ser ious problem wi th the re lat iv is t ic mean f ie ld ca lcu la tions is tha t m*/m drops rap id ly to avery smal l va lue , -0 .2 for p - 4 P n m . This gives a very st iff cold equ ation of s tate [10]. A t f irs t s ight thisappears to f i t the re lat iv is t ic heavy ion resu l ts , wh ich seem to requ i re such a la rge com press iona l energy .Ho we ver , as we sha ll d iscuss la te r , the heavy ion exper imen ts a re per form ed a t h igh energy and involvehigh mo me nta , so tha t they see a subs tan tia lly s ti f fe r EOS than the co ld one . Thu s such s ti ff co ld EO S'sappea r t o be ru l ed ou t .

In the re la t iv i s t ic theory , for cons is tency , e f fec t ive masses m* should be in t roduced in to thenegat ive-energy , as wel l as pos i t ive-energy s ta tes [18] . These g ive the so-ca l led nuc leo n loop te rms. Inchi ra l theor ies , g - loop te rm s should a l so be inc luded [20] . The loop te rms in t rodu ce new man y-bodyfo rce s , t h r ee -body , f ou r -body , . . , up t o i n f i n i t e -body . The pa r t i cu l a r way advoca t ed by Se ro t andW alecka [18 ] and im p lemen ted in t he f r amew ork o f t he D i r ac -B ruec kne r t heo ry by Horowi t z and Se ro t[10] , i s to car ry out the renormal iza t ion such tha t the three-body and four -body te rms d isappear , andone i s le f t wi th the remain ing express ion , which could be expanded out in to f ive-body and more-bodyte rms . To ob t a in t he Horow i t z and Se ro t exp re s sion [10 ] , one can add cou n te r t erms t o t he A BB CPexpress ion [15] (appendix A) for E ~ a c, s o as t o r emove t he t h r ee - and fou r -body t e rms when E / q ac isexpanded ou t i n power s o f & .

Even af te r removal of the three- and four -body te rms, the nuc leon loops a re very repuls ive .Inc lus ion of them is equiva len t to a den s i ty-depen dent increase in sca la r meson m ass , as shown in f ig . 3 .Since the scalar mean f ield Ups goes as

O o s = p s , ( 3 . 1 )mesuch an increase makes the scalar f ield smaller and brings m* closer to m, a l though the re la t ion

m* = m + / . )& (3 .2)no longer holds , once loops a re inc luded . Amusingly , m*/m in Horowi tz and Sero t , espec ia l ly a t thehigher dens i t ies , comes back c lose to the FP va lue . Fur thermore , the Horowi tz and Sero t co ld energyE/A i s very c lose to tha t of FP. In o th er wo rds , inc lusion of the loops in H orow i tz and S ero t br ings therelat ivist ic theory, at high densi t ies , back close to the nonrelat ivist ic one.


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CN( O ")

Fig. 3 . I nc lus ion o f nuc leon loops i s equ iva len t to a dens i ty -depend en t i ncr ease in a -mes o n mas s [20] . W i th the Sero t -W alecka r enorm al i za t ionprocedure th i s co r r es ponds to t e rm s ~m2 = C p 3 and o f h igher o rder i n p .

I n t h e r e l a t i v i s t i c t h e o r y , s o l u t i o n f o r m * i s c a r r i e d o u t t h r o u g h a s e l f - c o n s i s t e n t e q u a t i o n . W e s h a l ls h o w b y m o d e l p r o b l e m s i n a p p e n d i x A t h a t o n c e t h is s e l f- c o n s i st e n c y i s e n f o r c e d , t h e p r e c i s e w a y i nw h i c h l o o p s a r e i n t r o d u c e d ( i . e ., t h e w a y i n w h i c h r e n o r m a l i z a t i o n i s c a r r i e d o u t ) i s u n l i k e l y t o a f f ec tt h e a n s w e r m u c h . T h e s e l f -c o n s i s te n c y e n f o r c e s a g r e a t i n s e n s it i vi t y o f r e s u lt s t o i n p u t . T h u s , a l t h o u g hw e s e e a n u m b e r o f c h a n g e s ( i m p r o v e m e n t s ? ) w h i c h c o u ld b e m a d e i n th e H o r o w i t z a n d S e r o tc a l c u l a t i o n , w e d o n o t b e l i e v e t h a t t h e s e w i l l s t r o n g l y c h a n g e t h e i r r e s u l t s i n t h e r e g i o n o f d e n s i t i e sPnrn < P < 4 P n m "I t is o f i n t e r e s t t o f in d o u t i n m o r e d e t a i l w h a t t h e i n c l u s io n o f n u c l e o n l o o p t e r m s d o e s . T h e i r e f f e c to n k e e p i n g m * f r o m d r o p p i n g r a p i d l y i s c le a r , r e s u l t i n g f r o m d e c r e a s i n g t h e s c a la r f i e ld U P s - I t is le s sc l e a r t h a t t h e e n e r g y p e r p a r t i c l e E/A c a n b e k e p t l o w ; t h is r e s u l ts f r o m t h e e f f e c t o f l o o p s o n t h e v e c t o rm e a n f i e l d , a s w e n o w d i s c u s s .

I n t h e t e r H a a r a n d M a l f l ie t c a l c u l a t i o n t h e v e c t o r f ie l d V p f o l l o w s t h e r e l a t i v i s t ic m e a n f i e l dc a l c u l a ti o n c l o s el y , e v e n t h o u g h s h o r t - r a n g e c o r r e l a t i o n s a n d P a u l i e ff e c ts a r e i n c l u d e d i n t h e f o r m e r . I nt a b l e 1 b e lo w w e s h o w a c o m p a r i s o n . * ) S i n c e th i s is t h e v e c t o r c o m p o n e n t o f t h e o p t i c a l m o d e lp o t e n t i a l , w e h a v e c h o s e n t h e t e r H a a r a n d M a l f l ie t v a lu e s f o r P la b = 4 6 0 M e V / c ; w e n e e d t h is Pta b t o b ea b o v e k F a t p = 4 P n m . T h e m e a n f ie l d v a l u e s a r e , o f c o u r s e , l i n e a r in p. T h e y a r e c h o s e n b y B o g u t a [2 1]s o t h a t t h e e n e r g y d e p e n d e n c e o f t h e o p t i c a l m o d e l i s g iv e n c o r r e c tl y f o r p = P nm ( s ee s e c t i o n 4 ). F r o mt a b l e 1 w e s e e t h a t t h e D i r a c - B r u e c k n e r v a l u e s g o v e r y n e a r ly l i n e a rl y w i t h d e n si t y .

T h e v a l u e s o f l,~ p o f H o r o w i t z a n d S e r o t a r e m u c h l e ss t h a n t h o s e g i v e n i n t a b le 1 a t h i g h d e n s i ti e s .F r o m t h e f a c t t h a t t h e y o b t a i n m*/m


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1 7 4 Theory of supernovaeH a r t r e e m e a n f i e l d t h e o r y . T h u s , s h o r t - r a n g e c o r r e l a t i o n s h a v e a l a r g e e f f e c t h e r e . T h e s e ( t e n t a t i v e )c o n c l u s i o n s s h o u l d b e c h e c k e d b y c a l c u l a t i o n s .

4 . M o m e n t u m d e p e n d e n c e ( e n e r g y d e p e n d e n c e , v e lo c it y d e p e n d e n c e )I t h a s b e e n c l a i m e d t h a t r e l a ti v i st i c h e a v y i o n c o l l i s i o n s s h o w * ) s t i ff e q u a t i o n s o f s t a t e ( s e e f ig . 5 )

[ 23 ]. I t i s i m p o r t a n t t o u n d e r s t a n d t h e s e e x p e r i m e n t s , i n o r d e r to s e e w h a t i n f o r m a t i o n t h e y g i v e a b o u tt h e e q u a t i o n o f s t a t e a t h i g h d e n s i t i e s .

I t i s c l e a r t h a t t h e r e l a t i v i s t i c h e a v y i o n c o l l i s i o n s s h o u l d s e e a n e q u a t i o n o f s t a t e d i f f e r e n t f r o m t h ec o l d o n e . A t t h e h i g h e s t e n e r g i e s w e s h a l l c o n s i d e r t h e c . m . k i n e t i c e n e r g y p e r n u c l e o n , e cm =4 0 0 M e V / n u c l eo n , t h e m o m e n t u m P cm s mc, f ar a b o v e t h e F e r m i m o m e n t a . T w o t y p es o f m o m e n t u md e p e n d e n c e a r e i n v o l v e d .

T h e f i r s t t y p e h a s b e e n f a m i l i a r t o u s f o r a l o n g t i m e . F o r n u c l e o n s s c a t t e r e d b y n u c l e i , i t i s w e l lk n o w n t h a t t h e o p ti c a l m o d e l p o t e n t i a l h a s a n e n e r g y d e p e n d e n c e ( f o r O - O h m )

0 -~ - - 0 ~ s m - I - 0 . 3 e , ( 4 . 1 )w h e r e 0 ~ s m i s t h e s h e l l m o d e l p o t e n t i a l a n d e i s t h e n o n r e l a t i v i s t i c k i n e t i c e n e r g y . T h i s f o r m h o l d s f o re n e r g i e s n o t t o o f a r fr o m t h e F e r m i e n e r g y , s a y e -< 5 0 M e V ( b u t n o t a t th e F e r m i s u r fa c e , a s w e d i s c u ssi n a p p e n d i x A ) . A b o v e - 5 0 M e V t h e e n e r g y d e p e n d e n c e l e ss e n s so m e w h a t . ( L a te r w e s ha ll r e p la c e

>:E


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G . E . Br o w n , T h e eq u a tio n o f s tat e o f d en s e m a t t e r 175

0 . 3 e b y 0 . 2 5 e i n a n a p p r o x i m a t i o n t o h o l d o v e r a w i d e r e n e r g y r a n g e . ) I n t h e h e a v y i o n c o l li si o n s, th i sm e a n s t h a t t h e n u c l e o n s i n t h e p r o j e c t i l e i n c i d e n t o n t h e t a r g e t n u c l e u s ( o r v i c e v e r s a i n t h e c m s ) s e e as u b s t a n t i a l l y l e s s a t t r a c t i v e m e a n f i e l d q / a t h i g h e r e n e r g i e s t h a n a t l o w i n c i d e n t e n e r g y .

I n t h e n o n r e l a t i v i s t i c f o r m a l i s m t h i s m o m e n t u m d e p e n d e n c e i s j u s t t h a t e x p r e s s e d t h r o u g h t h ee f fe c t iv e m a s s . T h e m o m e n t u m d e p e n d e n t in t e r a c ti o n is

2 2~ q / ( p 2 ) = P P ( 4 . 2 )2 m * 2 m "I n t h e a p p r o x i m a t i o n

m * 1mm 1 + 0 . 2 5 u ( 4 . 3 )

w h e r e u = P / P n m , w e s e e t h a t2g o / /( p Z ) = 0 . 2 5 u P m = 0 . 2 5 u e . ( 4 . 4 )

I n t h e r e l a t i v i s t i c f o r m u l a t i o n , f r o m t h e m e a n f i e l d e q u a t i o n( E - ( / p ) % - ' g ' p - m + U p s = 0 , ( 4 . 5 )

o n e f i n d s- 2( E - 17"p) - p Z = ( m + Ups ) . ( 4 . 6 )

O n e c a n t a k e t h e s q u a r e r o o t t o o b t a i nE = IT"p+ ~ / m * z + p 2 , ( 4 . 7 )

s i n c e m * = m + 0 p s . I n t h e n o n r e l a t i v i s ti c e x p a n s i o n o f t h e s q u a r e r o o t , o n e r e c o v e r s t h e a b o v ee f fe c ti v e m a s s t e r m a n d ( 4 .2 ) . F r o m ( 4 .6 ) t h e r e a r e , h o w e v e r , s o m e a d v a n t a g e s t o e x p r es s t h e e n e r g yd e p e n d e n c e t h r o u g h [ 1 5 , 1 8 , 2 6 ] ,

= + ( 0 p s ) +- 2( v p s ) + e . ( 4 . 8 )2 m m

T h e l a s t t e r m h e r e g i v e s t h e s a m e e n e r g y d e p e n d e n c e a s ( 4 . 4 ) p r o v i d e d w e c h o o s e ( ( Z p ) / m = 0 . 2 5 .( S li g h tl y s m a l le r t h a n m e a n f ie ld s u su a l ly g i v e; t h e y u s u a ll y g iv e g 0 / / - 0 . 3 e . B u t , a s n o t e d a b o v e , w ew a n t a l it tl e sm a l l e r e n e r g y d e p e n d e n c e w h i c h w il l h o l d , o n t h e a v e r a g e , o v e r a w i d e r r an g e o fe n e r g i e s . )

S u p p o s e w e s e n d t w o n u c l e i o v e r e a c h o t h e r , d e s c r i b i n g t h e s i t u a t i o n b y r e l a ti v i s ti c m e a n f ie l dd y n a m i c s , a s d o n e b y C u s s o n e t a l. [2 7 ]. I n t h i s c a s e , d e n s i t i e s d o n o t b u i l d u p * ), s o o n e s h o u l d e v a l u a t e

*) We shall see that in the high-energy region (E~ab/A > 1 GeV) densities do not build up much over the Lorentz-contracted 2yp, where~ , = E / m , once the momentum dependence is included in the interaction.


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17 6 Theory of supernovae8g(p 2) for u = 2 . Th en

8 ~ ( p 2) = 0 . 5 e . ( 4 .9 )

For our benchmark ecru = 400 M eV , this gives~o//(p2) = 200 M eV . (4.10)

One conver t s these poten t ia l s in to energy per par t ic le byE p 2 i f~ - = 0 . 6 ~ m m + p [ ~ + 8 ~ ( P 2 ) ] d P

2 fP F 1: 0 . 6 ~ m + p O / / d p + g ? / (p 2 ) ,

(4.11)

wh ere in the last s tep i t has bee n assumed tha t 8~// is l inear in p as in eq . (4 .4) . [One must be carefu l tod is t inguish be tween the momentum and dens i ty dependence , as done in eq . (4 .4) . ] Thus , (4 .10)con t r i bu t e s - 100 MeV to E/A . I t i s c lear tha t th i s momentum dependence i s a l a rge e f fec t .

Recen t l y , two a r ti c le s pu tt i ng t he m om en tum depend ence i n to V UU ca lcu l at ions appea red [28 , 29 ].Both ind ica te tha t wi th inc lus ion of momentum dependence , re la t ive ly sof t nuc lear mat te r equa t ions ofstate can re pro duc e the gen eral features o f relat ivis t ic hea vy ion coll is ions. T he A ichel in et al . [28] w orkemploys t he momen tum dependen t i n t e r ac t i on

V(Ap, p) = 1.57(ln[1 + 5 10 -4 A p 2 ] ) 2 ( p / P n m ) M e V ,wi th the re la t ive m om entu m of the in te rac t ing par t ic les Ap = P l - P2 g iven in unit s of Me V/c . Thisi n t er ac ti on i s ob t a ined f rom the em p i ri c a l mo me n tum depend ence o f t he n uc l eon -n uc l eus s ca tt e ri ng fo rP = P ,m, and assume d to ex t rapola te l inear ly in the dens i ty , s imi la rly to o u r ~a/./(p2), eq. (4.4). In800 M eV/nu cleon La + La co l li sions , the max imum dens i ty reache d wi th the sof t EOS i s ->3 , tha t w i ththe ha rd one i s - 2 .5 , wh i l e t ha t w i th t he so ft EOS p lu s t he above mom en tum dep endenc e is on ly -2 .Thu s i t is clear that V(Ap, p) effect ively adds g reat s t if fness. Th e effect of V(Ap, p) on th e f low angle isnot so la rge ; the e f fec tive EO S i s so s ti ff tha t nu c le i b rush by each o ther on the sur face , wi thou t muchslowing dow n. B ecaus e the p aral lel mo m en tum Pll is kep t large, th e f low angle is small.

Gale e t a l . [29] cons ider 400 M eV/n ucleon he avy ion co ll is ions. W i th in an e f fec t ive mass f ram ewo rk ,t he se au tho r s sy s tema t ica ll y work ou t t he e f fec ts o f mom en tum depend ence . W i th r e spect t o m ax imumdens it y r eached , t hey f ind t he so f t EOS p lus m om en tum depend ence t o l i e hal fway be tween t he so f tand s t if f EOS s . W i th the soft EO S plus mom entu m d epen den ce they find Pmax o be s ligh tly grea te r than2p, m. In th is w ork the f low angle d i s t ribu t ion f rom the soft E OS plus mom entu m d epe nde nce i s qu i tes imi la r to tha t f rom th e hard EO S. I t i s c lear from the wo rk of re fs . [28] and [29] tha t the m ajord i sc r epanc ie s be tween t he s ti ff V U U cu rve shown in fig . 5 and t he FP and B CK EOSs a r e r emo ved byin t roduc t i on o f a r e asonab l e mo me n tum depen dence i n t he i n t e rac t ion .

But we have not f in i shed . There i s another e f fec t , o f comparable magni tude , which i s c lear in therelat ivis t ic theory, not so obvious in the nonrelat ivis t ic one. In the cms both project i le and target are


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1 7 8 Theory of supernovaew h e r e g v a n d g s a r e t h e v e c t o r a n d s c a l ar c o u p l i n g c o n s t a n t s , m v a n d m s a r e t h e m a s s e s , P 'l v, P 2 v, P 'ls a n dP2s a r e t h e L o r e n t z - b o o s t e d d e n s i t ie s . V l . V 2 c o m e s i n b e c a u s e t h e i n t e r a c t i o n f r o m v e c t o r - m e s o ne x c h a n g e i s

2Of fvec tor = (1 - a 1 o / 2 ) 4 ~ F e - m ' ~ r ' ( 4 . 1 6 )

l r

a n d u p o n a v e r a g i n g o v e r s p i n , t h e e x p e c t a t i o n v a l u e o f 0 / 1 0/2 is / 3 1 " / ' ) 2 'N o w

f ll v = ~ f l l v , f l 2 v = ~ f l 2 v ' ( 4 . 1 7 )

t h e v e c t o r d e n s i t ie s b e i n g L o r e n t z c o n t r a c t e d , s i n ce t h e y a r e t h e f o u r t h c o m p o n e n t s o f fo u r - v e c to r s .T h u s , o u r e f f e c t i v e i n t e r a c t i o n d e n s i t y i s

- Y - -5 PlvPZv 1 - - - -5 P l s P 2 s . ( 4 . 1 8 )m v c / m sN o w y 2 = 1 + p Z / m 2 a n d v 1 v z = P l " P 2 / m 2 , w h e r e p i s t h e c m s m o m e n t u m . T h e r e f o r e , w e c a n w r i te

Y ,2 u 2v ( p l - p 2 ) 2V - my2 Plv Pz v - --SinsPlsPls + --mv:~P lv P Z v 2 m ' ( 4 . 1 9 )

a n d t h e f i n a l t e r m j u s t g i v e s t h e m o m e n t u m d e p e n d e n c e . T h i s d e r i v a t i o n , a l t h o u g h u s i n g a s o m e w h a to v e r s i m p l i f i e d m o d e l , g i v e s a s i m p l e f i n al r e s u lt .U s i n g t h e r e l a t i o n [ 3 1 ]

( m - m *) = - - -5 ( 2Pnm ) ,m v ( 4 . 2 0 )

o n e c a n s h o w t h a t i n c l u s i o n o f t h e f i n a l t e r m i n ( 4 . 1 9 ) s i m p l y r e p l a c e s m b y m * i n t h e s i n g l e - p a r t i c l ee q u a t i o n s i m i l ar to e q . ( 4 . 7 ) . I n a n y c a s e , it i s n o w c l e a r th a t f o r t h e m o r e g e n e r a l c a s e , o n e s h o u l d t a k ec a r e o f t h e m o m e n t u m d e p e n d e n c e b y a d d i n g t h e r ea l p a r t o f t h e o p t i ca l m o d e l p o t e n t i a l,

V ( A p ) - V ( 0 ) , ( 4 . 2 1 )t o t h e s i n g l e -p a r ti c le e q u a t i o n , m u c h a s d o n e b y A i c h e l i n e t a l. [2 8] b u t w i t h o u t t h e e n h a n c e m e n t f a c to rP / P , m " I n o t h e r w o r d s , e a c h p a r t ic l e i n th e p r o j e c t i l e j u s t s e e s t h e m e a n f ie l d i n t e r a c t i o n f r o m t h e t a r g e ta s a s i n g l e p a r t i c l e w o u l d a t t h e s a m e e n e r g y .

I n t h e a b o v e a n a l y s i s i t i s a s s u m e d t h a t t h e r e i s n o a p p r e c i a b l e b u i l d u p o f d e n s i t i e s i n t h e h e a v y i o nc o l li s io n . O n t h e o t h e r h a n d o n e c o u l d a s s u m e t h a t t h e r e i s c o m p l e t e t h e r m a l i z a t i o n i n th e c o l l is i o n .W i t h t h is s c e n a r i o h i g h e r c e n t r a l d e n s i ti e s w o u l d b e a t t a i n e d . T h e q u a l i t a ti v e a n a l y si s o f A B B C Ps h o w e d f o r th i s ca s e a b o u t t h e s a m e a m o u n t o f r e p u l s i o n c o m i n g m a i n ly f ro m t h e e x p l ic it m o m e n t u m


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G.E . Brown, The equation of state of dense matter 1 7 9

d e p e n d e n c e o f e q . ( 4 . 8 ) . T h e t h e r m o d y n a m i c p r o p e r t i e s o f b u l k m a t t e r i n th i s l im i t h a v e b e e ni n v e s t i g a t e d i n m o r e d e t a i l i n r e f . [ 3 6 ] , w i t h n u m e r i c a l r e s u l t s g i v e n i n r e f . [ 3 1 a ] .

N o t e t h a t t h e p i c t u r e h a s c h a n g e d c o n s i d e r a b l y s i n c e t h e i n i t i a l p i c t u r e o f i n e r t i a l c o n f i n e m e n t [ 3 2 ] .T h e l a t t e r w a s l a r g e l y b a s e d o n c e n t r a l c o l l i s i o n s ( s e e f i g . 8 o f r e f . [ 3 2 ] ) , a n d t h e n u c l e o n - n u c l e o nc o l l i s i o n s w e r e u s e d t o b r i n g t h e m a t t e r t o r e s t a n d t o d e v e l o p t h e s h o c k w a v e , a l t h o u g h i t w a s c l e a r l yp o i n t e d o u t i n re f . [ 3 2 ] t h a t m a t t e r w o u l d n o t b e s t o p p e d a t t h e h i g h e r e n e r g i e s E - > 1 G e V w e a r ew o r k i n g w i t h .

B e c a u s e o f t h e p a u c i t y o f c e n t r a l c o l l i s i o n s , i n v e s t i g a t o r s h a v e n o w g o n e o v e r t o t h e f l o w a n g l e a n dt o t h e a v e r a g e o f t h e v e c t o r p e r p e n d i c u l a r m o m e n t u m , w h i c h w o u l d b e z e r o f o r a c e n t r a l c o l l i s i o n .T y p i c a l c o l l i s i o n s a r e a s s u m e d t o h a v e a n i m p a c t p a r a m e t e r o f - 2 f m [ 2 8 ] .

I t w a s n o t e d a l r e a d y in S o b e l e t a l. [ 3 2 ] t h a t t h e l o n g i t u d i n a l m o m e n t u m d e c a y l e n g t h ( t r a n s p o r tm e a n f r e e p a th ) A ( p 0 ) w a s - 4 f m f o r a n in c i d e n t l a b o r a t o r y e n e r g y o f 1 G e V / p a r t ic l e , a n d t h a t Ai n c r e a s e d s t e a d i l y w i t h i n c r e a s i n g m o m e n t u m . T h i s m e a n s t h a t a t e n e r g i e s E /A ~> 1 GeV, col l i s ion s w i l ln o t b e a b l e t o b r i n g t h e p a r t i cl e s to r e s t . H o w e v e r , t h e y w i l l b e p u s h e d s i d e w a y s b y t h e m e a n f i e l d( o p t i c a l m o d e l p o t e n t i a l ) , w h i c h i s r e p u l s i v e f o r t h e s e e n e r g i e s .

I n f a c t, w e c a n m a k e a v e r y s i m p l e e s t im a t e o f t h e s i d e w a y s fl o w i n t h e v e r y h i g h e n e r g y r e g i o n ( b u tb e l o w e n e r g i e s w h e r e q u a r k s t r ip p i n g a n d s t r in g f o r m a t i o n b e c o m e s i m p o r t a n t ) . T h e e m p i r i c a l b e h a v i o ro f t h e o p t i c a l m o d e l p o t e n t i a l i s s h o w n i n f i g . 6 .

N o t e t h a t R e V op = 25 M e V f o r E - 1 G e V a n d , s i n c e i t i s r a t h e r f l a t f o r h i g h e r e n e r g i e s , w e s h a l la do p t t h i s va l ue fo r E ~> 1 Ge V.L e t u s t a k e a s i m p l i f ie d p r o f i le o f t h e n u c l e u s , a s s h o w n i n fig . 7 , w i t h l i n e a r s u r f a c e o f t h ic k n e s s 4 a ,

w h e r e a - 0 . 5 f m . W e t a k e t h e o p t i c a l m o d e l p o t e n t i a l t o h a v e t h i s s h a p e . C o n s i d e r t h e s c a t t e r i n gp r o c e s s s h o w n i n f i g . 8 . T h e m a x i m u m s i d e w a y s p u s h o n t h e p r o j e c t i l e w i l l c o m e w h e n t h e i m p a c t

I / E (M eV )- 2 5 7- , 5 0

F i g . 6 . E m p i r i c a l b e h a v i o r o f t h e r e a l p a r t o f t h e o p t i c a l m o d e lp o ten t i a l a s a f u n c t io n o f th e s in g le - p a r t i c l e en e r g y .

P

4 aFig . 7 . S implif ied prof i le of the nucleus .

1"

1-Z

F i g. 8 . S c a t t e r i n g p r o c e ss w h e r e t h e i m p a c t p a r a m e t e r b c o m e s j u s t t o t h e n u c l e a r s u f r ac e .


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180 Theory of supernovaep a r a m e t e r b j u s t t o u c h e s t h e s u rf a c e, a s s h o w n . T h e m a x i m u m s id e w a y s m o m e n t u m t r a n sf e r is t h e n

f d V f c o s 0 d VA p l = c o s 0 - ~ r d t = d z (d -~ -d }) d r- r' ( 4 .22)w h e r e , i n t h e f in a l e x p r e s s io n , t h e i n t e g r a l is to b e c a r r i e d o u t o v e r t h e c h o r d o f l e n g t h 2 1 i n fi g. 8 a n dc o s 0 is t h e a n g l e s h o w n . N o w c o s 0 - > 0 .7 , I i s a b i t l a r g e r t h a n b f o r a t y p i c a l n u c l e u s a n d d z / d t = 0 .7f o r 1 G e V p a r t ic l e s , s o

(Ap)max > 2 b ( V / 4 a ) / c , (4 .23)s ince

d V / d r = V / 4 a . (4 .24)N o w t y p i c a l l y 2 b / 4 a - > 4 , so tha t

( A P i ) m a x -> 1 0 0 M e V / c . ( 4 . 2 5 )I t i s n o t c l e a r t h a t m u c h o f t h i s m e a n f i e l d s c a t t e r i n g i s s e e n i n e x p e r i m e n t s i n t h e r a n g e

E / A - < 1 G e V , b e c a u s e e x p e r i m e n t a l c u t s a n d b i a s e s a n d , e s p e c i a l ly , t r ig g e r i n g o n h i g h m u l t i p l i c it i e s,t e n d t o e x c l u d e b o t h p r o j e c t i l e a n d t a r g e t f r a g m e n t a t i o n , d e t e c t i n g c h i ef l y f lo w th r o u g h s i z a b l e a n g le s .J u s t i n t h is r e g i o n t h e r e s h o u l d b e l a rg e m e d i u m d e p e n d e n t e f f e c ts o n t h e c r o ss s e c ti o n w h i c h g re a t lyi n c r e a s e t h e s l o w i n g d o w n . ( T h e c o m m o n l y a d d u c e d P a u l i b l o c k i n g , w h i c h c u t s d o w n t h e s e c r o s ss e c t i o n s , w i l l n o t a c t i n i t ia l l y w h e n t h e m o m e n t u m s p h e r e s r e p r e s e n t i n g t h e p a r t ic l e s in t h e t a r g e t a n dp r o j e c t i l e a r e w e l l s e p a r a t e d . )T h e f ir st o f t h e s e m e d i u m e f f e c t s c a n b e o b t a i n e d f r o m t h e p r o c e s s o f fig . 2 . C u t t i n g o p e n o n e o f th el o o p s i n o r d e r t o p r o d u c e a t w o - b o d y c r o s s s e c t i o n , o n e s t i l l h a s t h e o t h e r l o o p , w h i c h p r o d u c e s am e d i u m d e p e n d e n t e f f e c t. T h i s c a n b e i n c o r p o r a t e d i n t o a n e f f e c ti v e m a s s f o r t h e p r o j e c t il e . J u s t i n t h er e g i o n o f e n e r g i e s 5 0 0 - 8 0 0 M eV , D i r a c p h e n o m e n o l o g y c h e c k s o u t t h a t r e la t iv i st ic n u c l e o n p r o j e c t il e se x p e r i e n c e a s c al ar f ie l d o f - 3 0 0 M e V i n t h e t a rg e t , g i v in g a n e f fe c t iv e m a s s m * - 0 . 7 m . N o w t h ee - m e s o n c o u p l e s t o t h e s c a la r d e n s i t y t~ qJ a s~ H = g ~o ~ O ( x ) . (4 .26)

I n v a c u u m , ~ O ( p ) = $ *Y o O= m / ~ / P 2 + m2, whereas in the t a rge t nuc leus , tO*yoO= m * / ~ / p 2 + m , 2 .T h e r e f o r e , t h e c o u p l i n g is d i m i n i s h e d b y t h e f a c t o r

m * / V p 2 + m .2R = (4 .27)m / ~ / p 2 + m 2T h i s is t y p i c al l y o n l y a - 2 0 % r e d u c t i o n , b u t s i n c e t h e r e a r e s t r o n g c a n c e l l a t i o n s b e t w e e n a t t r a c ti v es c a la r a n d r e p u l s i v e v e c t o r , w h i c h c o u p l e s t o ~0*q~ n d i s n o t c h a n g e d i n m e d i u m , t h i s m o r e t h a n d o u b l e st h e t w o - n u c l e o n c r o s s s e c t i o n .C u t t i n g b o t h o f t h e l o o p s i n t h e p r o c e s s o f f i g . 2 g i v e s a t h r e e - b o d y f o r c e . E s t i m a t e s o f t h i s


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G.E. Br ow n, The equation of state of dense matter 18 1

three-b ody force f ind it to be substantial ly larger than th e two-b ody one. Typical m om en tum transfersare -m , , . Therefore , the ra t io of three-body to two -body cross sec tion i s roughly1 3R 3 z = ~ ( m J k r ) ---2, (4.28)

wh ere the fac tor of comes because the propag ator (q2 + m 2)- i i s about (2m2,,) 1 a t the maxim um inthe integrand , w hereas i t i s only m~ 2 conn ect ing the bubb le in f ig. 2. This rough est imate shows thethree-body force to be more effect ive in s lowing down part icles than the two-body interact ions.Put t ing the me dium de pen den t cor rec t ion to th e tw o-body cross sect ions and the three-bod y forceswil l increase the s topping pow er in the central regions by a large factor. (G oing to mu ch higherenergies, these effects wil l drop , because w ith typical mom en tum transfers of q - m, , , the t ransportm ean free p ath will not b e substant ially affected. The scat tering will be m ost ly throu gh small angles.)Le t us cons ider energies E / A = 400 -800 MeV. In gross detai l , the s ideways f low does no t look m uchdifferent [34] at the se en ergies than th at sho wn in f ig. 9, the ma ximu m Ap bein g ->100 M eV/c. ( In fact ,the p plots corresponding to f ig. 9 look qui te s imilar for anything on anything up to E~ab/A - 2 GeV."Since al l the experimental biases and ineff iciencies are folded into this observable i t i s extremelydiff icult to co mp are the exp erime ntal results w ith theoret ical resul ts ." [34]) Since we f ind the s toppingpower to be severa l t imes tha t usua l ly employed, the prec i se s lowing down sequence i s no longerimpor tant an d w e switch o ver to the shock wave t rea tmen t of Sobel e t a l . [32] , assuming tha t the m at te rwh ich overlaps wil l be com pletely s topp ed and c om pressed. We now ta ke th e surface to be sharp.Th e si tuat ion we c onsider is shown in f ig. 10. Since the t ransport m ean free pa th A(p0) is now smallcom pared w i th the nuc lear rad ius , we can no w ta lk about pressure , e tc . , in the in te rsec t ing volume ofthe two nuc le i . The perpendicular force i s now

F = PA , ( 4 , 2 9 )

O)

J

1 O O

5 0

0

- - 5 0

- - 1 0 0

|IY T

{ I y :t {

I I I I I

- - 1 0 1R a p i d i t y y

F ig . 9 . M e a s u r e d [ 3 3 ] n - pl a ne r a n s v er s e n o m c n t u m f o r A r ( 1 8 0 0 c V / n u c l e o n ) o n K C I .


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p l a n e

Fig . 10 . As the top nucleus moves to the r ight and the lower one to the lef t , the pressure on the p lane across their in tersection pushes themsideways. The matter in the (shaded) in tersecting volume is assumed to be brought to rest .

w h e r e A i s t h e c r o s s s e c t i o n a l a r e a i n t e r c e p t e d b y t h e t w o n u c l e i o n t h e p l a n e i n f i g . 1 0 . O f c o u r s e t h i sa r e a i s a f u n c t i o n o f t i m e , t h e s i t u a t i o n a t o n e p a r t i c u l a r t i m e b e i n g s h o w n i n f i g . 1 0 .

T h e r e w i ll b e a c h a n g e i n m o m e n t u m o f t h e e n t i r e c h o p p e d - o f f s p h e r e b e l o w ( a b o v e ) t h e p l a n e o tf f F o r c e d z( m p ) t o t a l = F o r c e d t = ( d z / d t ) " ( 4 . 3 0 )

F o r o u r r o u g h e s t i m a t e , w e s e t

f Z d z = V , ( 4 . 3 1 )w h e r e V is th e v o l u m e o f th e c h o p p e d - o f f s p h e r e , a n d f in d t h e A p p e r p a r ti c le i n th a t s p h e r e t o b e

(Ap)to,al P- (4 . 32)n V n ( d z / d t ) 'w h e r e n i s t h e p a r t i c l e d e n s i t y . T h u s , o n t h e a v e r a g e , e a c h p a r t i c l e i n t h e c h o p p e d - o f f s p h e r e h a ss i d ew a y s m o m e n t u m i m p a r t e d t o i t o f t h e p r e s s u r e p e r p a r t ic l e in t h e c r o s s -h a t c h e d o v e r l a p p i n g r e g i o nd i v i d e d b y t h e a v e r a g e l o n g i t u d i n a l v e l o c i t y . F o r E l a b = 2 5 0 M e V / A , ( v / c ) ~ 0 .61 .

N o w

P /n = ( 7 - 1 ) e , ( 4 .3 3 )w h e r e 7 i s t h e a d i a b a t i c i n d e x a n d e i s t h e e n e r g y p e r p a r t i c l e . F u r t h e r m o r e t h e d e n s i t y c o m p r e s s i o nr a t i o i n a s t r o n g s h o c k w a v e i s g i v e n b y

P _ 3' +____11 (4 .3 4 )P n r n T - - 1 "

F o r m a t t e r a t n o r m a l n u c l e a r m a t t e r d e n s i t y a n d o p t i c a l m o d e l p o t e n t i a l e q u a l t o z e r o ( a s i t i s f o rElab/A ~ 2 50 M e V ) ,

eo = 0 . 6 k 2 / 2 m = 2 5 M e V . ( 4 .3 5 )


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18/37

1 8 4 Theory ofsupernova em * . I t i s k n o w n t h a t a t f i n i t e t e m p e r a t u r e t h e t h e r m a l p r e s s u r e f o l l o w s c o m p l e t e l y f r o m m * ( p ) ,d i f f e r i n g f r o m t h a t o f a n i d e a l g a s , w i t h m r e p l a c e d b y m * , b y t h e m u l t i p l i c a t i v e f a c t o r F =1 - 3 ( d I n m * ) / ( d I n p ) [ 3 2 , 36 ]. W e k n o w o f n o s i m i l a r c o n n e c t i o n f o r t h e c o l d E O S s , s t a ti n g a n e x a c tc o n n e c t i o n b e t w e e n t h e b e h a v i o r o f m * ( p ) a n d E / A a s a f u n c ti o n o f p , b u t t h e w o r k i n a p p e n d i x As h o w s t h a t E / A is ch i e fl y d e t e r m i n e d b y m * , g i v e n k n o w n c o u p l i n g c o n s ta n t s a n d m a s s e s .P h a s e t r a n s i t i o n s w i l l s o f t e n t h e E O S . I n a p p e n d i x A w e c o n s i d e r t h e s o r t o f c o n d i t i o n s t h a t aL e e - W i c k t r a n si ti o n [ 11 ], w h i c h w e c o n s i d e r t o b e t h e p h e n o m e n o l o g i c a l d e s c ri p t io n o f t h e t r a n s i ti o n t oq u a r k m a t t e r , o r , e q u i v a l e n t ly , t h e c h i r al r e s to r a t i o n t r a n s it io n , c o u l d p u t o n t h e b e h a v i o r o f t h e e n e r g yw i t h d e n s i t y i n t h e r e g i o n o f d e n s i t i e s w e c o n s i d e r . T h i s t r a n s i t i o n o c c u r s , h o w e v e r , o n l y a t v e r y h i g hd e n s i t i e s a n d t h e r e a r e a t l e a s t t h r e e t r a n s i t i o n s , w h i c h w e s h a l l n o w d e s c r i b e , w h i c h o c c u r a t l o w e rde ns i t i e s .

P i o n c o n d e n s a t i o n w i l l b e t h e f i r st p h a s e t r a n s i t i o n t o o c c u r w i t h in c r e a s i n g d e n s i t y . F o r m a n y y e a r s ,t h e p o s s i b i l i t y o f p - w a v e p i o n c o n d e n s a t i o n w a s c o n s i d e r e d . I t w a s d r i v e n , s u p p o s e d l y , b y t h e v e r ya t t r a c t i v e p i o n - n u c l e o n p - w a v e i n t e r a c t i o n . S h o r t - r a n g e c o r r e l a t i o n s b e t w e e n n u c l e o n s f r o m p - a n dt o - m e s o n e x c h a n g e g i v e , h o w e v e r , l o c a l f i e l d c o r r e c t i o n s o f a L o r e n t z - L o r e n z n a t u r e . T h e s e a r es u m m a r i z e d i n t e r m s o f a m o m e n t u m i n d e p e n d e n t p a r a m e t e r g ' . T h e s e c a n b e a c c o m m o d a t e d i n t h et h e o r y o f p i o n c o n d e n s a t i o n b y c h a n g i n g [ 3 7 ]

2g , , V ( g ) 2 = g , ( 1 - g ' ) , ( 5 . 1 )w h e r e g A i s t h e a x i a l - v e c t o r c o u p l i n g c o n s t a n t .

N o w t h e s - w a v e r r - - n e u t r o n i n t e r a c t i o n i s r e p u l s i v e a n d o p p o s e s ~ - c o n d e n s a t i o n i n n e u t r o n s t a r s .In t h i s c a se ,

( g,~ )2 _ 1 > 0 ( 5 . 2 )f o r t h e a t t r a c t iv e p - w a v e i n t e r a c t i o n t o p r e d o m i n a t e o v e r t h e r e p u l s i v e s- w a v e o n e [ 37 ]. S i n c e g ' i sc a l c u l a t e d [ 3 8 ] a n d m e a s u r e d [ 3 9 ] t o b e - >0 .9 , t h i s is h a r d l y p o s s i b l e .

I n t h e c a s e o f N = Z n u c l e a r m a t t e r , t h e a v e r a g e p i o n - n u c l e o n i n t e r a c t i o n i s z e r o , s o t h e r e i s n oo p p o s i t i o n f r o m t h e s - w a v e p i o n - n u c l e o n i n t e r a c t i o n , b u t p i o n c o n d e n s a t i o n w o u l d o c c u r o n l y a t m u c hh i g h e r d e n s i t i e s t h a n w e c o n s i d e r h e r e w i t h t h e s m a l l g ~ .

R e c e n t w o r k b y K a p l a n a n d N e l s o n [ 4 0 ] h a s , h o w e v e r , p o i n t e d u p t h e p o s s i b le i m p o r t a n c e o f c h ir a ls y m m e t r y b r e a k i n g i n t h e f o r m a t i o n o f c o n d e n s a t e s . I n t h e c a l c u l a t i o n o f p i o n c o n d e n s a t e s , o n eg e n e r a l l y a d d s t h e s y m m e t r y b r e a k i n g t e r m

2 2~ H s a = - f ~ , m ~ c o s 0 , ( 5 . 3 )

w h e r e f ~ is t h e w e a k d e c a y c o n s t a n t , 0 t h e c h i r a l a n g l e ( s e e a p p e n d i x A ) . F o r s m a l l 0 ,2 2 , - 2 2 ~ 2~nsB - f ~ m ~ , + 1 f , , m ~ t t

a n d s i nc e r t 2 = si n20- --02 , t h e l a s t t e r m i s j us t t h e p i on ma ss t e rm .N o w t h e p i o n m a s s c o m e s a t a m o r e f u n d a m e n t a l l e v e l f r o m

(5 . 3a )

2 1 +m , , - 2 f 2~ ( o l [ [ Q - d , H s B ] , Q s l l o ) , ( 5 . 4 )


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G.E . Brown, The equation of state of dense matter 1 85

w h e r e Q a r e t h e a x i al c h a r g e s a n d H s B , th e s y m m e t r y b r e a k i n g h a m i l t o n i a n , i sHSB = m u ~ u + m dd d + m j s ( 5 . 5 )

i n t e r m s o f t h e q u a r k f i e l d s u , d a n d s . A n e x p l i c i t c a l c u l a t i o n g i v e s2 = _ 1m r 2 f 2 ( m u + m d) ( 01d ( 0 ) u (0 ) + d ( 0 ) d ( 0 ) 10 ) . ( 5 . 4 ' )

T h e n u c l e o n a l so a c q u i r e s a m a s s f r o m c h i ra l s y m m e t r y b r e a k i n g ,a M - - ( P l n s B I P ) , ( 5 .6 )

w h i c h i s a p p r o x i m a t e l y e q u a l t o t h e ~ rN 2; t e r m ,Z ~ N = (m ~ + m d ) ( p l a u + d d l p ) . ( 5 . 7)

T h i s 2~ ~N c a n b e o b t a i n e d f r o m , r - N s c a t t e r in g [4 1] a n d t u r n s o u t t o b eZ ~,N = (57 --+ 10) M e V . (5 .8 )

S u c h m a s s t e r m s a r i s in g f r o m b a r e q u a r k m a s s e s t r a n s f o r m a s c o s 0 w i t h 0 , s o w e h a v e a t o t a l s y m m e t r yb r e a k i n g h a m i l t o n i a n

2 2H = ~ ~ P B COS 0 - - f~m~, c o s 0 , ( 5 . 9 )w h e r e P B i s t h e b a r y o n d e n s i t y . I n t h e l i m i t o f p i o n m o m e n t u m a n d f r e q u e n c y g o i n g t o z e r o , t h is i s a llw e h a v e i n t h e h a m i l t o n i a n . I t i s n o w e a s y t o s e e t h a t f o r

r 2 2 t ~ - ~ NPB > Pc = I~m~ /2, ( 5 . 1 0 )t h e e n e r g y o f t h e s y s t e m c a n b e l o w e r e d b y 0 g o i n g f r o m 0 t o z r /2 . T h e a m o u n t b y w h i c h it is l o w e r e d i s

b E = 2 ;~N( 1 - - pc~p).F o r ~ ,,r ~ = 5 7 M e V ( a n d f ~ = 9 3 M e V ) * ),

Pc = 2 " 2 Pn m ,bE(u = 4 ) = - 2 6 M e V .

(5 .11)

(5 .12)(5 .13)

W h a t h a p p e n s i s t h a t a b o v e P c t h e c h i ra l s y m m e t r y b r e a k i n g m a s s e s t o t h e p i o n a n d n u c l e o n a r e" r o t a t e d o u t " a s 0 g o e s f r o m 0 t o , r / 2 . T h u s t h e n u c l e o n i s l ig h t e r b y a n a m o u n t 2 ~r~ , o r b y - 6 % w i t ht h e v a l u e ( 5 . 8 ) .

*) In c o l l i s ions o f l a rge -A he a vy io ns , s uc h a s u ra n ium , one s hou ld do s om e wh a t be t te r w i th s -wa ve ~ r+ c onde ns a t ion , b r ing ing Pc down to~2pn m . For i s os p in s ym m e t r ic m a t te r th e r e pu ls ive ~ r+- p a nd a t t r a c t ive ~ r +-n in te ra c t ions c a nc e l ou t , bu t in ne u t ro n- r ic h m a t te r the l a t t e r one wins .


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1 86 Theory of supernovaeT h e s t ra n g e n e s s c o n d e n s a t i o n , c a l c u l a te d b y K a p l a n a n d N e l s o n [ 40 ] t o o c c u r a t p - 2 . 7 p , m , is a

m o r e d r a m a t i c p h e n o m e n o n . A s a t r a n s i t i o n a l c a l c u l a t io n , w e f ir st c o n s i d e r ,n , ~q c o n d e n s a t i o n [ 4 2] ,w h i c h w i ll il l u s tr a te t h e m a i n p o i n t s , s t a r t i n g f r o m o u r r r c o n d e n s a t i o n a b o v e .

F i r s t, l e t u s g o b a c k t o e q s . ( 5 . 7 ) a n d ( 5 . 8 ) a n d f i g u r e o u t h o w a ~ ~N t e r m a s la r g e a s 57 M e V c o m e sa b o u t . I n t h e M I T b a g m o d e l

< p l d d l p ) = 0 . 4 8 , ( 5 . 1 4 )t h e i n t e g r a l i n v o l v i n g t h e d i f f e r e n c e o f t h e s q u a r e o f u p p e r a n d l o w e r c o m p o n e n t s b e c a u s e o f t h e 3;0 i n5 = u ty 0 . G i v e n ( m u + m d ) / 2 - - 7 M e V , t h e t h r e e v a l e n c e q u a r k s w o u l d c o n t r i b u te o n l y - 1 0 M e V t o2 ~N . T h i s m e a n s t h a t - 5 / 6 o f 2 " N m u s t c o m e f r o m s o m e w h e r e el se . W e n o w s h o w t h a t t h e m a j o r p a r to f ~ ~N c o m e s f r o m c l e a r in g t h e v a c u u m .

I m a g i n e t h e n u c l e o n t o b e a b a g ( o r b u b b l e ) i n w h i c h q u a r k s a r e c o n f i n e d , im m e r s e d in a m e d i u m( i. e . Q C D v a c u u m ) . T h e b u b b l e m u s t c l ea r o u t a t l e as t s o m e o f t h e c o n d e n s a t e i n t h e p h y s ic a l v a c u u mi n o r d e r t o e n a b l e v a l e n c e q u a r k s t o e x i s t i n t h e r e g i o n . T h e 2 ' ~ N s h o u l d b e c o n s i d e r e d a s t h e differenceb e t w e e n t h e e n e r g y l o d g e d within t h e v o l u m e V o c c u p i e d b y t h e q u a r k s ( o r s o u r ce s ) a n d t h e e n e r g yl o d g e d i n t h e s a m e v o l u m e without s o u r c e s . T h i s i s d e s c r i b e d i n f i g . 1 1 .

F r o m t h e W e i n b e r g s u m r u l e [ 4 3 ]2 2f~m~ = - ( m u + m ) ( O l ff u l O ) . ( 5 . 1 5 )

W i t h [4 4] m u + m d = 1 4 M e V ( e v a l u a t e d a t t h e r e n o r m a l i z a t i o n p o i n t q = 1 G e V ) o n e c a n f in d t h a t * )< O l u lO > = < O l d d lO > = - ( 2 3 0 MeV ) 3 . ( 5 . 1 6 )

L a t e r w e w i l l u s e [ 4 5 ]( 0 l s l0 > = 0 . 8 < 0 1 u 1 0 > . ( 5 . 1 7 )

T h e q u a r k c o n d e n s a t e ( 0 It~ ul0 ) i s b u i l t i n t o t h e v a c u u m w i t h a n e g a t i v e s i g n . C l e a r l y m , ( 0 1 ti u l0 )

/ / / / / / / / /

Fig . 11 . T he e f f e c t o f ins e r t ing the bubb le in the va c uum i s to c le a r ou t a t l e a s t s om e o f the qua rk c onde ns a te .

* ) A l tho ugh th e re m a y be s om e a m bigu i ty in m d + m u , no te tha t the p rod uc t on th e r igh t -ha nd s ide o f e q . (5 .15) i s we l l de te rm ine d s inc e f~ a ndm ~ a re a c c ura te ly known,


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G.E. Br ow n, The equation of state of dense matter 18 7

m u s t b e n e g a t i v e f o r t h e p h y s i c a l v a c u u m t o h a v e a l o w e r e n e r g y t h a n t h e p e r t u r b a t i v e o n e . N o t e t h a t_ 4 7r(1 fm) 3(01 a u l 0 ) = 6 . 6 7 . ( 5 . 1 8 )

T a k i n g o n t h e a v e r a g e * ) flu = u'u~2 a s f o r t h e l s ~ / 2 s t a t e o f t h e b a g m o d e l , t h i s m e a n s t h a t t h e r e a r e- 1 3 u p q u a r k s , 1 3 d o w n q u a r k s a n d 1 0 s t ra n g e q u a r k s c o n d e n s e d i n e a c h s p h e r e o f ra d i u s in t h ev a c u u m , 3 6 u p , d o w n , s t r a n g e q u a r k s i n t h e p r o t o n i n a ll , i f o n e a d d s t h r e e v a l e n c e q u a r k s .

S u p p o s e w e n e e d t o o b t a i n 4 7 M e V o f 2 " N f r o m a c o m p l e t e c l ea r in g o f t h e v a c u u m . W h a t b u b b l er a d i u s i s n e c e s s a r y ? I t c a n b e o b t a i n e d f r o m

4 7r R3 (1 4 M e V ) ( 2 3 0 M e V ) 3 = 4 7 M e V , ( 5 . 1 9 )g i v i n g R = 0 . 8 f m . F o r a b a g o f r a d i u s R ~ 1 .1 fl n , o n l y 3 8 % o f t h e v a c u u m i s c le a r e d ; s u c h b a g s a r e s ti lld i r ty i n s id e . M a y b e t h is e x p l a in s w h y t h e R u s s i a n s u m r u l es [ 4 5 ] d o n o t n e e d t o d i s ti n g u is h b e t w e e nv a c u u m p h a s e s i n s i d e a n d o u t s i d e b a g s . B a g s o f d i f f e r e n t r a d i i s h o u l d b e a c c e p t a b l e f o r m o s tp u r p o s e s - t h e C h e s h i r e C a t p i c t u r e [ 4 6] . N o t e t h a t t h e p r e s e n t ( m u + m d ) ( O l a u l O ) i s d e t e r m i n e dd i r e c tl y b y t h e k n o w n f, , a n d m ~ , e q . ( 5 . 1 5 ) , s o t h a t t h e r e i s n o u n c e r t a i n t y i n o u r d e r i v a ti o n h e r e .

O b v i o u s l y a s m a l l e r b a g , w i t h r a d i u s * * ) ~ 0 . 6 f m , t h a t o f t h e t o p o l o g i c a l c h i r a l b a g , w i ll c o m p l e t e l yc l e a n s e t h e v a c u u m . T h e l it tl e b a g i s a n e x c e l l e n t " v a c u u m c l e a n e r " ! t) D e c r e a s i n g t h e b a g v o l u m e d o e sn o t , h o w e v e r , d e c r e a s e t h e ~ N t e r m p r o p o r t i o n a t e l y , b e c a u s e t h e r e i s a c o n t r i b u t io n **) t o t h e ,V ~Nt e r m f r o m t h e m e s o n c l o u d [ 4 7 ] ,

( 8 2 ~ N ) = f d r 4 7 rr 2( 1 - c o s 0 ) ,Rbag

( 5 . 2 0 )

w h e r e 0 is t h e c h i r a l a n g l e . H e r e 0 m u s t b e c a l c u l a t e d w i t h n o n z e r o p i o n m a s s m ,~ b e c a u s e t h e r e s u l t iss e n s i ti v e t o t h e r e s u l t in g e x p o n e n t i a l d r o p o f 0 w i t h r f o r la r g e r . F o r 0 ( R b a g ) = '7 7" /2 , t h i s c o n t r i b u t i o nc a n e a s i ly b e - 5 0 % o f ,~ ~N . I f 5 0 % o f ,~ ~N h a s t o c o m e f r o m a c o m p l e t e c l e a n si n g o f t h e v a c u u m b y t h eb a g , t h e n R = 0 .6 3 f m r a t h e r t h a n t h e 0 . 8 f m f o r p r o d u c i n g a ll o f ,~ ~N .*)

I n c o r p o r a t i o n o f t h e S k y r m i o n i n to S U ( 3 ) S U ( 3 ) h a s n o t b e e n s a t is f a ct o ri ly c a r r i e d o u t t o d a t e .T h e r e f o r e , w e w i ll s t ic k w i t h a b a g r a d i u s R - 0 . 8 f m , w h i c h h a s v e r y l it tl e t o p o l o g i c a l c l o u d in o u rc o n s i d e r a t io n s , w h i c h i n c l u d e s tr a n g e n e s s a n d , r e l y in g o n t h e C h e s h i r e c a t m o d e l , h o p e f o r t h e b e s t. I nc l e a ri n g o u t s t r a n g e q u a r k s f r o m a li tt le b a g o f ra d i u s ( 0 . 5 - 0 . 6 ) f m o n e w o u l d h a v e t o t a k e i n to a c c o u n tt h e c l e a n i n g b y t h e c l o u d , w h i c h i s k n o w n t o b e l a r g e - l o o k a t t h e S k y r m i o n [ 4 9 ] .

W e a r e n o w r e a d y t o d i s c u ss k a o n c o n d e n s a t i o n . T o d o t h is w e i n t r o d u c e V - sp in . T h e t r a n s i ti o n f r o m*~ Th is m ay be an unde re s t ima te o f u'u, and the re fo re , an unde re s t ima te o f the number o f condensed qua rks , because fo r s ta te s o f h igh

mom en ta , whe re lower componen ts become equa l to uppe r ones in magn i tude , f i u ~O.** ) The app rop r ia te r ad iu s to u se he re m ay be a l i t tl e b i t l a rge r than the s ta t ic r ad ius . Because o f ze ro po in t m o t ion o f the bag su r face and

because of surface effects , the vacuum may be c leansed to a somewhat larger radius.t ) I am indeb ted to Jo e Suche r fo r th i s obse rva t ion .~*) I am g rate fu l to M ano j B ane r jee and Tom Cohen fo r po in t ing th i s ou t to me .~) An explic i t ca lcula tion of the c loud contr ibutio n to ,~"N has bee n carr ied out for # (R) = n-/2 by Ulf MeiBner (private comm unication) , g iv ing

"~doud = 27 MeV. Me~6ner took th e c loud beyo nd R to be that o f Jackson an d R ho [48] for m , = 0 . T he contr ibu tion of the c loud to ,~ ~r~ is then thed i f fe rence be tween c loud ene rg ie s fo r p ion masses m~ and 0 .


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1 88 Theory of supernovaei sosp in to V-sp in i s made by chang ing down quarks to s t range quarks , a s shown be low:

+ a u

o f l u - d d' r r - V ~"rr = - r i d ,

K + = - g u ,_ _ V o _ f l u - ~ s

'

K - = - f i s .- - = o + X/-3~, (5 .21 )

Th e ( m a s s ): o f t h e q u a s ip a r t i cl e d e s c r i b e d b y V i s m ~ b e c a u s e o f t h e G e l l - M a n n - O k u b o f o r m u la: t : 3 : (5.22 )K = z m ~ , + z m ~ .

W e c a n n o w d is cu ss V = ( x r r + V ~'q ) c o n d e n s a t i o n in t h e s a m e w a y a s w e d i sc u ss e d ~rc o n d e n s a t i o n , m a k in g a f o u r - v e c to r u p o u t o f t h e g - f ie l d a n d t h e f ie l ds c o r r e s p o n d in g t o V - sp in .Co m p le t e ly a n a lo g o u s ly t o t h e 0 c o n d e n s a t i o n , o n e f i n d s t h e t h r e s h o ld f o r V c o n d e n s a t i o n t o b e

, ' 2 2 t* - , KNPc = ]Km K/z (5 .23)2 2A t f i r s t s i g h t t h i s t h r e s h o ld w o u ld s e e m to b e v e r y h ig h , b e c a u s e m K / m ~ = 12 .5 . ( fK i s know n to be

(1 - 1 .2 ) t imes f~ , .) Ho we ver , l e t u s look a t ,VKN:2 K N = ( m u + ms)(plau + gslp) (5 .24)

Here [43, 44]m s - 2 5 , ( 5. 25 )-l(mu2 + m d)

a n d i f t h e v a c u u m i s c o m p l e t e l y c l e a n se d , t h e n * ) (p l g sl p ) w ill b e - - 0 . 8 ( 0 [ a u [ 0 ) . F r o m t hi s a n d o u re a r l i e r n u m b e r s , w e f i n d

2 KN/z~N -- 10. (5 .26)Bo th "rr and V conden sa t ions invo lve chem ica l po te n t ia l s /z 0 = 0 , beca use

n ~ n + ,r r n ~ n + V ( 5. 27 )F o r K - c o n d e n s a t i o n [4 0 ] t h e c h e m i c a l p o t e n t i a l / z is s u b st a nt ia l ,

/~_ = /z , - ~p -> m~, (5.2 8)in n e u t r o n s t ar s . N o w th e m 2 i n t h e n u m e r a to r o f ( 5 .2 3) a c tu a ll y c o m e s * *) f r o m th e i n v e r s e p r o p a g a to r

*) Th i s impl i es tha t ~1 /5 o f t he q uarks in the p ro ton a r e s t r ange [49].** ) C onde ns a t ion deve lops when th e r e l evan t co r r e l a t ion func t ion deve lops a po le . The empi r i ca l K-nu c leo n in t e r ac t ion i s apprec iab le , s ome of

t h e s c a t t e ri n g l en g t h s be i n g l ar g e r t h a n t h e i n d iv i d u al p i o n - n u d e o n o n e s . H o w e v e r , t h e s e g o b y w a y o f t h e v e c t o r m e s o n e x c h a n g e , a s in t h eW einberg r e l a t ion fo r p ion-nuc leon s ca t t e r ing l eng ths . Thus , t hey a r e p ropor t iona l t o toK, which bec ome s equa l t o ~ ,K in the conden s a te . S ince#K/tOK"~ I , t hes e e f f ec t s a r e muc h r educed in the cond ens a te .


23/37

G.E . Brown, The equation of state of dense matter 1 8 92[50] (k 2 + m K ]..2), SO the thre sh old f or s-wave K - co nd en sat i on i s

2 2Pc = f K(mK --/X2)/2KN , (5.29)s li g h tl y ( - 1 0 % ) l o w er t h an t h a t fo r V co n d e n sa t i o n . O f co u r se , s t r an g en es s h a s to b e v i o l a ted i n o rd e rt o e s t ab li sh K - co n d e n sa t i o n . T h e re is p l en t y o f t im e i n n eu t ro n s t a r s o r i n s te l la r co l lap se fo r V co n d en sa t e s t o r e l ax i n t o K - o n es b y w eak i n t e r ac t i o n s , a l t h o u g h t h e re w o u l d n o t b e i n r e l a t i v i s t i ch eav y i o n co l li si o n s. I n t h e l a t te r , V co n d en sa t i o n co u l d b e ach i ev ed , h o w ev e r . T h e en e rg y g a i n s inthese con den sa t ions i s g iven by (5 .11) wi th ~ '~N rep lace d by ~ KN, the m ass of t he b aryo n dr opp ing by->2 5 % b e t w een P c an d u = 4 w i t h o u r ab o v e n u m b er s . O n ce ag a i n , a s d i s cu s sed w i t h t h e p i o nco n d en sa t e fo l l o w i n g eq . (5 . 1 3 ) , s i m p l y ch an g i n g t h e n u c l eo n m ass w i ll n o t ch an g e t h e en e rg y o f t h ed en se n u c l ea r m an y -b o d y sy s t em m u ch , s i n ce t h e f eed b ack s d i s cu s sed i n ap p en d i x A co m e i n t o p l ay .

A s k ao n co n d en sa t i o n p ro ceed s , t h e s i t u a t i o n m ay b e d i f f e r en t f ro m t h a t w i t h p i o n s , h o w ev e r ,b ecau se t h e k ao n s can h av e i n t e r ac ti o n s w i th t h e n u c l eo n s . W i t h in c rea s in g d en s i t y , t h e p ro p o r t i o n o fp ro t o n s i n t h e n eu t ro n s t a r w i l l i n c rea se an d t h e ch em i ca l p o t en t i a l / z_ = / z , - / Z p w i l l b e b ro u g h t t oze ro . T h i s m ean s t h a t t h e k ao n s w i l l f o rm b o u n d s t a t e s w i t h t h e b a ry o n s , so m ew h a t i n t h e C a l l an -K l eb an o v [5 1 ] s en se . T h e se au t h o r s sh o w ed t h a t t h e n u c l eo n -k ao n i n t e r ac t io n i s su f f ic i en t ly s t ro n g t op ro d u ce b o u n d s t a te s . A t l e a s t so m e o f t h e se e f f ect s h av e b een i n c l u d ed fo r m an y y ea r s b y t h ei n t ro d u c t i o n o f s t r an g e h y p e ro n s i n t o t h e co m p o s i t i o n o f n eu t ro n s t a rs a t h i g h d en s it y . T h i s i n t ro d u c t i o no b v i o u sl y s o f te n s t h e E O S s o m e w h a t .

I t s eem s c l ea r t h a t K - co n d en sa t i o n w ill o ccu r i n n eu t ro n s t a r s. A l t h o u g h t h e , ~ KN t e rm m ay b eso m ew h a t o v e re s t i m a t ed b ecau se t h e ,~ ~N t e rm is l es s t h an 5 7 M eV , an d w e sca l ed f ro m t h i s , o n e w o u l dexp ec t fK to d ecrease wi th increas ing dens i ty , jus t as one expec t s [52] for f~ . Thu s , we d o no t be l i eveo u r m o d e l t o u n d e re s t i m a t e t h e t r an s i ti o n d en s it y . A n ex c i t in g p ro sp ec t fo r o b se rv i n g K - co n d en sa t i o n ,a t l e a st t h e e f f ec t s o f i t, i s i n t h e co o l i n g o f n eu t ro n s ta r s. H e re K - co n d e n sa t i o n sh o u l d tak e o v e r t h ero l e fo re seen fo r p -w av e p i o n co n d en sa t e s [5 3 ], w h i ch w e n o w b e l iev e , f o r r e a so n s g i v en ab o v e , n o t t oo c c u r. T h e K - w o u l d d e c a y t h r o u g h K - ~ I ~ - + v p r e d o m i n a n t ly . T h e a v a il ab le e n e r g y is th e r m a len e rg y . B ecau se t h e n o rm a l n eu t ro n s t a r co o l in g i n v o lv i n g n eu t r i n o s em i t t ed i n n e u t r o n -n e u t r o nco l li si o n s is k i n em a t i ca l ly b l o ck ed , co o l i n g w ill b e sp eed e d u p b y m an y o rd e r s o f m a g n i t u d e b y k ao nco n d en sa t i o n a s g i v en b y t h e t h e o ry ( i f n o t p r ac t ic e ) o f p i o n co n d en sa t i o n [53 ].

T h e re a r e t w o q u es t i o n s co n ce rn i n g n eu t ro n s t a r co o li n g . T h e f ir st o n e i s a s to w h e t h e r o n e n e ed san y t h i n g in ad d i t i o n t o co n v en t i o n a l m ech an i sm s . T h e w o rk o f N o m o t o an d T su ru t a [5 4 ] i n d i ca te s th a ti n t h e ca se s w h e re t h e re a r e k n o w n co m p a c t so u rce s , co n v en t i o n a l m e ch an i sm s co o l t h e s ta r ssu f fi c ien t ly . O f co u r se , t h e m easu r em en t s m ay b e u p p e r l im i t s, p o s s ib l y r e f e r r i n g t o t em p e ra t u re s o f t h esu r ro u n d i n g p l a sm a r a t h e r t h a n t h a t o f t h e n eu t ro n s t a r i ts e lf . T h e s eco n d q u es t i o n i s a s to w h e t h e r t h econduct iv i ty i s suf f i c i en t so tha t t he sur face of t he neut ron s t a r wi l l l ea rn of t he cool ing dur ing thel i fe t ime of t he s t a r [55] . W ork on the l a t t e r q ues t ion i s p roc eed ing [56] .

A l t h o u g h , ch i e f l y d u e t o u n ce r t a i n t i e s i n t h e p rec i s e v a l u e s o f t h e 2 t e rm s , t h e re a r e l a rg eu n ce r t a i n t i e s in t h e n u m er i ca l e s ti m a t e s i n t h is s ec t io n , w e b e l i ev e o u r w o rk t o e l u c i d a te t h e su g g es t edp o ss i b le n ew co n d en s a t e s b y th e w o rk o f K ap l an an d N e l so n [4 0 ], w h i ch w o u l d b e ex p ec t ed t o o ccu r a tso m e d en s i t y .

I t s eem s c l ea r t h a t i m p o s i n g t h e co n d i t i o n s n ece s sa ry so t h a t a t h i g h e r d en s i t ie s n u c l ea r m a t t e r w i llm ak e t h e L ee -Wi ck t r an s i t i o n i n ap p en d i x A w i l l l o w er t h e E / A , t h e en e rg y p e r n u c l eo n , a t h i g h e rd en s i t i e s an d i t i s n o t u n rea so n ab l e t h a t t h e E /A f ro m t h e co n v e n t i o n a l n o n re l at i v is t ic c a l cu la t i o n o fF r i ed m an an d P an d h a r i p an d e sh o w n in f ig . 5 w ill b e b ro u g h t d o w n i n t o t h e c ro s s h a t ch e d a r ea i n th a tf i g u re w h i ch d em arca t e s t h e B C K reg i o n i n w h i ch su p e rn o v ae w i l l w o rk .


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190 Theory of supernovaeOu r m ain conclus ions ma y be th a t the s ideways f low in h igh-energy heavy ion reac t ions does not g ive

informat ion on the h igh-dens i ty nuc lear EOS and tha t the low -energy exper im ents have not ye t beenproper ly ana lysed .

AcknowledgmentsIn addi t ion to my col labora tors , l i sted in the beginning , I would l ike to thank Subal Das G upta , Phi l

Siemens, Reinhard Stock, Horst St6cker and Steve Wallace for discussions, help and cri t icism. I amespec ia l ly gra te fu l to Subal Das Gupta and Hors t S t6cker for ear ly prepr in ts of the i r work put t ingveloc i ty depend enc e of the e f fec t ive in te rac t ion in to the i r ca lcu lat ions and to H ors t S t6cker for de ta i leddiscussions of s topping and s ideways f low. I wo uld l ike to tha nk the W .K. Kel logg Labora tory a t CalTech for warm hospi ta l i ty whi le th i s paper was comple ted . I would espec ia l ly l ike to thank MannqueRh o for he lpfu l c ri ti ci sm and m any suggest ions. The wo rk on kao n cond ensa t ion descr ibed here wasin i t ia ted by h im. F ina l ly , bu t no t leas t , th is pap er ow es muc h to ex tens ive d iscussions wi th Hans Bethe .

Appendix AIn th i s appendix I wish to out l ine work done wi th T .L. Ainswor th , M. Prakash and W. Weise , *)

a ime d a t in t roducing the cor rec t beha vior of the c r -meson mass wi th la rge dens ity p . As has been kno wnsince the L ee- W ick theo ry [11] , the sca la r f ie ld 6 em ploy ed in boson exchange m odels is the f luc tua t ionabou t the scalar me son f ield, an d th e m ass of this f luctuat ion f ield is related to th e cur vatu re of the f ieldenergy a t the re levant dens i ty . Fur thermore , the energy of the sys tem Ep(&) should be minimized*) ateach dens i ty p wi th respec t to the f luc tua t ion f ie ld , i . e . ,

o E . ( 6 )06 o = 0 . (A .1)When nonl inear i t ies in 6 come in to the ca lcu la t ions , as when loops a re in t roduced , th i s minimiza t ionplays an important role. (Of course, as long as the energies are quadrat ic in 6, i t is t r ivial . )

In ch i ra l models one s ta rt s ou t f rom a f ield energy wi th no m at te r pres ent of **)(A.2)

Tak ingor = f.~ + 6 , (A .3)

f~ be ing the m ean v a lue of the c r- field in vacuum , we ex pand in 6 , f ind ing2 2 2V = A f , , 6 + A Z f~ 6 3 + 1 / ~ 2 6 4 " (A .4 )

t~ No w publish ed in Phys . L e tt . B 200 (1988) 413.* ) I t c a n be s how n tha t a n e qu iva le n t p roc e dure i s to s o lve fo r the nuc le on e f f e c t ive m a s s m*(p) se lf-consis tently as in re f . [18] .**) W e s ho u ld r e a l ly inc lude the p ion f ie ld , o" 2 ~ o 2 + l r z. Ho we v e r , s inc e we do no t c on s ide r p ion c onde n s a te s he re , we d ro p i t fo r s im pl ic ity .


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G . E . Br own, The equat ion o f s ta te o f dens e m at t er 19 1

F r o m h e r e i t i s c l e a r t h a t f o r t h e s m a l l - ~ b r e g i m e2 = 2 A z f 2 , ( A . 5 )~

n a m e l y , m ~ , i s t h e c o e f f i c i e n t o f th e ( ])2 te r m .A s m a t t e r i s a d d e d ( s ee fig . A . 1 ) t h e r e i s a m a t t e r e n e r g y o f pgtr, w h i c h m u s t b e p u t t o g e t h e r w i t h

t h e a b o v e f i e ld e n e r g y . T h i s c o m e s a b o u t f r o m t h e s p o n t a n e o u s s y m m e t r y b r e a k i n g i n t h e t r- fi e ld , t h em a s s e n e r g y b e i n g

~ H = g ~ t r ~ . ( A . 6)T h e t o t a l V (o -) fo r i n t e r m e d i a t e d e n s i t y is s h o w n i n f ig . A . 1 , b y t h e d a s h e d l i n e . m~(p) f o r t h i s d e n s i t yw i l l , a g a i n , b e r e l a t e d t o t h e c u r v a t u r e o f t h i s t o t a l e n e r g y a t m i n i m u m ,

d 22 = _ _ ( A . 7 )m ~ d o , 2 E p ( r ) m i n i m u m 'a s i t w as w i t h t h e v a c u u m c u r ve . U l t i m a t e l y , w i t h i nc r e a si n g d e n si t y , t h e m i n i m u m i n t h e c o m p o s i t ec u r v e w i l l m o v e t o o - = 0 . W e b e l i e v e , f o r s e v e r a l r e a s o n s , t h a t t h i s t r a n s i t i o n t o t r = 0 ( m a s s l e s sn u c l e o n ) w i l l b e a s m o o t h o n e , n o t a s t r o n g f i r s t - o r d e r t r a n s i t i o n . F i r s t o f a l l , t h e r e i s t h e B 6 g - S h e it h e o r e m [5 7], w h i c h s ta t es " t h a t t h e n a t u r e o f s y m m e t r y r e a li z a ti o n ( W i g n e r - W e y l v e r s u s N a m b u -G o l d s t o n e ) i s i r r e l e v a n t i n d i s c u s s i n g t h e s h o r t - d i s t a n c e s y m m e t r y . " T h u s , a t l e a s t t h e i n t e r a c t i o n ss h o u l d m e r g e s m o o t h l y f r o m o n e p h a s e i n t o t h e o t h e r . * ) S e c o n d l y , c a l c u l a t io n s w i t h t h e c h i r a l b a gm o d e l [ 2] s h o w a r a t h e r s m o o t h m e r g i n g o f n u c le o n s t o q u a r k s a s t h e m e s o n c l o u d o f e a c h n u c l e o n i ss q u e e z e d o u t w i t h i n c r e a s i n g d e n s i t y , l e a v i n g o n l y t h e q u a r k s .

I n a s m o o t h t r a n s i t i o n , t h e t r- m a s s , o r c u r v a t u r e i n V ( t r) , g o e s t o z e r o a s t r y 0 .I n t h e e a r l y L e e - W i c k w o r k [1 1], p o s s i b i li t ie s o f fi r s t- o r d e r p h a s e t r a n s i t io n s w e r e d i s c u s s e d . I n af i rs t - o rd e r t r a n s i t io n , m ~ n e e d n o t g o s m o o t h l y t o z e r o . W e b e l i e v e t h a t a f ir s t -o r d e r tr a n s i t i o n w o u l d b e

V ( o ' )o

~ t rons i t i on

f r r c r

Fig. A.1 . B eha vior of the f ield ene rgy V(tr ) versu s t r . Sol id lines: n omat t e r p r es en t , das hed l ine : i n the p r es ence o f mat t e r~ The s t r a igh tl ine g ives the con t r ibu t ion f rom the mat t e r . T he d o t t ed l i ne s hows them i n i m u m h a v i n g m o v e d t o t r = 0 .

Fig. A.2. The chiral circle.

* ) T h e m a s s i v e n u c l e o n s a r e i n t h e N a m b u - G o l d s t o n e p h a s e . W e t h i n k o f t h e m a s s l e s s n u c l e o n s a s a p h e n o m e n o l o g i c a l d es c r ip t io n o f t h eW i g n e r - W e y l , o r q u a r k / g l u o n p h a s e .


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1 92 Theory of supernovaea n a r t i fa c t r e s u l t i n g f r o m t h e i n c o r r e c t c h a r a c t e r o f t h e t h e o r y t h a t i t i s n o t a s y m p t o t i c a l y fr e e . I t w o u l ds e e m m o r e c o n s i s t e n t w i t h th e k n o w n t r a n s it i o n t o c h ir a l r e s t o r a ti o n t o m a k e t h e m e r g i n g s m o o t h .

H o w e v e r t h e p h a s e t r a n s it i o n g o e s, t h e a b e c o m e s d e g e n e r a t e w i t h t h e p i o n a s f ~ , ~ 0 a t t h et r a n s i t io n a n d t h e c h i r a l c ir c le c o l la p s e s i n w a r d s . T h i s m e a n s , q u i t e g e n e r a l l y , t h a t m , , m u s t d e c r e a s et o w a r d s m , , . T h e l a t t e r w o u l d b e e x p e c t e d t o r e m a i n s m a l l , a s t h e d e n s i t y i n c r e a s e s , s in c e it ar is e s f r o mt h e e x p li ci t c h ir a l s y m m e t r y b r e a k i n g - t h e n o n z e r o , s m a l l u p a n d d o w n c u r r e n t q u a r k m a s s e s .

F i r s t l y , w e n o t e t h a t t h e s c a l a r f i e l d u s e d i n b o s o n e x c h a n g e m o d e l s i s a h i g h l y e f f e c t i v e o b j e c t . T h eo r i g i n a l s c a l a r f i e l d i n t h e c h i r a l m o d e l i s , i n a s e n s e , t h e p r o g e n i t o r . H o w e v e r , t h i s i s k n o w n f r o m t h es m o o t h n e s s i n s o f t - p i o n e x t r a p o l a t i o n t o b e a h e a v y o b j e c t , m ,~ -> m . F r o m d i s p e r s i o n r e l a t i o n s [ 5 8] o n ef i n d s t h a t t h e e f f e c t i v e a i n v o l v e s p r o c e s s e s s u c h a s s h o w n i n f i g . A . 3 .

L e t u s a s a p r o t o t y p e c h o o s e t h e s i m p l e r p r o c e s s s h o w n i n f ig . A . 4 . T h i s is u s e d i n , e . g . , t h e B o n np o t e n t i a l [ 59 ], t o g e t m u c h o f th e e f f e c t iv e a - e x c h a n g e . I n m a t t e r , t h i s e ff e c t iv e a - e x c h a n g e w i ll b em o d i f i e d b e c a u s e o f m e d i u m c o r r e c ti o n s . T h e s p i n - i s o s p i n i n t e r a c ti o n i n t h e n u c l e o n s p a c e is st ro n g l yr e p u l si v e , a n d w i ll n o t c o n t r i b u t e m u c h . M o s t o f t h e a c t io n w i ll c o m e f r o m t h e N A c h a n n e l a s s h o w n i

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