C H IN ．P H Y S．L E T T V o1．22N o．12 (2005)3036
D o e s T h e r e E x ist O n ly O n e S o lu t io n o f th e D y s o n ——S ch w in g e r E q u a t io n fo r th e
Q uark P ropagator in th e C ase of N on—Z ero C urrent Q uark M ass
ZO N G H ong．Shi(宗红石) SU N vei．M in(孙 民) vP IN G Jia—Lun(加仑)
LU X iao．Fu(吕晓夫) vW A N G Fan(~ )
D ep artm en t of P hysicsN anjing U niversityN anjing 21 0093
0 C C A S T (W orld L aboratory)P 0 B ox 8730B eijing 100080
0 Join t C en ter for P article．N uclear P hy sics and C osm ologyN anjing 21 0093
D ep artm en t of P TsJcsN anjing N orm al U niversityN anjing 210097
D ep artm en t o f P hy sics S ich u an U n io n U n iv ersity C h en g d u 61 0 06 4
(R eceived 17 A ugust 2005)
It is sh o w n o n g en era l g ro un d th a t th er e ex ist tw o q ua lita tiv ely d is tin ct so lu tio n s of th e D y son S ch w in g er eq u a tion
for th e q u ark p rop a ga tor in th e case o f n on —z ero c urren t q ua rk m a ss ．O n e so lu tion c orresp on d s to th e ＇N am b u —
G olds ton e’’p h ase an d th e o th er on e corresp on d s to th e W ig n erp h ase in th e ch ira l lim it．
P A C S ： 1 1 ．1 5 ．T k 1 2 ．3 8 ．A w 12 ．3 8 ．L g 12 ．3 9 ． x
It is g e n era lly b e liev ed th a t w ith in crea sin g t em —
p e ra tu re a n d b a ry o n n u m b er d e n sity th e h a d ro n ic
m a tte r u n d e rgo es a p h a se tran sitio n to th e q u ark ——
gluon plasm a (q c P ) w hich is expected to appear in
u ltra reia tiv istic h e av y io n co llisio n s．T h ese tw o p h a ses
a re g en era lly refe rred to a s th e N a m b u — G o ld sto n e
p hase fch aracterized by d yn am ical ch iral sym m etry
b reaking and con fin em ent of d ressed quarks) and th e
W ignerphase (corresponding to Q G P w here chi—
r al sy m m etry is resto red a n d q u a rk s a re n o t co n fi n e—
m ent1．T heoretically th ese tw o p hases are describ ed
by tw o d ifi erent solu tion s fth e N am bu G old ston e
a n d th e W ig n e r so lu tio n of th e q u a rk p ro p a g a —
to r an d th e ex iste n ce o f th e se tw o so lu tio n s in th e
ch ira l lim it h a s b ee n rig o ro u sly p ro v en in th e fra m e—
w ork of D ysonSchw inger equation (D SE ) approach
of Q C D ．H ow everit is a general view in th e litera—
tu re th a t w h en th e c u rren t q u a rk m a ss is n o n —ze ro
th e D y so n S ch w in g er eq u a tio n fo r t h e q u ark p rop a —
g a to r h a s o n ly o n e so lu tio n w h ich c o rre sp o n d s to th e
N a m b u — G o ld sto n e p h a se ．w h ile th e so lu tio n co rre—
sp o n d in g to th e W ig n e r p h a se d o e s n o t ex ist
H o w ev era s fa r a s w e k n o w th is cla im h a s n ev er b e en
p ro v ed ．D o e s th ere rea lly ex ist on ly o n e so lu tio n of th e
D y so n S ch w in g er eq u a tio n fo r th e q u a rk p ro p a g a to r
in th e ca se o f n o n —z ero c u rren t q u a rk m a ss
In or d e r to in v e stig a te th is p ro b le m ．1et u s fi rst re—
ca ll th e u su a l a rg u m en ts w h ic h ex c lu d e th e ex iste n c e
o f th e W ig n e r so lu tio n o f th e D y so n S ch w in g er
e q u a tio n fo r th e q u a rk p ro p a g a to r in th e ca se of n o n
z ero cu rren t q u a rk m a ss ． T h e D S E sa tisfi e d b y th e
q u a rk selfen erg y is
X (pm )
．
d 4q
夕 2 ab (
7ut9 (qm ) (pq) (1)
w here 夕 2 a b(p) and (pg) are the fullnonp er·
tu rb a tiv e g lu o n p ro p a g at or a n d th e q u a rk ·g lu o n v e r’
te x re sp ect iv e ly a a n d b a re th e c o lo u r in d ic es w ith
t： f0r the standard G ellM ann SU (3) represen 0
tationand is the current quark m ass．
T he quark propagator 9 (pm ) and the quark self
energy X (pm ) are related by
9 (pm ) 三io＇·P + m + X (pm )
i7 ．pA (p )+ (p )+ m (2)
w here A (p ) and S (P ) are the quark self energy func—
tio n s in th e c a se o f n o n ．zer o cu rre n t q u a rk m a ss．If m
is set to be zero9 (pm ) goes into the dressed quark
propagator in the chiral lim it G (p) 三 9 (pm 0)
w h ich h a s th e d ec o m p o sitio n
G (p)三i7 ．P + X (P仇 0) i7 ·pA (p )+ B (p)
In order to handle E q．(1) it is necessary to m ake
ce rta in sim p lifi ca tio n s a n d tru n ca tion s ． O n e c o m —
m only used truncation of E q．(1)called the rainbow
a p p ro x im a tio n in v o lv es rep la cin g th e fu ll v e rt ex in
Eq．(1) by the bare vertex
(pq)
In th is ca se th e q u a rk selfen erg y fu n c tio n s
B (p )are determ ined by the rainbow D SE
ra l lim it：I1— 3
[A (p )1]p 4
3 ／． y s J ／(2丌)
(4)
A (p)and
in th e ch i．
S u p p o r te d in p a rt b y t h e N at io n a l N a t u ra l S c ie n c e F o u n d a tio n o f C h in a u n d e r G ra n t N o s 1 0 17 5 0 3 3 1 0 1 3 5 0 3 0 a n d 1 0 5 7 5 0 5 0
a n d th e R es e a rc h F u n d fo r t h e D o ct o r a l P ro g r a m o f H ig h e r d u c a t io n u n d er G r a n t N o 2 0 0 3 0 2 8 4 0 0 9 ．
E m ail：z0n gh s@ ch enw an g ．n ju ．edu ．cn
⑥ 200 5 C hin ese P hy sical S ociety a nd IO P P ub lish in g L td
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N o ．12 Z O N G H o n g —S h i e t a l 3 0 3 7
A (q)
qA (q) + B (q)
lP ·q + 2 L P ·(P — q)q ·(P — q)
(P — q)
B∽4／ d4q 2(p＇q)
B (q) qA (q)+ B (q)’ (5)
w h ere w e h a v e u se d L a n d a u g a u g e ．It is rea d ily seen
that B (p ) in E q．(5)has tw o qualitatively distinct so—
lu tio n s ．T h e N a m b u — G o ld sto n eso lu tio n ．fo r w h ich
B (p ) ≠ 0describes a phasein w hich (1) chiral
sy m m etry is d y n a m ic ally b ro k en ．b eca u se o n e h a s a
n onzero qu ark m ass fun ction and f21 th e dressed
qu a rk s a re co n fi n ed b e ca u se th e p ro p ag a to r d e scrib e d
b y th e se fu n ctio n s d o es n o t h av e a L e h m a n n rep —
re sen ta tio n ． T h e o th er so lu tio n ．th e W ig n er on e ．
B (p ) 三0describ es a phasein w hich the chiral sym —
m etry is n o t b ro k en a n d th e d ressed —q u a rk s a re n ot
confined．[23H ow everw hen 仇 ≠ 0the selfenergy
functions A (p ) and S (p) are determ ined by the fol—
lo w in g ra in b ow D S E ：
)＇l 罟／ g2~D(p—q) (q) q (q)+[ (q)+m
IP ·q + 2 P ·(P — q)q ·(P — q)
(P — q)
(p)：4 t／．研d4q 2(p
(q)+ m q (q)+ [ (q)+ m (6)
Com paring E q．(6) w ith Eq．(5)it is apparent that
(p2) 三 0 is not a solution to E q．(6)． From this
ob serv a tio n o n e o ften co n c lu d e s th a t in th e c a se o f
non—zero current quark m ass E q．(6) has only one so—
lution (B (p ) ≠0)w hich corresponds to the N am bu—
G o ld sto n e p h a se in th e ch ira l lim it．w h ile th e so lu tio n
co rresp o n d in g to th e W ig n er p h a se sim p ly d o es n o t
ex ist． It sh o u ld b e n o ted th a t in o b ta in in g th is co n —
clu sio n o n e h a s a ssu m e d th a t th e d re ss ed g lu o n p ro p —
a ga to rs in th ese tw o p h a ses a re th e sa m e ． H o w e ve r
sin ce th e fe a tu res o f th ese tw o p h ase s ar e so d iff ere n t
it is re a son a b le to e x p ect th a t th e b e h a v io u r o f th e
d resse d g lu o n p ro p a g a tors in th e se tw o p h a ses s h o u ld
also b e di ren t．In ad dition ．th e fa ct that g (p 1 三0
is not a solu tion to E q ．f61 does n ot necessarily m ean
that there exists only one solution ( (p ) ≠ 0) fo r
Eq．(6)． A s is show n in the fo llow ingthere do ex—
ist tw o qualitatively distinct B (p ) ≠ 0 solutions in
Eq．(6)． ．
In o rd e r to d em o n stra te th is p o in tle t u s sep a —
ra te t h e co n trib u tio n s o f d y n a m ica l a n d ex p licit ch ira l
sym m etry breaking (driven by current quark m ass m )
ex p lic itly ．B y d iff e ren tia tin g th e d re sse d q u a rk p ro p —
agator 9 (Pm ) w ith respect to 仇w e find that the
d resse d q u ark p ro p a g a to r is rela ted to th e v ertex fo r
th e sca lar o p e ra tor q b y
(pm ) —a0 1(p m )
In te g ra tin g th is eq u a tio n w e h a v e
01(pm) 9 (Pm + 厂 (p0m m
0
G (p)+ 9云 (pm ) (8)
w here G (p)an integration constantis the contri—
b u tio n of d y n a m ic al ch ira l sy m m etry b re ak in g a n d is
independent ofthe current quark m ass m 9云 (Pm )is
th e c o n trib u tio n o f ex p licit ch ira l sy m m e try b reak in g
a n d v a n ish e s if t h e c u rren t q u a rk m a ss 仇 e q u a ls z ero ．
W ithout loss of generality9 (Pm ) can be w ritten
a S
9云 (pm ) m [i~／·pE (p)+ F (p)． (9)
Substituting E qs．(8) and (9)into Eq．(2)w e have
A (p) 三A (p)+ m E (p)
(p) 三B (p)+ m F (p)m
P utting E q．(10) into E q
fled by E (p) and F (p)
(6)w e have the D SE satis—
B(p2)+m )m4／研d4q 2(p＇q)
B (q)+ m F (q)
q2[A (q)+ m E (q)+ [B (q) + m F (q)’
[A (p) + m E (p) lip
4 I d4q
3 (27r)
g2D (p — q)[A (q) + m E (q)
q2[A (q)+ m E (q)+ [B (q)+ m F (q)
p．q+2 ]
w here A (p ) and B (p ) are determ ined by E q．(5)．
For a given m odel gluon propagator g J[)(p)w e
can solve consistently E qs．(5) and (1 1) to obtain the
four scalar functions A (p)B (p)E (p)and F (p)．
In the treatm ents of the previous literature[1—6] one
o fte n a d o p ts th e sa m e fo rm o f th e m o d e l g lu o n p ro p —
a g a to r a s in p u t to so lv e fo r th e se sc a la r fu n ctio n s co r—
r esp on d in g to d ifi eren t p h a ses ． A s w a s p o in ted o u t
b e fo re th is is p ro b lem a tic ．T h e co rrect w ay is to u se
d iff eren t m o d el g lu o n p ro p ag a to rs as in p u t in so lv in g
fo r different phases．To be specificwe use g~D (p — q)
to d e n o te th e d re ssed g lu o n p rop a g a to r in th e N a m b u
G oldstone phase and g82 J[) (p — q) to the dressed gluon
p ro p a ga to r in th e W ig n e r p h a se ．
A s is sh o w n a b o v e th ere ex ists tw o so lu tion s
to E q．(5)i．e． the N am bu—G oldstone solution
1 ●● ●● J
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3 0 3 8 Z O N G H o n g —S h i e t a l V o 1．2 2
(B (p) ≠o) and the W ignersolution (B (p) 三0)．
Substituting these two solutions into Eq．(11)w e can
obtain tw o different solutions for E (p) and F (p)．
For exam plesubstituting B (p) 三 0 and the m odel
gluon propagator g D (p q)into E qs．(5)and (11)w e
have the W ignersolution A )E ) and F (p)
an d
[A (p)1]p d4q 2
D＇(p_q)[p．q
— q) q ·(P — q)]
(p — q) J q2A ＇(q)’ (12)
p)14／ d4q g p_q)
F (q)
q[A (q)+ m E (q)]+ [m F (q)]
(p)+ (p)_1]p2 4／ d4q
g D (p — q)[A (q) + m E (q)]
q[A (q) + m E (q)]+ [m F (q)]
p．q+2 ]．(13
S o fa r at th e ra in b o w a p p rox im a tio n to th e D S E
w e h a v e co m p le ted th e d e riv a tio n o f th e d ep en d e n c e
of (Pm ) on m in the N am buG oldstoneand
W ig n erp h a se s sep a ra tely
(ⅣG) (pm ) i7 ．P A (p)+ B (p)
+ m [i~／·pE (p)+ F (p)]fo r B (p) ≠0
(14)
( ) (p m ) i7 ．P A (p)+ m ．pE (p)
+ F(p)1 for B(p)≠0． (15) J
Just as is show n by E qs．(14)(15)there does exist
tw o qualitatively distinct solutions w ith ( ) ≠ 0 in
th e c ase o f n o n —ze ro c u rren t q u ark m a ss ．In a d d itio n
w e w a n t to stress th at th e a b o v e a p p ro a ch is g en e ra l
in th e sen se th at it d o es n o t d ep e n d o n th e ra in b ow
a p p rox im a tio n u se d h e re ． It ca n a lso b e a p p lied to
th e c ase o f th e B a ll C h iu [71 an d C u rtis— P en n in g to n [ l
v e rtice s．
In o rd e r to h av e a q u a lita tiv e u n d ersta n d in g o f th e
a b o v e tw o q u a lita tiv e ly d istin c t so lu tio n s a p a rtic—
u la rly sim p le an d u sefu l m o d el o f th e d re ssed g lu o n
tw o —p o in t fu n ctio n [9j in th e N a m b u ～G o ld sto n e p h a se
is e m p lo y ed ：
g D (p — q) 47r 町
( )(P — q)
(P ～q) (p ]
(P — g)
w h e re th e sca le p a ra m eter r／ is a m e a su re o f th e
stre n g th o f th e in fra red slav ery eff ec t． T h is m o d e l
h a s th e a d v a n ta g e th a t th e in teg ra l eq u a tio n o f D S E
re d u c es to alg e b ra ic eq u a tio n s．
H e re w e sh o u ld also sp ec ifv t h e d ressed g lu o n tw o —
p o in t fu n c tio n in th e W ig n er p h ase ． 昭 h av e n o a p ri·
or i k n ow le d g e o f th is fu n c tio n e x ce p t th at it sh o u ld b e
d ifrere n t fro m th e o n e in th e N a m b u G o ld ston e p h a se ．
In o rd er to sh o w th a t th e re d o e x is t tw o q u a lita tiv ely
d istin c t so lu tio n s o f th e D y so n ——S ch w in g er eq u a tio n fo r
th e q u a rk p ro p a g a to r in th e c as e o f n o n —ze ro c u rren t
q u a rk m ass w e c h o o se th e fo llow in g a n sa tz
gD (p — q)4丌 町 5
． ( )(P — q)
(P — g) g)
(P — q)0
w ith a sca le p a ra m e ter r／ d iff e ren t fro m 町．H ere w e
em p h a size th a t w e ch o o se th e a b o v e an sa tz o n ly fo r
co n v en ien ce a n d th is w ill n o t aff ect o u r m a in co n clu —
sio n ．
Substituting E q．(16) into E q．(5)w e have the
N a m b u — G o ld sto n e so lu tio n in ch ira l lim it
8 (p) (町～4p)
A (p) 2 fo r P<
B (p) 0
叩2
4
( +
Sim ilarlysubstituting E q．(17)into E q．(12)w e have
th e W ig n e r so lu tio n in ch ira l lim it
B (pz)三A (pz)： 1[1+(1+2pr／2／~ ]．(19)
W ith th e m o d el o f th e d res sed g lu o n p ro p a g ato r
specified in E qs．(16)and (1 7)and the explicit expres—
sion fo r A (p)B (p)and A (p)given in E qs．(18)and
(19)E qs．(11)and (13)entail that the scalar functions
u (p) and F (p) satisfy
A (p)+ rnE (p) 2[B (p)+ m F (p)
B (p)+ m F (p) + m
+mFcp m { 竽
+[B(p)+mF(p)])町 (20)
and the scalar functions E (p) and F (p) satisfy
A (p)+ m E (p)
)1]{ +rn2F＇( )
8
1
／
>
2 p
r
1 ● ●● J
l 2
＼ ＼
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N o ．1 2 Z O N G H o n g —S h i e t a l 3 0 3 9
U sin g M a th e m a tica it is n o t d iffi cu lt to v e rify th at th e
solutions of E q s．(20) an d f2 1) do ex ist．T h e fuU ex—
pressions for E (p )F (p)E (p)and F )are too
len g th y a n d w ill n o t b e g iv en in th is stu d y ．It sh o u ld
be noted that the m odel gluon propagator (16) is an
in fra red —d o m in a n t m o d e l th a t d o es n o t rep re sen t w e ll
the behaviour of g J[) (p) aw ay from P 0．N ev—
erth elessth is sim p le m o d el ca n p ro v id e a rea so n ab le
g u id e to th e n o n D e rtu rb ativ e p rop erties o f m ore so —
p h istic ate d D S E —m o d el o f O C D ．
In su m m a ry w e sh o w on g en era l g ro u n d th at th e re
ex ist tw o q u alita tive ly d is tin ct so lu tio n s o f th e D y son ——
S ch w in g er eq u a tio n fo r th e q u a rk p ro p ag a to r in th e
case of non—zero current quark m ass (one of w hich
co rresp o n d s to th e N a m b u ——G o ld sto n e p h a se a n d th e
oth er c o rresp o n d s to th e W ig n er p h a se in th e ch ira l
lim it)．T h is app roach h as the a dvan tage th at w e can
a n aly se th e effect s of ex p lic it a n d d y n am ic a l ch ira l
sy m m etry b rea k in g se p a ra tely ． T h e b a sic e q u a tio n s
u sed here are E q s．f5) an d f11)．In order to have a
q u a lita tiv e u n d e rs tan d in g o f th e a b o v e tw o q u a lit a
tiv ely d istin ct so lu tio n s w e ch o o se a sim p le co n fi n —
ing m odel (16) in solving E qs．(5) and (11)． From
th is th e d re sse d q u ar k p ro p a g a to r in th e N am b u
G o ld ston e p h a se an d th e W ig n e r p h a se are d erive d ．
W ith th ese tw o p h a ses ch a ra cte riz ed b y q u a lita —
tiv e ly d iff e ren t m om en tu m —d ep en d en t q u a rk p ro p a g a
torsone can stud y th e Q C D p hase stru ctu re in a d ef
in ite w a y ．
R e fe r e n c e s
1C ahill R T and R oberts C D 1985 P hysR D 32 2419
[2] T andy P C 1997 P rog．P art． N uc1．P hys．39 1 17 and
re fe re n c es t h e r ein
l3 l R ob erts C D a nd W illiam s A G 1994 P rog．尸art． Ⅳcf．
P h y s ．3 3 4 7 7 a n d re fer e n c e s t h er ein
[4] R oberts C D and Schm idt S M 2000 P rog．P art．肌 cf．
P h y s ．4 5 S 1 1 a n d re fere n ce s t h e re in
[5] Zong H SP ing J L Y ang H T L x F and W ang F 2003
P h y s ．R e u ．D 6 7 0 7 4 0 0 4
Z on g H S Q i S C h en W S un W M a nd Z hao E G 200 3
P h y s ．L ett．B 5 7 6 2 8 9
Y a n g H T Z o n g H S P in g J L a n d W a n g F 2 0 0 3 P h y s ．
L e tt．B 5 5 7 3 3
[6] Zong H SC hang L H ou F Y Sun W M and Liu Y X 2005
P h y s ．R e u ．C 7 1 0 1 5 2 0 5
Z o n g H S H o u F Y S u n W M P in g J L a n d Z h ao E G
2 0 0 5 P h y s R e ．C 7 2 0 3 5 2 0 2
H o u F Y C h a n g L S u n W M Z o n g H S a n d L iu Y X 2 0 0 5
P h y s ．R e ．C 7 2 0 3 4 9 0 1
7l B all J S and C hiu T W 1980 P hys．R eu．D 22 2542
[8l C urtis D C and Pennington M R 1990 Phys．R eu．D 42
4 1 6 5
9I M unczek H J and N em irovsky A M 1983 P hys．Reu．D 28
8
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