A quark-antiquark potential from a superconducting model of confinement

Nut.lear Ph)sl¢,, B226 (19g~;) 299 ~;()g ' North-Holland P u b h s h m g C o m p a n )

A QUARK-ANTIQUARK P O T E N T I A L FROM A S U P E R C O N D U C T I N G M O D E L OF C O N F I N E M E N T J W AI.COCK. M J B U R F I T T and W N C O T T I N G t l A M t t I t 14/tll~ Ph~ su ~ I ahoratom. Unu erstt~ of Bristol I ngland

Reccwed 4 Januar~ 1983

The Landau-Om.,burg phenomenologlcal theory of super¢onducll~lt~, is used as a modcl of flux confinement A monopole pa~r of sources ~,, included to smmlate a quark-anuquark s~,,,lcm The interaction encrg) ~s found m the siahc approxamat~on appropriate for hca',,x quark svstcm,, and equated ".alth the mterquark potential Thl,, potential is ,..omparcd ~.lth other ,,uggestcd phcnomenolog.cal potcnhals and succeeds m reproducing hca~,,, quark spectra

!. Introduction

Quantum-chromodynamlcs has proved to be a compelling theory for particles, successfully, though mostly qualitatively, combining and explaining the varlou,, theoretical ideas in earlier theories. The 'stretched' model of a gluonlc tube of flux between quarks recalls the earlier string models which arose from Reggc theory [1]. Inherent in this picture is the non-appearance of free quarks, and the hadronlzatlon of fast-parting quark-antlquark pairs as the 'string' binding them breaks. At the other extreme is the 'slack" string, where the colour forces are so weak at small separation, that the quarks are effectively free, reminiscent of the successful free quark and bag models [2]. QCD is a complicated theory to solve quantltatwely, particularly with the inclusion of the quark degrees of freedom. Some success. In particular with lattice calculations, has been obtained in the no-quark sector where, plausibly, string-like confinement has been demonstrated and the glueball masses extracted [3]. At a phenomenologlcal level, the use of potentials motivated by QCD ideas has proved successful for the heavy quark states. The addition of sphttlngs arising from the spin-spin aspects of one-gluon exchange, has been successful in explaining some, and predicting other of the charmomum and bottomonium states I4, 51. The possible relationship of the Landau-Ginzburg model of superconductivity to QCD was introduced by Nielsen and Olsen [6] who investigated the infinite vortex solution as a model of the string. The model was further investigated by Nambu [7]. In this paper we consider the Landau-Glnzburg equation as a phenomenology of 299

300

J 14 4ho~l, et al

/ ( onfinement

Q C D . but include a monopolc-antlmonopole pmr to simulate a quark-antlquark pmr. The expectation, confirmed b3' our results, is that a small-scparauon coulombtc. and large-separauon hnear, potenual will result, thereby generating the features used successfull3 by the Corncll potenual [4] The outhne of the paper ,s as follows' m sect. 2 we introduce the l.andauGmzburg equations and review the infinite vortex solution. In the follo~ mg scct~on. ~ e introduce the sources, and the method for computing the mteracuon energy at various separations. In sect. 4 we present the results, w~th a summary and addmonal comments.

2. The Landau-Ginzburg equations We consider first the Landau-Gmzburg phenomenolog,cal theory of magnensm in a superconductor. The energy funcnonal is

l t : = f d~r -71.~[2+~/3[@1a+3-~

-thV"-

"c a

~ -+

Y " × A ) e , c2.1

3', /J, m and e are parameters, .g, a complex scalar order parameter tleld and A the electromagnetxc vector potentml. The complex field + can be ehmmated m terms of its magmtude and phase ~ ( r ) = e'*l~,[ to ymld E =

f

(

' l 4 + 2,~-~ 1 [ h 2 ( v ' l ~ l ) 2 + hV'O d ~r [ - Y l ' b l 2 + ~BIwl

eA (

2

11

+_,(V-XA)

1

2

(22) J'he minimum (vacuum) energy of the theory, corre,,pondmg to normal superconductor has energ~ density - e where e = ~y2//~ and there are also the ~ell-known Abr|kosov stationary values of the funcuonal that correspond to a flux tube w~th flux strength

f A • dr =

2~-ch

(2 3)

e

These are associated with the phase g~ changing b~ 2~r upon enclrchng the tube. Such tubes could terminate on magnetic monopoles of strength ch v/4~ 2-7

]a, =~

(2.4) •

The characteristic length of the theory, the London length

d ~

4hml, elul / (onhnement

301

characterizes the flux tube radius. It is convenient also to introduce the dimensionless constant D1C

(26)

l, = 7 ¢ 2 # . If lengths are measured in umts of X i.e. i," =

(2 7)

/~3

and the functional written in terms of dlmensmnless fields f and g

#

(28) £,4-

g - ¢2mv "

1

(2.9)

~. we;,.

then

E = 2px'f

d~ [_/2 + !f~+

,

--( v/):+/:(~):+

(

1

v'x g+ ~ we?)

'] (2.10)

In terms of the dimensionless p a r a m e t e r k, the ~acuum energy' denslt,, P and length ~k. the m o n o p o l e strength 4 ~a"

¢4rrP A )t2"

(2.11)

(We include a factor of ~ so as to idenufy the a, of Q C D [8].) The lnfimte flux tube has cyhndncal s y m m e t r y and m cylindrical coordinates s = (p, O, ,~), f l s a function only of p such that f ( 0 ) = 0. We? has magn.tude l / p and is .n the azimuthal direction as is g g = igJ is also a function only of p. A finite vector potential ~mplles that g(p) has a singularity g(p) - l / A p at small O and the Abrlkosov infinite flux tube solution is obtained by minimizing the functlonal T whmh is a shght modification of E:

f:[

If4

I

df' :

('1 d ( p g ) ) 2 ] p do + f 2 9 2 . (2 12) ,

subject to the above constraints on f and g at O -- 0. T ]s the energy per unit length of the flux tube and can be called the string tensmn. The m l m m l z a t l o n yields two second order ordinary differenual equations

/ W Also, l, eta/ / ( onfmement

302

for f(p) and g(p) and the addittonal boundary conditions at mflmty g ( v c ) = 0, f ( ~ ) = 1. We have solved these equauons numerically for ~alues of/~ between 0.5 and 2.0 and the results for the dimensionless quantity T/4~rP~ 2 are shown in fig. 1 In the apphcation of the above formahsm as a phenomenology of the gluon flux tubes which bind heavy quarks we have to make a different interpretation of the parameters The string tension T ~s a reasonably well-determmed parameter The phenomenologtcal value of 5, depends upon the heavy quark system being studied Even for charmonlum ~ts empirical value is not well-determined but for example the Cornell phenomenologlcal potential [4] of 1980 ts 4 Of,,

V(r)

-

3 r + Tr

(2 13)

and the charmomum spectrum indicates reasonable values to be (with T = 0.91 GeV fm l). ~a, = 0.5 = (0.1 GeV fm)

(2.14)

or a, = (0.075 GeV fm). The vacuum energy density P (the pressure m the bag model) Is also not well-determined empirically at present. Johnson and Heller [9] have a small pressure P = 0 055 GeV fm ~ and a large a, = 0.44 GeV fm which with

T

41~p~2

1

h

L

I

2k F~g 1 The dependence of the stnng tension Y on the parameter k for the mfimte vortex

J 14' Aho¢l~et a l / Confinement

303

T = 0.91 w o u l d i m p l y m our m o d e l k = 0.8 a n d ~ = 0.86 fm, while H a s e n f r a t z et al. [10] q u o t e P = 0.381 G e V fm ~ a n d a , = 0.077 GeV fm which i m p l y / , = 0.95 and ~ = 0 3 8 fm.

3. The monopole-antimonopole pair T h e m a t h e m a t i c a l t e r m i n a t i o n of the flux tube on a p o i n t m o n o p o l e is e a s d y carried out m the W u - Y a n g f o r m a h s m [11 ]. C o n s i d e r explicitly a pole situated on the tube axis at ~5= a a n d an a n t l p o l e at f = - a . We then retain c y h n d n c a l s y m m e t r y and a reflexlon s y m m e t r y m the p l a n e f = 0. Dw~de space into three regions. I ,~>~a, lI - a ~ < + a , III ,~< - a . In regions I and III there is no net flux entering since the flux tube terminates on the poles, and the phase q, will not change on encircling the ~ ax~s, (without loss of generality we can take q~ = 0 in these regions). In the W u - Y a n g f o r m a h s m the vector potential d is not c o n t i n u o u s over the b o u n d a r i e s of the regions but if we keep eq. (2.9) as defining g m region lI and have

1 g=

e

vf2rn~ c

.4

(3.1)

m 1 and III then g is a c o n t i n u o u s dlffcrentlable function across the b o u n d a r i e s The a d & u o n a l phase in the definition of g m the central region exactl~ cancels the W u - Y a n g gauge t r a n s f o r m a t i o n that ~s responsible for the d l s c o n t m u l D in A. We now see that the p r o b l e m of d e t e r m i n i n g the energy of a p o l e - a n t l p o l e pa~r s e p a r a t e d by a d~stance 2 a ~ is the p r o b l e m of minimizing the functional

E = 4~'P~. ~ pdo f

df

-1)2 + k~

-d-riO) + ( -~ ) )

(3.2)

where f and g are scalar functions of p and ~ subject to the b o u n d a r y c o n & t l o n s

f ( o , ~) = o

g(p,~)=l/kp, Of _ O, Op

g(p,

£) = 0.

p=0t-a<~
1~51 > a .

(3 3)

In eq. (3.2) we have taken the energy relative to the total b a c k g r o u n d energy (which is infinite). H o w e v e r there is still an infinity in the p r o b l e m due to the energy m the

304

J W Ahoclt et al / Confinement

fields near the point poles, the mfimte mass renormahzauon. This can be exphc~tly removed by considering the ordinary Coulomb field around a dipole. When treated by the vector potential [11] this field yields

g"(°'~)=

1 [ ~+a _ ] 2~0 ~/0,+(~+o)2 V'p- + ( 4 - a) ~ "

=4~rP)k a mpdo

= M-

~ d~

Og° 2

1

(3.4)

O0

4 %

(3.5)

3 r

where M is the (infinite) constant independent of a and the second term Is the ordinary Coulomb potential. We now write the functional m terms of a new function h(o, 4) related to g(o, ~) by

(3.6)

g(p, 4 ) = h(p, ~ )go(p. ~ ). Subtracting the self-energy M then yields the energy functional E

43 2h~ a,

+ 47rpx3W(k" a),

W(k.a)=fo pdpf~d~

(3.7)

[

_f2+ ~(f4+i)+~_~2

+

a(o~hgo)

o,,)2

(¢~p, + ( O f '2'

)-'_( _y()Ogo-' +(01 0(0hg,,)a0)~,

1 a(vg")) 2+ / % ( h ~--1) ] ao

v

(3.s)

and W has to be minimized with respect to the funcuons f and h subJect to the boundary conditions of eq. (3 3) for f , and for h h ( 0 , ~ ) = 1,

~O (h 0 ,. ~ )

= O,

I~]
I~1 > a .

(3.9)

It can be seen that this method of subtraction of the self-energies also makes explicit the Coulomb part of the potential. At the minimum, l:(r) is a function of

J W

Alcm/~

et al / ('onfmement

305

four parameters, the quark-ant~quark separation distance r = 2Xa and any three of pressure P, tension T, pole strength a,, d~stance scale ~, and k (see eq. (2.11) and fig. 1, which relate the~e five). However, for practical purposes we only have to minimize the funcuonal W(I,, a) to obtain the value of W at minimum as a function of k and a. The other two parameters in the theory can be simply included through eq. (3.7). We have found the minimum value of W approximately by imposing the plausible forms o f f and h f= I - exp(-clkP) }

h=exp(_blp2)

h

I~1
(3.10)

= exp(-bip 2 - b2(~¢- a) 2)

and imposing the reflexlon symmetry m the plane ,~= 0 and mlmmlzmg the functional numerically with respect to the four parameters h 1, b2. c~ and ~~. For

1 ~"

P

Gev/f

"-

1 2 --

/

8

--

~--

I 05

I 0

7~

10

/

GeVfm

-02

I

I

I 5

075

" ~

1~

f

I 10

I 1

I

25

: 5

K

1 ig 2 The d e p e n d e n c e of the p r e s s u r e P a n d c o u p l i n g c~, o n tile p a r a m e t e r / , The c e n t r a l point,, with e r r o r b a r s a r e c a l c u l a t e d v.~th I = 1 02 (Je~ fill 1 T h e e r r o r s repre,.ent o u r exlmlate of the nument_al u n e e r t m n t v T h e s o h d h n e ,,hov,,, ,.'.here the c e n t r a l ,..dues m o ~ e t o if T is i n c r e a s e d b y 1()% a n d sHnflarl~. the d o t t e d c u r v e for a r e d u c t i o n b,, 10',

~06

J W Ahocl~ et al / Confinement

large s e p a r a t i o n s the m i n i m a l W increases linearly with a slope which matches the energy per u m t length calculated i n d e p e n d e n t l y in sect. 2 to within 10%. Th~s gwes us c o n f i d e n c e that our functional forms are a d e q u a t e at least for d e s c r i b i n g the central regions of the flux tube. A t small a the m i n i m a l W(k, a) tends to zero: th~s is b e c a u s e the m a g n e t i c field is then c o n c e n t r a t e d as in free space only at distances a r o u n d the poles of o r d e r of the pole s e p a r a t i o n distance, which is small. The '" v a c u u m " function f can then go from zero between the poles to its n o r m a l value of 1 within these small distances, the largest energy cost in m a k i n g this " h o l e in the v a c u u m " comes from the d e r i v a t w e s (Of/Ot~) 2 a n d (OrlOn) 2. However when i n t e g r a t e d over three d i m e n s i o n s these give vanishing c o n t r i b u t i o n s p r o p o r t i o n a l to a, as a tends to zero.

4. Results and di~ussion W e identify the q u a r k - a n t l q u a r k p o t e n t i a l energy V(r) as the m i n i m a l value of the functional E(r). In a d d i t i o n to being a function of the quark s e p a r a t i o n r, tt d e p e n d s on three of the p a r a m e t e r s discussed m the previous sections a n d has the form V(r)=

-4c~/r+

Tr+ U ( r ) ,

(4 1)

where from the n u m e r i c a l calculations U(r) a p p e a r s to be a m o n o t o m c function with U ( 0 ) = 0 a n d U ( ~ e ) = a constant. W e fred numerically for k less than some value k 0 = 0.8 that U(r) IS posltxve a n d the force - d U / d r attractive; for k >/,() it is repulswe; a n d that U(r) is consistent with zero for k = k o. T h u s we can r e p r o d u c e the simple Corneli p o t e n t i a l of eq (2.13) with k --- 0.8. If the string tension T ~s fixed, then a , P ~s a c o n s t a n t for each value of k. I'akmg T = 1.02 + 0.1, values of a , P are gwen m table 1 for k m the range 0.5 to 1.5. The c o m p u t a t i o n a l u n c e r t a i n t y in these n u m b e r s we believe to be within 7%. P h e n o m e n o l o g l c a l potentials, though h a w n g different forms at both largc and small r [12], nevertheless are r e q m r e d to be m r e a s o n a b l e agreemcnt between r = 0.2 a n d r = 0 8 fm so that they can generate the c h a r m o m u m spectrum. In p a r t i c u l a r all

TAB112 1 T h e ",alue of the c o m b i n a t i o n a,P for d i f f e r e n t values of k, the c e n t r a l ",aluc u,,es a "~alue of I = 1 0 2 ( ; e V fm ~ a n d the h m l t s arc for l0% c h a n g e m Y /,

a , t'

0 5 0 7 10 1 25 14

0 016 z 0(X)3 0 0 3 0 + 0 (X)6 0 047 -_ 0 (X)9 0061+_0012 0 076 ± 0 015

,I 1~ A k o d , et a l /

Confinement

307

of the p o t e n t i a l s have a slope of the o r d e r 1.5 G e V fm l a t r - 0 . 4 fm We use this further to restrict o u r class of p o t e n t i a l s so that a , a n d P are s e p a r a t e l y d e t e r m i n e d at each value of k. In fig. 2 P and a , are shown for different values of k with an error b a r which is our e s t i m a t e of the c o m p o u n d e d numerical uncertainties. W e also show the result for different choices of the string tension. If T = 0.92 G e V fm ~ the central values of P and a , move in a correlated way to lie on the solid curve. S l m d a r l y if T is increased to 1.12 the points move to the d a s h e d curve As r e m a r k e d previously the s~mple Cornell potential of 1980 is included in our m o d e l when k = 0.8, the values of T and a , he near our d a s h e d curve. In bag-like models the values of H a s e n f r a t z et al. P = 0.38 G e V fm 3, [10], a , = 0.077 G e V fm a n d T = 0.99 G e V f m - l are close to our values with k s o m e w h a t less than 1 0. W e agree with H a s e n f r a t z a n d exclude the M I T bag values [2, 9]. As an i n d e p e n d e n t check on our class of models, we have calculated the charm o n l u m a n d upsilon spectra w~th the potential p a r a m e t e r s a , = 0 0 4 G e V fm, T = 1 00 G e V fm l, k = 0.5, (with these values U(ao) = 0.175 GeV). The small value of a , was chosen to be in better accord with Q C D values a n d a small value of oq r e q m r e s k to be small in o r d e r to m a k e up the extra s h o r t - r a n g e attraction. There are three p a r a m e t e r s remaining, the quark masses and an u n k n o w n c o n s t a n t c o n t r i b u t i o n C to the p o t e n t i a l that we c a n n o t calculate. W i t h the values rn, = 1.84 G e V / c 2,

m b = 5.18 G e V / c 2,

C = - 1.18 G e V .

a n d the h a m d t o n l a n h2

H=

---

V"2+ m

V(r)+2mc2 +

C,

we o b t a i n the spectra as given in table 2. The levels are in the same sequence as for the Cornell potential. T h e r e are 6T states that couple to the e ' e - system a n d there are only four states listed in the particle d a t a tables. T h e D - s t a t e c o u p l i n g should be weak and p e r h a p s have not yet been observed. The a g r e e m e n t of model with data, in p a r t i c u l a r for the c h a r m o n i u m s p e c t r u m is good, especially bearing in mind that relativistic c o r r e c t i o n s can be expected to bring the higher states d o w n s o m e w h a t in energy. It is also worth r e m a r k i n g that In o u r calculation the root mean square d i s t a n c e between the quarks for the ~(4S), the most energetic of the o b s e r v e d c h a r m o n m m states, is 1.34 fm a n d that the energy is sensitive to the p o t e n t i a l at even greater d~stances; the linear p o t e n t i a l characterized by the old string tension of Regge theory works very well. A l s o in our model the mean radius of the energy d i s t r i b u t i o n m the string ~s - 0.3 fm. This distance is a characteristic d i s t a n c e of the c o n f i n i n g field a n d as stated by Nielsen and Olesen [6] small characteristic distances o f this field Imply large masses for the field excitations or large glue-ball masses. It

3{}8

J IV Aho¢l, et al / (_onfmement TABI I 2 The ma,,.,es of the charmonlum and bottonlonll.lm stale', preda.ted by our potential for parameter,, / = 1 0 (ieV fin l a, = 0 04 {Je%' fin and/, = 0 5. the as,s~gnment of le,,cN ~lth the ph',s~cal states ~,, the ,,ame as E~c_lltcn ct al [4] C h a r l l l O n l u n l lllaSs

UpMlon llla,,s

State

theory ({ JeV)

exp

theor~

e\p

IS 2S 1D 3S 2D 4S

3 097 3 682 3 781 4 112 4 179 4 479

3 097 3 686 3 770 4 030 4 159 4 415

9 456 9 957 1(} 064 1(} 295 10 :~66 1{} 574

9 456 1{}016

I

1{) ~,47 1() 569

would be interesting to fred other experimental data whtch refer to the trans',crse size of strings. In th~s context ~t is also interesting that m the stung model of baryons, as opposed, for example, to the bag model, although the quarks can explore a large volume of space (radius about 1 fm) about the barBm centre of mass the confining strings do not occup) the whole of the volume for all the time: a fact which might go some way m explaining why baryon-baryon forces see no ~lgn of a bag boundary at a 2 fm separation d~stance.

References [1] Y Nambu. Lectures at the Copenhagen Summer S'vmpo,,mm 1970. P (Joddard, J Goldstone, C Rebbl and ( B Thorn, N u d Ph',s B56 11971)109 [2] A {hodos, R L Jaffe, K J o h n , , o n , ( ' B Thorn a n d V F Welsskopf, Ph'.s Re~ D9{197413471. I)ll) (1974) 2599, A Thomas, ('ERN prepnnt TII 3368 11982) [3] M Creutt, {. omment,, on Nut.lear and Partl~.le Ph``slt.s 10 (1981) 163 [4] k. Elchten, K (Jottfrled, T Kmo,,h]ta, J Kogut, K I) l a n e and T M ~'an. Ph,,s Re'. l ett z;4 (1975) 369, E Izlchten, K (Jottfned T Kmoshita, K I) I ane and T M Yah. Ph}s Re`` I)17 {.19781 3090 I)21 (1980) 2{}z~ [51 A Bradle``,J Ph~s G4(1978} 1517 J I. Richardson. Phys Lctt 82B (1979)272, R McClar',, and N B?ers, UCLA prepnnt (1982) [6] H B Nielsen and P Olesen, Nucl Phvs B61 (1973) 45 [7] Y Nambu, Phvs Re', DI0 (1974) 4262 [8] T Appelqmst and H D Pohtzer, Ph?s Re', Lett 34 (1975) 43 [9] L lteller and K Johnson, Phys Lett 84B (1979) 501 [10] P llasenfratz, R R Horgan, J Kut~ and J M Rtchard, Phvs l.ett 94B 1198{.)1 401. 95B (19801 299 [11] T T W u a n d C N Yang, Nucl Phys B107(1976) 365 [12] K J Miller and M G Oleson Phys Lett 109B (1982) 314