A consistent derivation of the quark--antiquark and three quark potentials in a Wilson loop context

33 Pages·0.24 MB·English

by  N. Brambilla

IFUM 452/FT, December 1993
A consistent derivation of the quark-antiquark
and three-quark potentials
in a Wilson loop context
4 N. Brambilla, P. Consoli and G. M. Prosperi
9
9
Dipartimento di Fisica dell’Universita` – Milano
1

n
INFN, Sezione di Milano – Via Celoria 16, 20133 Milano
a
J

2
1

1 Abstract
v
1
5
0 Inthispaperwegiveanewderivation of thequark-antiquarkpotential inthe
1
0
Wilson loop context. This makes more explicit the approximations involved
4
9
/ and enables an immediate extension to the three-quark case. In the qq case
h
t
- we find the same semirelativistic potential obtained in preceding papers but
p
e
h for a question of ordering. In the 3q case we find a spin dependent potential
:
v
identical to that already derived in the literature from the ad hoc and non
i
X
r correct assumption of scalar confinement. Furthermore we obtain the correct
a
form of the spin independent potential up to the 1/m2 order.
PACS numbers: 12.40.Qq, 12.38.Aw, 12.38.Lg, 11.10.St
Typeset using REVTEX
1
I. INTRODUCTION
The aim of this paper is twofold.
First we give a simplified derivation of the quark-antiquark potential in the context of
the so called Wilson loop approach [1] in which the basic assumptions, the conditions for
the validity of a potential description and the relation with the flux tube model [2] can be
better appreciated.
Secondly we show how the procedure can be extended to the three-quark system [3]
obtaining consistently not only the static part (stat) of the potential but also the spin
dependent (sd) and the velocity dependent (vd) ones at the 1/m2 order.
For what concerns the qq potential the result is identical to that reported in [4,5] (see [6]
forthespindependent potential)except foraproblemoforderingofminorphenomenological
interest:
Vqq = Vqq +Vqq +Vqq (1.1)
stat sd vd
where
4α
Vqq = s +σr, (1.2)
stat −3 r
1 1 1 4α
Vqq = + 2 s +σr +
sd 8 m21 m22!∇ (cid:18)−3 r (cid:19)
1 4α σ 1 1
s
+ S (r p ) S (r p ) +
2 (cid:18)3 r3 − r(cid:19)"m21 1 · × 1 − m22 2 · × 2 #
1 4α
s
+ [S (r p ) S (r p )]+
m m 3 r3 2 · × 1 − 1 · × 2
1 2
1 4 1 3 8π
+ α (S r)(S r) S S + δ3(r)S S , (1.3)
m m 3 s r3 r2 1 · 2 · − 1 · 2 3 1 · 2
1 2 (cid:26) (cid:20) (cid:21) (cid:27)
1 4α
Vqq = s(δhk +rˆhrˆk)phpk
vd 2m m 3 r 1 2 −
1 2 (cid:26) (cid:27)W
2 1 1
σrp2 σrp p . (1.4)
− 6m2{ jT}W − 6m m { 1T · 2T}W
j=1 j 1 2
X
Obviously in Eqs. (1.2)-(1.4) r = z z denotes the relative position of the quark and the
1 2
−
antiquark and p the transversal momentum of the particle j, ph = (δhk rˆhrˆk)pk where
jT jT − j
2
ˆr = (r/r); the symbol stands for the Weyl ordering prescription among momentum
W
{ }
and position variables (see Sec. IV). Furthermore in comparison with [5] the terms in the
zero point energy C have been omitted, since they should be reabsorbed in a redefinition of
the masses in a full relativistic treatment.
For the 3q potential the result is
V3q = V3q +V3q +V3q (1.5)
stat sd vd
with
2α
V3q = s +σ(r +r +r ), (1.6)
stat j<l −3rjl! 1 2 3
X
1 2 α 2 α
V3q = 2 s s +σr +
sd 8m2∇(1) −3r − 3r 1
1 (cid:18) 12 31 (cid:19)
1 2 α 2 α σ
s s
+ S (r p ) +(r p ) (r p ) +
(2m21 1 ·" 12 × 1 3r132! 31 × 1 −3r331!− r1 1 × 1 #
1 2 α 1 2 α
s s
+ S (r p ) + S (r p ) +
m1m2 1 · 12 × 2 −3r132! m1m3 1 · 31 × 3 3r331!)
1 2 1 3 8π
+ α (S r )(S r ) S S + δ3(r )S S
m1m23 s(r132 "r122 1 · 12 2 · 12 − 1 · 2# 3 12 1 · 2)
+ cyclic permutations, (1.7)
1 2α 3 1
V3q = s(δhk +rˆhrˆk)phpk σr p2
vd j<l 2mjml (3rjl jl jl j l)W −j=1 6m2j{ j jTj}W −
X X
3 1 3 1
σr z˙2 σr p z˙ . (1.8)
− 6{ j MTj}W − 6m { j jTj · MTj}W
j=1 j=1 j
X X
Again r = z z denotes the relative position of the quark j with respect to the quark l
jl j l
−
(j,l = 1,2,3) and r = z z the position of the quark j with respect to a common point
j j M
−
M such that 3 r is minimum. As well known, if no angle in the triangle made by the
j=1 j
P
quarks exceeds 120 , the three lines which connect the quarks with M meet at this point
◦
with equal angles of 120 like a Mercedes star (I type configuration, see Fig. 1A). If one of
◦
the angles is 120 , then M coincides with the respective vertex and the potential becomes
◦
≥
a two-body one (II type configuration, see Fig. 1B). Furthermore 2 is the Laplacian with
∇(j)
3
respect to the variable z and now vh = (δhk rˆhrˆk)vk where ˆr = (r /r ). The quantity
j Tj − j j j j j
z˙ in (1.8) is given by
M
N 1 3 p /r m I type configuration
z˙ =  − j=1 jTj j j (1.9)
M (cid:16) (cid:17)
 pl/mPl II type configuration : zM zl ,
≡
N being a matrix with elements Nhk = 3 1(δhk rˆhrˆk).
j=1 rj − j j
P
Finally Eq. (1.7) properly refers to the I configuration case. In general one should write
3 1
VLR = S VLR p . (1.10)
sd − 2m2 j ·∇(j) stat × j
j=1 j
X
In comparing (1.10) with (1.7) one should keep in mind that the partial derivatives in z
M
of VLR vanish due to the definition of M.
stat
We observe that the short range part in Eqs. (1.6)-(1.8) is of a pure two body type: it
is identical to the electromagnetic potential among three equal charged particles but for the
colour group factor 2/3 and it is well known. Even the static confining potential in Eq. (1.6)
is known [7,1,3]. The long range part of Eq. (1.7) coincides with the expression obtained by
Ford [8] starting from the assumption of a purely scalar Salpeter potential of the form
σ(r +r +r )β β β , (1.11)
1 2 3 1 2 3
but at our knowledge it has not been obtained consistently in a Wilson loop context before.
Eq. (1.8) is new. It should be stressed that (1.11) corresponds to the usual assumption of
scalar confinement for the quark-antiquark system. As well known from this assumption
Vqq and Vqq result identical to (1.2) and (1.3), but Vqq turns out different from (1.4).
stat sd vd
The important point concerning Eqs. (1.1)-(1.8) and (1.10) is that they follow from
rather reasonable assumptions on the behaviour of two well known QCD objects, W and
qq¯
W , related to the appropriate (distorted) quark-antiquark and three-quark “Wilson loops”
3q
respectively.
For the qq case the basic object is
1
W = TrPexp ig dxµA (x) . (1.12)
qq µ
3
(cid:28) (cid:18) IΓ (cid:19)(cid:29)
4
Here the integration loop Γ is assumed to be made by a world line Γ between an initial
1
position y at the time t and a final one x at the time t for the quark (t < t ), a similar
1 i 1 f i f
world line Γ described in the reverse direction from x at the time t to y at the time
2 2 f 2
t for the antiquark and two straight lines at fixed times which connect x to x , y to y
i 1 2 2 1
and close the contour (Fig. 2). As usual A (x) = 1λ Aa(x), P prescribes the ordering of
µ 2 a µ
the color matrices (from right to left) according to the direction fixed on the loop and the
angular brackets denote the functional integration on the gauge fields.
The quantity ilnW is written as the sum of a short range contribution (SR) and of a
qq
long range one (LR): ilnW = ilnWSR +ilnWLR. Then it is assumed that the first term
qq qq qq
is given by the ordinary perturbation theory, that is at the lowest order
4
ilnWSR = g2 dxµ dxνiD (x x ) (1.13)
qq 3 1 2 µν 1 − 2
ZΓ1 ZΓ2
(D being the usual gluon propagator and α = g2/4π the strong interaction constant) and
µν s
the second term by the so called “area law” [9,1,4]
ilnWLR = σS , (1.14)
qq min
where S denotes the minimal surface enclosed by the loop (σ is the string tension).
min
Obviously Eq. (1.13) is justified by asymptotic freedom, Eq. (1.14) is suggested by lattice
theory, numerical simulation, string models and other types of arguments.
Up to the 1/m2 order, the minimal surface can be identified with the surface spanned
by the straight line joining (t,z (t)) to (t,z (t)) with t t t ; the generic point of this
1 2 i f
≤ ≤
surface is [4]
u0 = t umin = sz (t)+(1 s)z (t) (1.15)
min 1 − 2
with 0 s 1 and z (t) and z (t) being the positions of the quark and the antiquark at
1 2
≤ ≤
the time t.
We further perform the so called instantaneous approximation in (1.13), consisting in
replacing
5
+
D (x) Dinst(x) = δ(t) ∞dτ D (τ,x) (1.16)
µν −→ µν µν
Z−∞
and use (1.15) and (1.16) at an early stage in the derivation procedure. In this way we shall
obtain Eqs. (1.1)-(1.4) in a much more direct way and without the need of assuming a priori
the existence of a potential as done in [4]. So, once that Eqs. (1.13) and (1.14) have been
written, Eqs. (1.15) and (1.16) give the conditions under which a description in terms of a
potential actually holds.
Notice that, while (1.12), (1.14), and even (1.13) in the limit of large t t, are gauge
f i
−
invariant quantities, the error introduced by (1.16) is strongly gauge dependent. The best
choice of the gauge at the lowest order in perturbation theory is the Coulomb gauge for
which the above error is minimum. To this choice Eq. (1.4) does refer.
For the three-quark case the quantity analogous to (1.12) is
1 a1b1
W = ε ε Pexp ig dxµ1A (x)
3q 3! * a1a2a3 b1b2b3(cid:20) (cid:18) ZΓ1 µ1 (cid:19)(cid:21)
a2b2 a3b3
Pexp ig dxµ2A (x) Pexp ig dxµ3A (x) . (1.17)
µ2 µ3
(cid:20) (cid:18) ZΓ2 (cid:19)(cid:21) (cid:20) (cid:18) ZΓ3 (cid:19)(cid:21) +
Here a ,b are colour indices; Γ denotes a curve made by: a world line Γ for the quark
j j j j
j between the times t and t (t < t ), a straight line on the surface t = t merging from
i f i f i
a point I (whose coordinate we denote by y ) and connected to the world line, another
M
straight line on the surface t = t connecting the world line to a point F with coordinate x
f M
(Fig. 3). The positions of the two points I and F are determined by the same rules which
determine the point M above.
The assumption corresponding to (1.13),(1.14) is then
2
ilnW = g2 dxµ dxνiD (x x )+σS , (1.18)
3q 3 i j µν i − j min
i<jZΓi ZΓj
X
where now S denotes the minimum among the surfaces made by three sheets having the
min
curves Γ , Γ and Γ as contours and joining on a line Γ connecting I with F.
1 2 3 M
Weshallsee thatEqs. (1.5)-(1.8)followif wesubstitute (1.16)in(1.18)andagainreplace
S with the surface spanned by the straight lines
min
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u0 = t umin = sz (t)+(1 s)z (t) (1.19)
j min j j − M
with j = 1,2,3, s [0,1], z (t) being again the point for which 3 z (t) z (t) is
∈ M j=1| j − M |
P
minimum.
The plan of the paper is the following one. In Sec. II we shall report the simplified
derivation of the quark-antiquark potential as sketched above. In Sec. III we shall report
the derivation of the three-quark potential. In Sec. IV we shall make some remarks and
discuss the connection with the flux tube model.
II. QUARK-ANTIQUARK POTENTIAL
As usual the starting point is the gauge invariant quark-antiquark (q ,q¯ ) Green function
1 2
(for the moment we assume the quark and the antiquark to have different flavours)
1
G(x ,x ,y ,y ) = 0 Tψc(x )U(x ,x )ψ (x )ψ (y )U(y ,y )ψc(y ) 0 =
1 2 1 2 3h | 2 2 2 1 1 1 1 1 1 2 2 2 | i
1
= Tr U(x ,x )SF(x ,y A)U(y ,y )C 1SF(y ,x A)C . (2.1)
3 h 2 1 1 1 1| 1 2 − 2 2 2| i
Here c denotes the charge-conjugate fields, C is the charge-conjugation matrix, U the path-
ordered gauge string
b
U(b,a) = Pexp ig dxµA (x) (2.2)
µ
Za !
(theintegration pathbeing thestraight linejoiningato b), SF andSF thequark propagators
1 2
inanexternal gaugefieldAµ; furthermoreinprinciple theangularbrackets shouldbedefined
as
[A]M (A)f[A]eiS[A]
f
f[A] = D , (2.3)
h i [A]M (A)eiS[A]
R f
D
R
S[A]being thepuregaugefieldactionandM (A) thedeterminant resulting fromtheexplicit
f
integration onthe fermionic fields. Inpractice assuming (1.14) corresponds to take M (A) =
f
1 (quenched approximation).
7
Summarizing the first part of the procedure followed in Ref. [4] (see such paper for
details) first we assume x0 = x0 = t , y0 = y0 = t with τ = t t > 0 and note that SF are
1 2 f 1 2 i f − i j
4 4 Dirac indices matrices type. Then performing a Foldy-Wouthuysen transformation on
×
G we can replace SF with a Pauli propagator K (a 2 2 matrix in the spin indices) and
j j ×
obtain a two-particle Pauli-type Green function K. We shall show that in the described
approximations this function satisfies a Schr¨odinger-like equation with the potential (1.1)-
(1.4).
One finds (see [4]) that, up to the 1/m2 order, K satisfies the following equation
j
∂
i K (x,y A) = H K (x,y A) :=
∂x0 j | FW j |
1 1 g
= m + (p gA)2 (p gA)4 S B+gA0
" j 2mj j − − 8m3j j − − mj j · −
g g
(∂ Ei ig[Ai,Ei])+ εihkSk (p gA)i,Eh K (x,y A) (2.4)
− 8m2j i − 4m2j j{ j − }# j |
with the Cauchy condition
Kj(x,y A) x0=y0 = δ3(x y) (2.5)
| | −
where εihk is the three-dimensional Ricci symbol and the summation over repeated indices
is understood. By standard techniques the solution of Eq. (2.4), with the initial condition
(2.5), can be expressed as a path integral in phase space
zj(x0)=x x0
K (x,y A) = [z ,p ]Texp i dt[p z˙ H ] ; (2.6)
j j j j j FW
| Zzj(y0)=y D ( Zy0 · − )
here the time-ordering prescription T acts both on spin and gauge matrices, the trajectory
of the quark j in configuration space is denoted by z = z (t), the trajectory in momentum
j j
space by p = p (t) and the spin by S . Then, by performing the translation
j j j
p p+gA, (2.7)
−→
we obtain an equation containing the expression dt(gA0 gz˙ A) gdxµA , which is
µ
− · ≡
8
formally covariant. 1 It is also useful to have an expression for K in which the tensor field
j
Fµν and its dual Fˆµν appear. To this end we make the further translation
g
p p (E S) (2.8)
−→ − m ×
and, apart from higher order terms, we obtain
K (x,y A) = zj(x0)=x [z ,p ]Texp i x0dt p z˙ m p2j + p4j
j | Zzj(y0)=y D j j ( Zy0 " j · j − j − 2mj 8m3j−
g g g
gA0 + S B+ S (p E) S (z˙ E)+
− m j · 2m2 j · j × − m j · j ×
j j j
g
+gz˙ A+ (∂ Ei ig[Ai,Ei]) . (2.9)
j · 8m2 i − #)
j
Thus we obtain the two-particle Pauli-type propagator K in the form of a path integral
on the world lines of the two quarks
z1(tf)=x1 z2(tf)=x2
K(x ,x ,y ,y ;τ) = [z ,p ] [z ,p ]
1 2 1 2 1 1 2 2
Zz1(ti)=y1 D Zz2(ti)=y2 D
tf 2 p2 p4
exp i dt p z˙ m j + j
 Zti jX=1" j · j − j − 2mj 8m3j#
1  2 ig 
TrT Pexp ig dxµA (x)+ dxµ
*3 s  IΓ µ jX=1 mj ZΓj
1 1
SlFˆ (x) SlεlkrpkF (x) DνF (x) . (2.10)
j lµ − 2mj j j µr − 8mj νµ !)+
Here T is the time-ordering prescription for spin matrices, P is the path-ordering prescrip-
s
tion for gauge matrices along the loop Γ and as usual
Fµν = ∂µAν ∂νAµ +ig[Aµ,Aν], (2.11)
−
1
Fˆµν = εµνρσF , (2.12)
ρσ
2
1 More precisely, sincethe Ah arematrices, thestep d3pf(p gA)= d3pf(p)can bejustified
−
R R
byexpandingf(p gA)inpowersofg; apartfromthezerothorderterm,alltheothertermsinvolve
−
derivatives of f(p) and do not contribute to the integral.
9
DνF = ∂νF +ig[Aν,F ] (2.13)
νµ νµ νµ
and εµνρσ is the four-dimensional Ricci symbol.
Furthermore as in Eq. (1.12) Γ denotes the path going from (t,y ) to (t ,x ) along
1 i 1 f 1
the quark trajectory (t,z (t)), Γ the path going from (t ,x ) to (t,y ) along the antiquark
1 2 f 2 i 2
trajectory (t,z (t)) and Γ is the path made by Γ and Γ closed by the two straight lines
2 1 2
joining (t,y ) with (t,y ) and (t ,x ) with (t ,x ) (see Fig. 2). Finally Tr denotes the trace
i 2 i 1 f 1 f 2
on the gauge matrices. Note that the right-hand side of (2.10) is manifestly gauge invariant.
What we have to show is that the angular bracket term in Eq. (2.10) can be expressed
as the exponential of an integral function of the position, momentum and spin alone taken
at the same time t:
1 tf
TrT Pexp... T exp i dtVqq(z ,z ,p ,p ,S ,S ) ; (2.14)
s s 1 2 1 2 1 2
3 ≃ −
(cid:28) (cid:29) (cid:20) Zti (cid:21)
indeed then we can conclude that
∂ 2 p2 p4
i K = m + j j +Vqq K , (2.15)
∂t j=1 j 2mj − 8m3j! 
X
 
Vqq playing the role of a two-particle potential. To this aim, expanding the logarithm of the
left-hand side of (2.14) up to 1/m2 order, we should have
2 ig 1
ilnW +i dxµ Sl Fˆ (x) Slεlkrpk F (x)
qq m jhh lµ ii− 2m j j hh µr ii−
j=1 j ZΓj (cid:18) j
X
1 1 ig2
DνF (x) T dxµ dxσSl Sk
− 8mj hh νµ ii(cid:19)− 2 j,j′ mjmj′ sZΓj ZΓj′ ′ j j′
X
tf
Fˆ (x)Fˆ (x) Fˆ (x) Fˆ (x) dtVqq , (2.16)
lµ kσ ′ lµ kσ ′
hh ii− hh iihh ii ≃
(cid:18) (cid:19) (cid:20)Zti (cid:21)
with the notation
1 TrP[exp(ig dxµA (x))]f[A]
f[A] = 3 h Γ µ i (2.17)
hh ii 13 hTrPexp(Hig ΓdxµAµ(x))i
H
and W defined in Eq. (1.12).
qq
At this point in Ref. [4] we assumed that a quantity Vqq satisfying (2.16) existed and
derived its form. Here we no longer make such an a priori assumption but start directly
from (1.13) and (1.14).
10