Confining Bethe--Salpeter equation from scalar QCD

22 Pages·0.21 MB·English

by  N. Brambilla

IFUM 485/FT, December 1994
Confining Bethe–Salpeter equation from scalar QCD
N. Brambilla and G. M. Prosperi
Dipartimento di Fisica dell’Universita` – Milano
INFN, Sezione di Milano – Via Celoria 16, 20133 Milano
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We give a nonperturbative derivation of the Bethe–Salpeter equation based

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on the Feynman–Schwinger path integral representation of the one–particle
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2 propagator in an external field. We apply the method to the quark–antiquark
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0 systeminscalarQCDandobtainaconfiningBSequationassumingtheWilson
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area law in the straight line approximation. The result is strictly related to
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the relativistic flux tube model and to the qq¯semirelativistic potential.
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PACS numbers: 12.40.Qq, 12.38.Aw, 12.38.Lg, 11.10.St
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Typeset using REVTEX
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I. INTRODUCTION
Various attempts have been done in the literature to apply the Bethe-Salpeter equation
to a study of the spectrum and the properties of the mesons. The hope was to obtain an
unifiedandconsistent descriptionofthequark-antiquarkboundstates envolving lightquarks
as well as heavy ones. In all such attempts, at our knowledge, the choice of the kernel of
the equation was purely conjectural and only made in such a way that the successful heavy
quarks potential could be recovered in the non relativistic limit. While a kind of derivation
of the qq¯potential from QCD can be given in the Wilson loop context, even if at the price
of not completely proved assumptions [1]– [5], no similar derivation seems to exist for the
BS equation.
In this paper we want to consider this problem for the simplified model of the scalar
QCD.
As it is well known a satisfactory semirelativistic potentialfor theheavy quark–antiquark
system can be obtained from the assumption
ilnW = i(lnW) +σS (1.1)
pert min
where: W stands for the Wilson loop integral
1
W = TrPexpig dxµA ; (1.2)
3h {I µ}i
Γ
Γ denotes a closed loop made by a quark world line (Γ ), an antiquark world line (Γ )
1 2
followed in the reverse direction and two straight lines connecting the initial and the final
points of the two world lines; (lnW) and S are the perturbative evaluation of lnW and
pert min
the minimum area enclosed by Γ respectively; finally the expectation value in (1.2) stands
for the functional integration on the gauge field alone. More sofisticate evaluations of ilnW
have also been attempted [5], [4], [7], but they shall not be considered here.
In the derivation of the potential, S is further approximated by the surface spanned
min
by the straight lines connecting equal time points onthe quark and the antiquark worldlines,
i.e. by the surface of equation
2
x0 = t , x = sz (t)+(1 s)z (t) , (1.3)
1 2
−
t being the ordinary time, z (t) and z (t) the quark and the antiquark positions at the time
1 2
t, and s a parameter with 0 s 1 . The result is
≤ ≤
Smin ∼= Ztitf dtrZ01ds[1−(sz˙1T +(1−s)z˙2T)2]21 = (1.4)
where z˙h stands for (δhk rhrk)dzk and r(t) = z (t) z (t).
jT − r2 dt 1 − 2
In fact it can be shown that (1.4) is correct up to the second order in the velocities
[2], [3] and so is perfectly appropriate for a derivation of the semirelativistic potential. For
a full relativistic extention, Eq.(1.4) cannot be correct as it stands, due to the privileged
role played by the time in it. However one can try to assume (1.4) in the center of mass
frame and then obtain implicitly the corresponding equation in a general frame simply by
Lorentz transformation. In this way the assumption would become equivalent to the so
called relativistic flux tube model [8], [3].
InthispaperwewanttoshowthatinscalarQCD(i.e. neglectingthespinofthequarks) a
Bethe Salpeter (BS) equation which include confinement can be derived from (1.1) and (1.4)
in the center of mass frame. In the momentum representation and after the factorization of
the four–momentum conservation δ, the resulting kernel turns out in the form
Iˆ(p ,p ; p′,p′) = Iˆ (p ,p ; p′,p′)+Iˆ (p ,p ; p′,p′) (1.5)
1 2 1 2 pert 1 2 1 2 conf 1 2 1 2
(p′ +p′ = p +p ), where Iˆ is the usual perturbative kernel which can be written, at the
1 2 1 2 pert
lowest order in the strong coupling constant,
16 p +p′ p +p′
Iˆ (p ,p ;p′,p′) = g2( 1 1)µD (p′ p )( 2 2)ν , (1.6)
pert 1 2 1 2 3 2 µν 1 − 1 2
while in the center of mass system Iˆ is given by
conf
p +p′ p +p′
Iˆ (p ,p ;p′,p′) = d3rei(k′−k)·rJ(r, 1 1, 2 2) (1.7)
conf 1 2 1 2 Z 2 2
(p = p = k, p′ = p′ = k′) with J expressed as an expansion in σ
1 − 2 1 − 2 m2
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σr 1
J(r,q ,q ) = 4 [q2 q2 q2 +q2 q2 q2 +
1 2 2 q +q 20q 10 − T 10q 20 − T
10 20
q2 q2 q q
+ 10 20(arcsin| T| +arcsin| T|)]+... (1.8)
q q q
T 10 20
| | | | | |
Notice that by a usual instantaneous approximation, from the kernel defined by (1.5)-(1.8)
one can obtain the following hamiltonian
(r,q) = m2 +q2 + m2 +q2 +
H q 1 q 2
σr 1 m2 +q2 m2 +q2
+ v 2 m2 +q2 +v 1 m2 +q2 +
2 m2 +q2 + m2 +q2nuum21 +q2q 1 r uum22 +q2q 2 r
1 2 t t
q q
m2 +q2 m2 +q2 q q
+ q 1 q 2 arcsin T +arcsin T +...
(cid:16) qT (cid:17)(cid:16) m2 +q2 m2 +q2(cid:17)o
1 2
q q
+ (r,q) (1.9)
pert
V
with an appropriate ordering prescription. In Eq.(1.9) q stands now for the momentum in
the center of mass frame, q = (ˆr q)ˆr, qh = (δhk rˆhrˆk)qk, while is the ordinary
r · T − Vpert
perturbative Salpeter potential. We stress that Eq.(1.9) is identical to the hamiltonian for
the already mentioned relativistic flux tube model and consequently is strictly connected to
the semirelativistic potential as given in [2], [3] when the spin dependent part is neglected.
Notice however that the ordering in (1.9) corresponding to (1.7) is not identical to the Weyl
prescription.
Thepresentpaperhasmainlyapedagogicalpurposeandwehavenotdoneanyattemptto
explicitly apply the kernel (1.5)–(1.8) to an evaluation of the spectrum and of the properties
of the mesons. Notice however that very interesting results have been obtained in this
direction by the relativistic flux tube model [9] and in a sense we may consider our paper
also as providing a more fundamental justification to that model.
As in potential theory the starting object in our derivation is the gauge invariant quark-
antiquark propagator
Ggi(x ,x ; y ,y ) = 0 φ∗(x )U(x ,x )φ (x )φ∗(y )U(y ,y )φ (y ) 0 =
4 1 2 1 2 h | 2 2 2 1 1 1 1 1 1 2 2 2 | i
1
= Tr U(x ,x )∆(1)(x ,y ,A)U(y ,y )∆(2)(y ,x ,A) , (1.10)
−3 h 2 1 F 1 1 1 2 F 2 2 i
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where U(b,a) = P exp(ig bdxµA (x)) is the path ordered gauge string (the integration
ba a µ
R
path is over the straight line joining a to b), while ∆(1) and ∆(2) denote the Feynman
F F
propagators for the two quarks in the external field A . In contrast with the potential case,
µ
however, no semirelativistic expansion is used, but the propagators are treated exactly using
the covariant Feynman-Schwinger path integral representation.
Notice that in establishing the BS equation we have to neglect in (ilnW) the con-
pert
tributions from the two extreme lines x x and y y and in S the border contribution
1 2 1 2 min
corresponding to x = x and y = y , as in the potential case. This is correct for x0 y0
10 6 20 10 6 20 1− 1
and x0 y0 large with respect to x x , y y , x0 x0 and y0 y0. So, strictly, we
2 − 2 | 1 − 2| | 1 − 2| 1 − 2 1 − 2
obtain a BS equation but for a quantity G which coincides with Ggi in the above limit.
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This is immaterial for what concerns bound states or asymptotic states. Naturally out of
the limit situation, G is no longer gauge invariant as it should be talking of a BS equation.
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The significance of the Feynman–Schwinger representation in the framework of QCD has
already been appreciated in [10] and particularly in [4]. With respect to Ref. [4] we have
however a different attitude on the role of the BS equation.
The plan of the paper is the following one: in Sec. 2 we illustrate our method on the
example of a one dimensional particle in a velocity dependent potential. In Sec. 3 we apply
the method to the derivation of the Bethe–Salpeter equation for a system of two scalar
particles interacting via a scalar field and discuss the various complications related to the
perturbative kernel. In Sec.4 we derive the BS equation for a quark-antiquark system in
scalar QCD under the assumption discussed above and obtain the kernel reported. Finally
in Sec.5 we derive the Salpeter potential from the BS equation.
II. ONE DIMENSION POTENTIAL THEORY
Let us consider the model made by a nonrelativistic particle in one dimension with the
Hamiltonian
p2
H = +U(x,p) = H +U (2.1)
2m 0
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and the corresponding Schr¨odinger propagators
K(x,y,t) = x e−iHt y , K (x,y,t) = x e−iH0t y . (2.2)
0
h | | i h | | i
From the operatorial identity
t
e−iHt = e−iH0t i dt′eiH0(t−t′)Ue−iHt′ , (2.3)
− Z
0
we obtain the equation
t
′ ′ ′
K(x,y,t) = K (x,y,t) i dt dξ dηK (x,ξ,t t) ξ U η K(η,y,t). (2.4)
0 − Z Z Z 0 − h | | i
0
which is somewhat analogous to the nonhomogeneous Bethe-Salpeter equation in the con-
figuration space.
We want to derive Eq.(2.4) by the path–integral formalism.
Let us be for definiteness
U = V(x)+(W(x)p2) , (2.5)
ord
where () stands for some ordering prescription. In terms of path–integral we can write in
ord
the phase space
x t p′2
K(x,y,t) = z pexp i dt′[p′z˙′ V(z′) W(z′)p′2] (2.6)
Zy D D { Z0 − 2m − − }
with z′ = z(t′), p′ = p(t′), z˙′ = dz(t′). In Eq.(2.6) the functional “measures” are supposed to
dt′
be defined by
m iε
N N
z = ( )2 dz1...dzN−1, p = ( )2 dp1...dpN−1dpN
D 2πiε D 2πm
1
DzDp = (2π)Ndp1dz1...dpN−1dzN−1dpN (2.7)
where ε = t , the limit N is understood and the end points x and y stand for the
N → ∞
condition z = y, z = x. As well known (see e.g. [6]) the ordering prescription is concealed
0 N
under the particular discretization adopted in the limit procedure implied in the definition
of (2.6).
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Let us first consider the case W = 0. In this case there is no ordering problem and
it is possible to calculate explicitly the integral in p in (2.6) obtaining the path-integral
representation in the configuration space
K(x,y,t) = x zei 0tdt′(mz˙2′2−V(z′)). (2.8)
Zy D R
Then using the identity
e−i 0tdt′V(z′) = 1 i tdt′V(z′)e−i 0t′dt′′V(z′′) (2.9)
R − Z R
0
one obtains
K(x,y,t) = K0(x,y,t)−iZ0tdt′ZyxDzV(z′)eiRtt′dt′′mz˙′2′2+iR0t′dt′′mz˙′2′2−iR0t′dt′′V(z′′) (2.10)
which, taking into account that
x x ξ
z... = dξ z z... (2.11)
Zy D Z Zξ D Zy D
(having identified ξ = z(t′)), can be rewritten in the form (2.4) with ξ U η = V(ξ)δ(ξ η).
h | | i −
In the general case W(x) = 0 it is convenient to work with the original path-integral
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representation in the phase space (2.6) and it is necessary to use discretized expressions
explicitly. For Weyl ordering in Eq.(2.5) the correct discretization is the mid–point one and
we can write
1
K(x,y,t) = (2π)N Z dpNdzN−1dpN−1...dz1dp1
expiXN {pn(zn −zn−1)−ε[2pm2n +V(xn +2xn−1)+W(xn +2xn−1)p2n]}. (2.12)
n=1
Then it is convenient to introduce the Fourier Transform of K(x,y,t)
K˜(k,q,t) = dx dye−ikxK(x,y,t)eiqy (2.13)
Z Z
and to use the discrete counterpart of (2.9)
exp( iε) N [V(xn +xn−1)+W(xn +xn−1)p2] = 1 iε N [V(xR +xR−1)+W(xR +xR−1)p2]
− X 2 2 n − X{ 2 2 R
n=1 R=1
exp( iε)R−1[V(xr +xr−1)+W(xr +xr−1)p2]. (2.14)
· − X 2 2 r
r=1
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Replacing (2.14) in (2.12) and Fourier transforming we have
1 N N
K˜(k,q,t) = K˜0(k,q,t)−iε(2π)N X Z dzNdpN ...dp1dz0e−ikzN expi X pn(zn −zn−1)
R=1 n=R
[V(zR +2zR−1)+W(zR +2zR−1)p2R]expiRX−1[pn(zn −zn−1)−
n=1
ε( p2n +V(zn +zn−1)+W(zn +zn−1)p2)]eiqz0 =
2m 2 2 n
N 1 1
= K˜0(k,q,t)−iε X (2π)2nZ dpR+1Z dpR−1(2π)N−R−1 Z dzNdpN ...dpR+2dzR+1e−ikzN
R=1
N
exp i pn(zn zn−1) eipR+1zR+1
{ X − } o
n=R+2
n21π Z dzRdpRdzR−1e−i(pR+1−pR)zR[V(zR +2zR−1)+W(zR +2zR−1)p2R]ei(pR−1−pR)zR−1o
1 R−2
n(2π)R−2 Z dzR−2...dz0e−ipR−1zR−2expi X{pn(zn −zn−1)
n=1
ε[ p2n +V(zn +zn−1)+W(zn +zn−1)p2] eiqz0 (2.15)
− 2m 2 2 n } o
which in the continuous limit reads
t dk′ dq′
K˜(k,q,t) = K˜ (k,q,t) i dt′ K˜ (k,k′,t t′)I˜ (k′,q′)K˜(q′,q,t′) (2.16)
0 − Z Z 2π Z 2π 0 − W
0
with
1 z′′ +z′ z′′ +z′
I˜ (k,q) = dz′′dpdz′e−ikz′′eipz′′[V( )+W( )p2]e−ipz′eiqz′ =
W 2π Z 2 2
k +q 1
= V˜(k q)+W˜ (k q)( )2 k (V(x)+ p, p,W(x) ) q . (2.17)
− − 2 ≡ h | 4{ { }} | i
Eq.(2.16) is equivalent to Eq.(2.4) and the kernel reduces to the Fourier transform of the
case W(x) = 0.
Had we considered the symmetric ordering in Eq.(2.5) we should have replaced in (2.14)
and (2.15) [V(xn+2xn−1)+W(xn+2xn−1)p2n] with [21(V(xn)+V(xn−1))+12(W(xn)+W(xn−1))p2n]
and the result would be
1 V(z′′)+V(z′) W(z′′)+W(z′)
I˜ (k,q) = dz′′dpdz′e−ikz′′eipz′′[ + p2]e−ipz′eiqz′ =
S 2π Z 2 2
k2 +q2 1
= V˜(k q)+W˜ (k q) k (V(x)+ p2,W(x) ) q (2.18)
− − 2 ≡ h | 2{ } | i
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III. SPINLESS PARTICLES INTERACTING THROUGHT A SCALAR FIELD
Let us consider two scalar “material” fields φ and φ interacting through a third scalar
1 2
field A with the coupling 1(g φ2A+g φ2A). Then, after integration over φ and φ , the full
2 1 1 2 2 1 2
one particle propagator can be written as
AeiS0(A)M(A)i∆(j)(x,y;A)
G(j)(x y) = 0 Tφ (x)φ (y) 0 = i∆(j)(x,y;A) D F (3.1)
2 − h | j j | i h F i ≡ R AeiS0(A)M(A)
D
R
where ∆(j)(x,y,A) is the propagator for the particle j in the external field A, S (A) is the
F 0
free action for the field A and the determinantal factor M(A) comes from the integration of
the fields φ
j
det(∂µ∂ +m2 g A) −1
M(A) = µ j − j 2 =
j=Y1,2h det(∂µ∂µ +m2j) i
1 1
= 1 g d4xA(x)∆(j)(0) g2 d4xd4yA(x)∆(j)(x y)A(y)∆(j)(y x)+... (3.2)
− 2 X {− jZ F − 2 j Z F − F − }
j=1,2
(j)
The covariant Feynman-Schwinger representation for ∆ reads
F
i ∞ x τ 1 1 1
∆(Fj)(x,y;A) = −2 Z0 dτ Zy DzDpexp{iZ0 dτ′[−pµ′z˙′µ + 2p′µp′µ − 2m2j + 2gjA(z′)]}
i ∞ x τ 1
= dτ zexp i dτ′ [(z˙′2 +m2) g A(z′)] , (3.3)
−2 Z0 Zy D {− Z0 2 j − j }
where the pathintegralsareunderstood tobeextended over allpaths zµ = zµ(τ′) connecting
y with x expressed in terms of an arbitrary parameter τ′ with 0 τ′ τ. In Eq.(3.3) z′
≤ ≤
stands for z(τ′), p′ for p(τ′), z˙′ for dz(τ′) and the “functional measures” are assumed to be
dτ′
defined as
1 iε
Dz = (2πiε)2Nd4z1...d4zN−1, Dp = (2π)2Nd4p1...d4pN−1d4pN
1
z p = ( )4Nd4p1d4z1...d4pN−1d4zN−1d4pN. (3.4)
D D 2π
Replacing Eq.(3.3) in (3.1) we obtain
1 ∞ x i τ ig τ
G(j)(x y) = dτ zexp dτ′(z˙′2 +m2) exp j dτ′A(z′) (3.5)
2 − 2 Z0 Zy D {−2 Z0 j }h 2 Z0 i
where, suppressing the tadpole term in (3.2),
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ig τ ig2 τ τ′
exp j dτ′A(z′) = exp j dτ′ dτ′′[D (z′ z′′)+
h 2 Z i 4 Z Z F −
0 0 0
g2
+ i d4ξ d4ηD (z′ ξ)(∆(i)(ξ η))2D (η z′′)+...] (3.6)
X 2 Z Z F − F − F −
i
Now let us consider the two particle propagator
G (x ,x ;y ,y ) = 0 Tφ (x )φ (x )φ (y )φ (y ) 0 = G(1)(x ,y ;A)G(2)(x ,y ;A) =
4 1 2 1 2 h | 1 1 2 2 1 1 2 2 | i h 2 1 1 2 2 2 i
1 ∞ ∞ x1 x2
= ( )2 dτ dτ z z
2 Z0 1Z0 2Zy1 D 1Zy2 D 2 ×
exp −i τ1 dτ′(z˙′2 +m2)+ τ2 dτ′(z˙′2 +m2) exp i g τ1dτ′A(z′)+g τ2dτ′A(z′) . (3.7)
2 {Z 1 1 1 Z 2 2 2 }h 2{ 1Z 1 1 2Z 2 2 }i
0 0 0 0
In this case
i τ1 ′ ′ τ2 ′ ′
exp g dτ A(z )+g dτ A(z ) =
h 2{ 1Z 1 1 2Z 2 2 }i
0 0
exp igj2 τj dτ′ τj′ dτ′′[D (z′ z′′)+ gi2 d4ξ d4ηD (z′ ξ)
X 4 Z0 jZ0 j F j − j X 2 Z Z F i − ×
j=1,2 i=1,2
(∆(i)(ξ η))2D (η z′′)+...]+ ig1g2 τ1 dτ′ τ2 dτ′[D (z′ z′)+
F − F − i 4 Z 1Z 2 F 1 − 2
0 0
g2
+ i d4ξ d4ηD (z′ ξ)(∆(i)(ξ η))2D (η z′′)+...] (3.8)
X 2 Z Z F i − F − F − i
i=1,2
If we replace M(A) by 1 in (3.1) and (3.7), i.e., if we retain only the lowest order terms in
the exponent in (3.8) (quenched approximation), we have no “material” fields loops in the
evaluation of the gauge field average (3.6) and (3.8) and then we can write exactly
1 ∞ ∞ x1 x2
G (x ,x ;y ,y ) = ( )2 dτ dτ z z
4 1 2 1 2 2 Z0 1Z0 2Zy1 D 1Zy2 D 2
exp −i[ τ1 dτ′(z˙′2 +m2) g12 τ1 dτ′ τ1′ dτ′′D (z′ z′′)]
2 Z 1 1 1 − 2 Z 1Z 1 F 1 − 1 ×
0 0 0
exp −i[ τ2 dτ′(z˙′2 +m2) g22 τ2dτ′ τ2′ dτ′′D (z′ z′′)]
2 Z 2 2 2 }− 2 Z 2Z 2 F 2 − 2 ×
0 0 0
exp ig1g2 τ1dτ′ τ2dτ′D (z′ z′). (3.9)
4 Z 1Z 2 F 1 − 2
0 0
Proceding as in Sec.2, using the identity
exp ig1g2 τ1dτ′ τ2dτ′D (z′ z′) =
4 Z 1Z 2 F 1 − 2
0 0
= 1+ ig1g2 τ1dτ′ τ2 dτ′D (z′ z′)exp[ig1g2 τ1′ dτ′′ τ2dτ′′D (z′′ z′′)] (3.10)
4 Z 1Z 2 F 1 − 2 4 Z 1 Z 2 F 1 − 2
0 0 0 0
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