- Email: [email protected]
Quark confinement and scaling structure functions
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
Q U A R K C O N F I N E M E N T A N D S C A L I N G S T R U C T U R E F U N C T I O N S ¢r S. BLAHA Laboratory o f Nuclear Studies, Cornell University, Ithaca, New York 14850, USA Received 25 September 1974 We give examples of a class of four-dimensional gluon field theories of hadron binding in which the Schwinger mechanism manifestly prevents the appearance of free quarks in the physical particle spectrum. The deep inelastic structure functions are shown to have Bjorken scaling.
The paradoxical failure to observe quarks despite apparently successful quark-parton model descriptions of hadronic processes has led to a number of investigations of hadronic binding and quark confinement [1 ]. While these studies have been suggestive, a four dimensional field theory with manifest quark confinement and wel-defined procedures for calculating Green's functions and S-matrix elements has not appeared. We will study a class of Lorentz invariant, unitary, indefinite metric gluon theories which may fill that gap. For simplicity, our models will only attempt to describe the large distance component of quark interactions, neglecting short distance components and internal quark quantum numbers. Since the hypothetical gluons responsible for quark binding have not been identified we shall take the economical view that they have no independent dynamical existence but rather are only an embodiment of the quark interaction. Therefore our general procedure will be to introduce a second quantized gluon field in a model Lagrangian, establish all operator relations and expressions (e.g. the expansion of the S-matrix in time ordered products of the interaction Lagrangian) and, instead of associating a set of states with the gluon field operator, impose sufficient operator conditions on the gluon field to reduce all q-number expressions to C-number expressions. The conditions we impose are suggested by the identity
~(x)~fy) = ½ [~(x), ~(.v)]+ + ½ [~(x), ~(.v)l_,
(1)
with [A, B] ± = A B +BA and the observation (for bosons) that the commutator is uniquely determined by the
Supported in part by the National Science Foundation.
requirements that it be consistent with the equations of motion and the equal time canonical commutation relations, and that it vanish for space-like distances. Consequently, we require all gluon field operator expressions be expanded using eq. (1) with the commutator replaced by the appropriate invariant function and then set all anti-commutators equal to zero. Finally for terms with an odd number of gluon field operators we set the field operator to zero. Our procedure is well defined and clearly does not violate any of the initial operator relationships. We shall verify the Lorentz invariance, unitarity, etc., of the resulting S-matrix in our models. It should be remarked that the application of our procedure to the electromagnetic field results in a model QED whose classical limit is the adjunct field theory of action-at-a-distance electrodynamics [2] (without absorber). We consider two models which appear to offer attractive prospects for consideration as the hadron binding mechanism. The Lagrangian in both has the general form
.6?= Z?i - g A 1 j u + ~Oq,
(2)
where Z?q is a conventional free quark Lagrangian, Ju is the quark current and in model I ±wl F2UV__t~2A2a2u (3) and in model II .~. - _ i F 1 ~-3~u+lw2 F 2 ~ v _ ~ 2 A 2 A 3 t ~ tI-
2 /.*v~
4"~v
(4)
with A/u a massless gluon field, F/uv = buAiv - ~uAiv and ), a coupling constant with dimensions of mass. Z?I is ~nilar in form to a dipole ghost scalar model of Glaser [3]. Recently dipole ghosts have ,found physical 373
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
application as a mechanism for pion condensation in neutron stars [4]. The gluon equations of motion are
a~Fl~v+ X2A 2v = 0 , O~F2UV+gjV = 0 ,
(5,6)
and OuFI~v+ X2A2v = O, alaF2uu-
(7, 8)
X2A 3v = 0 ,
a~F3~V+ j v = 0 ,
(9)
in modelsI and II respectively. A 1 is the only gauge invariant field in both models. The equal time commutation relations in the radiation gauge (V "A 1 = 0) are
[F~oi(X),A~(y)I =ih ab f
-igor +iGIv=iX2 g.ve/k 4 , k 2 + ie
(11)
iDF/w ~ iG /av II = iX4gtwp/k6 (12) in the respective models. The time ordered product, T(Alu(x)AIO,)), is the Fourier transform of the quantities on the right side of eqs. (11) and (12). The gluon potential of alstatic quark charge at "short" distances is -gX2lrl[2 in model I and -gX4[r[3/12 in model II in contrast to e/Irl in electromagnetism. It appears that the models are consistent with the roughly harmonic oscillator character of the hadron binding force which has been suggested by some experimental analyses [5]. At larger distances the gluon potential is drastically modified by vacuum polarization effects. The full gluon propagator can be expression in terms of the proper gluon self-energy, H(k), as '
-
gm'X2
Gluv(k) k4 +g2XEk2II(k)'
(13)
in model I (with a similar expression in model II) where II(k) is normalized to equal the corresponding quantity in QED in lowest order (fig. 1). Since II is a con- p o ~ 374
Fig. 2. Quark self-energy diagram.
stant up to logarithms in lowest order and proportional to inverse powers of k 2 in higher order, clearly, important screening effects occur at large distances with the scale being set by (gX) -1 . The denominator ~:erm g2 X2k 2 II (k) w.ill control this region. These same polarization effects preclude the existence of free quarks. In the Coulomb gauge of Alv the total charge is given by
Q =fd3x J°(x)
k~k/
d3k exp[ik'(x-y)] (5i] i2 ) ,(10) (270 3 lk with h 12= h 21 = 1 and zero otherwise in model I, and h 13 = - h 22 = h 31 = 1 and zero otherwise in model II. We quantize the gluon fields as disaussed previously, and the quark field as a conventional spin 1/2 particle. As a result we may use the perturbation theory rules of QED if the photon propagator is replaced with
iDFuv(k)
Fig. 1. Lowest order vacuum polarization diagram.
fdS.
(14)
v (~ Al°(x)),
(15)
after taking account of the equations of motion. Now
A~(x)=fd4y GV(x_y)jO(y).
(16)
Because of vacuum polarization (eq. (13)) the surface term integration in eq. (15) is necessarily zero and thus a free quark would be totally screened. Only neutral bound states exists in this model (and for model II using a similar argument). We now establish the Lorentz invariance and unitarity of the S-matrix. The Lorentz invariance of any Feynman diagram amplitude is established by expressing it as a Feynman parameter integral of functions of Lorentz invariant quantities through the use of such identities as oo
P[k4 = - ½
fd4k
f
d a a e(a) exp [iak2],
exp (iCk 2) = ilr2
e(C)/C 2 ,
(17)
(18)
where e(x) = -+1 f o r x <> 0. Unitarity is established if, for any Feynman diagram amplitude, there are no contributions from intermediate states containing gluons to the unitarity sum for its absorptive part in a given channel. As one naively expects, having a principal value gluon propagator rather then a Feynman propagator removes the gluon
Volume 56B, number 4
PHYSICS LETTERS ! I
J
/
/
\
I I I I
\
b
i i
I
i •
J
I ¢
d
Fig. 3. Forward Compton amplitude diagrams giving the lowest order contributions to the inelastic structure functions. The dashed line indicates the contributing absorptive part. pole and no such contributions arise. We have verified this explicitly in perturbation theory by calculating the lowest order quark self-energy (fig. 2) in models I and II. In model II we have ~(q) - 16rt2q X4g2p4 {~[ [ln m (A~22)
2q2m 2 (q2_ m2)2
+ 3q4m2-q6 In (-~2)1
+2ml-q22+m2-2'm2 q )ll _ (q2_ m2)2
(q2_ m2)3 In ~
,
where q is the quark four-momentum, m is the quark mass, the cutoffA 2 results from a subtraction in the quark propagator, and P signifies that all singularities are taken in principal value (Y, (q2+ ie) = ~;(q2 _ ie)). In contrast to QED where an off shell electron can decay into an electron and photon we see that no such process is possible here. It should be noted that the second order self-energy has a quadratic divergence in model I. A principal value propagator increases the degree of divergence of a diagram. On the other hand, it does not generate infrared divergences. A more general unitarity proof can be given by decomposing a principal value propagator into a Feynman propagator plus a delta function term which opens up the loop where the gluon occurs. Thus an amplitude in our models can be expanded in terms of conventional Feynman amplitude integrals and the absorptive
12 May 1975
part of that sum shown to receive no contributions from intermediate states with gluons (which now have ordinary Feynman propagators). A similar procedure was used by Feynman [6] in a study of unitarity problems in Yang-Mills theories. Since the commutativity of field operators at spacelike distances is maintained microscopic causality is not violated in our models. Advanced effects will occur. But free quarks do not exist and vacuum polarization reduces the gluon interaction to short range, so that advanced effects are unobservable, Although our models are not super-renormalizable they furnish a framework for understanding the success of the parton model of deep inelastic electroproduction. The lowest order diagrams contributing to the structure functions are given in fig. 3. The contribution of fig. 3a is dominant. Figs. 3b, c, d contribute terms of O(q -4) to vW2 in model I and O(q -8) in model II where q2 is the virtual photon mass. The gluon propagators sharply damp contributions from large transverse momenta. In conclusion, we have proposed models for hadron binding which appear to account for some of the principal qualitative features of phenomenological quark models of hadronic processes. A study of bound state properties in these models is underway to determine which agrees best with experimental results. I am grateful to the members of Newman Laboratory for helpful discussions and to Professor L. Susskind for suggesting the quark confinement argument of eq. (15).
References [1] K. Johnson, Phys. Rev. D6 (1972) 1101; A. Casher, J. Kogut and L. Susskind, Cornell Univ. preprint CLNS-251 ; K.G. Wilson, Cornell Univ. preprint CLNS-262; A. Chodos, R. Jaffe, K. Johnson, C. Thorn and V. Weisskopf, M.I.T. preprint ~ 387. [2] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 17 (1945) 157; 21 (1949) 425; [3] M. Froissart, Suppl. Nuovo Cimento 14 (1959) 197. [4] S. Blaha, Cornell Univ. preprint.
[5] H.J. Lipkin, Physics Reports (1974); R. Feynman, M. Kislinger, F. Ravndal, Phys. Rev. D3 (1971)
2706. 375
PHYSICS LETTERS
12 May 1975
Q U A R K C O N F I N E M E N T A N D S C A L I N G S T R U C T U R E F U N C T I O N S ¢r S. BLAHA Laboratory o f Nuclear Studies, Cornell University, Ithaca, New York 14850, USA Received 25 September 1974 We give examples of a class of four-dimensional gluon field theories of hadron binding in which the Schwinger mechanism manifestly prevents the appearance of free quarks in the physical particle spectrum. The deep inelastic structure functions are shown to have Bjorken scaling.
The paradoxical failure to observe quarks despite apparently successful quark-parton model descriptions of hadronic processes has led to a number of investigations of hadronic binding and quark confinement [1 ]. While these studies have been suggestive, a four dimensional field theory with manifest quark confinement and wel-defined procedures for calculating Green's functions and S-matrix elements has not appeared. We will study a class of Lorentz invariant, unitary, indefinite metric gluon theories which may fill that gap. For simplicity, our models will only attempt to describe the large distance component of quark interactions, neglecting short distance components and internal quark quantum numbers. Since the hypothetical gluons responsible for quark binding have not been identified we shall take the economical view that they have no independent dynamical existence but rather are only an embodiment of the quark interaction. Therefore our general procedure will be to introduce a second quantized gluon field in a model Lagrangian, establish all operator relations and expressions (e.g. the expansion of the S-matrix in time ordered products of the interaction Lagrangian) and, instead of associating a set of states with the gluon field operator, impose sufficient operator conditions on the gluon field to reduce all q-number expressions to C-number expressions. The conditions we impose are suggested by the identity
~(x)~fy) = ½ [~(x), ~(.v)]+ + ½ [~(x), ~(.v)l_,
(1)
with [A, B] ± = A B +BA and the observation (for bosons) that the commutator is uniquely determined by the
Supported in part by the National Science Foundation.
requirements that it be consistent with the equations of motion and the equal time canonical commutation relations, and that it vanish for space-like distances. Consequently, we require all gluon field operator expressions be expanded using eq. (1) with the commutator replaced by the appropriate invariant function and then set all anti-commutators equal to zero. Finally for terms with an odd number of gluon field operators we set the field operator to zero. Our procedure is well defined and clearly does not violate any of the initial operator relationships. We shall verify the Lorentz invariance, unitarity, etc., of the resulting S-matrix in our models. It should be remarked that the application of our procedure to the electromagnetic field results in a model QED whose classical limit is the adjunct field theory of action-at-a-distance electrodynamics [2] (without absorber). We consider two models which appear to offer attractive prospects for consideration as the hadron binding mechanism. The Lagrangian in both has the general form
.6?= Z?i - g A 1 j u + ~Oq,
(2)
where Z?q is a conventional free quark Lagrangian, Ju is the quark current and in model I ±wl F2UV__t~2A2a2u (3) and in model II .~. - _ i F 1 ~-3~u+lw2 F 2 ~ v _ ~ 2 A 2 A 3 t ~ tI-
2 /.*v~
4"~v
(4)
with A/u a massless gluon field, F/uv = buAiv - ~uAiv and ), a coupling constant with dimensions of mass. Z?I is ~nilar in form to a dipole ghost scalar model of Glaser [3]. Recently dipole ghosts have ,found physical 373
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
application as a mechanism for pion condensation in neutron stars [4]. The gluon equations of motion are
a~Fl~v+ X2A 2v = 0 , O~F2UV+gjV = 0 ,
(5,6)
and OuFI~v+ X2A2v = O, alaF2uu-
(7, 8)
X2A 3v = 0 ,
a~F3~V+ j v = 0 ,
(9)
in modelsI and II respectively. A 1 is the only gauge invariant field in both models. The equal time commutation relations in the radiation gauge (V "A 1 = 0) are
[F~oi(X),A~(y)I =ih ab f
-igor +iGIv=iX2 g.ve/k 4 , k 2 + ie
(11)
iDF/w ~ iG /av II = iX4gtwp/k6 (12) in the respective models. The time ordered product, T(Alu(x)AIO,)), is the Fourier transform of the quantities on the right side of eqs. (11) and (12). The gluon potential of alstatic quark charge at "short" distances is -gX2lrl[2 in model I and -gX4[r[3/12 in model II in contrast to e/Irl in electromagnetism. It appears that the models are consistent with the roughly harmonic oscillator character of the hadron binding force which has been suggested by some experimental analyses [5]. At larger distances the gluon potential is drastically modified by vacuum polarization effects. The full gluon propagator can be expression in terms of the proper gluon self-energy, H(k), as '
-
gm'X2
Gluv(k) k4 +g2XEk2II(k)'
(13)
in model I (with a similar expression in model II) where II(k) is normalized to equal the corresponding quantity in QED in lowest order (fig. 1). Since II is a con- p o ~ 374
Fig. 2. Quark self-energy diagram.
stant up to logarithms in lowest order and proportional to inverse powers of k 2 in higher order, clearly, important screening effects occur at large distances with the scale being set by (gX) -1 . The denominator ~:erm g2 X2k 2 II (k) w.ill control this region. These same polarization effects preclude the existence of free quarks. In the Coulomb gauge of Alv the total charge is given by
Q =fd3x J°(x)
k~k/
d3k exp[ik'(x-y)] (5i] i2 ) ,(10) (270 3 lk with h 12= h 21 = 1 and zero otherwise in model I, and h 13 = - h 22 = h 31 = 1 and zero otherwise in model II. We quantize the gluon fields as disaussed previously, and the quark field as a conventional spin 1/2 particle. As a result we may use the perturbation theory rules of QED if the photon propagator is replaced with
iDFuv(k)
Fig. 1. Lowest order vacuum polarization diagram.
fdS.
(14)
v (~ Al°(x)),
(15)
after taking account of the equations of motion. Now
A~(x)=fd4y GV(x_y)jO(y).
(16)
Because of vacuum polarization (eq. (13)) the surface term integration in eq. (15) is necessarily zero and thus a free quark would be totally screened. Only neutral bound states exists in this model (and for model II using a similar argument). We now establish the Lorentz invariance and unitarity of the S-matrix. The Lorentz invariance of any Feynman diagram amplitude is established by expressing it as a Feynman parameter integral of functions of Lorentz invariant quantities through the use of such identities as oo
P[k4 = - ½
fd4k
f
d a a e(a) exp [iak2],
exp (iCk 2) = ilr2
e(C)/C 2 ,
(17)
(18)
where e(x) = -+1 f o r x <> 0. Unitarity is established if, for any Feynman diagram amplitude, there are no contributions from intermediate states containing gluons to the unitarity sum for its absorptive part in a given channel. As one naively expects, having a principal value gluon propagator rather then a Feynman propagator removes the gluon
Volume 56B, number 4
PHYSICS LETTERS ! I
J
/
/
\
I I I I
\
b
i i
I
i •
J
I ¢
d
Fig. 3. Forward Compton amplitude diagrams giving the lowest order contributions to the inelastic structure functions. The dashed line indicates the contributing absorptive part. pole and no such contributions arise. We have verified this explicitly in perturbation theory by calculating the lowest order quark self-energy (fig. 2) in models I and II. In model II we have ~(q) - 16rt2q X4g2p4 {~[ [ln m (A~22)
2q2m 2 (q2_ m2)2
+ 3q4m2-q6 In (-~2)1
+2ml-q22+m2-2'm2 q )ll _ (q2_ m2)2
(q2_ m2)3 In ~
,
where q is the quark four-momentum, m is the quark mass, the cutoffA 2 results from a subtraction in the quark propagator, and P signifies that all singularities are taken in principal value (Y, (q2+ ie) = ~;(q2 _ ie)). In contrast to QED where an off shell electron can decay into an electron and photon we see that no such process is possible here. It should be noted that the second order self-energy has a quadratic divergence in model I. A principal value propagator increases the degree of divergence of a diagram. On the other hand, it does not generate infrared divergences. A more general unitarity proof can be given by decomposing a principal value propagator into a Feynman propagator plus a delta function term which opens up the loop where the gluon occurs. Thus an amplitude in our models can be expanded in terms of conventional Feynman amplitude integrals and the absorptive
12 May 1975
part of that sum shown to receive no contributions from intermediate states with gluons (which now have ordinary Feynman propagators). A similar procedure was used by Feynman [6] in a study of unitarity problems in Yang-Mills theories. Since the commutativity of field operators at spacelike distances is maintained microscopic causality is not violated in our models. Advanced effects will occur. But free quarks do not exist and vacuum polarization reduces the gluon interaction to short range, so that advanced effects are unobservable, Although our models are not super-renormalizable they furnish a framework for understanding the success of the parton model of deep inelastic electroproduction. The lowest order diagrams contributing to the structure functions are given in fig. 3. The contribution of fig. 3a is dominant. Figs. 3b, c, d contribute terms of O(q -4) to vW2 in model I and O(q -8) in model II where q2 is the virtual photon mass. The gluon propagators sharply damp contributions from large transverse momenta. In conclusion, we have proposed models for hadron binding which appear to account for some of the principal qualitative features of phenomenological quark models of hadronic processes. A study of bound state properties in these models is underway to determine which agrees best with experimental results. I am grateful to the members of Newman Laboratory for helpful discussions and to Professor L. Susskind for suggesting the quark confinement argument of eq. (15).
References [1] K. Johnson, Phys. Rev. D6 (1972) 1101; A. Casher, J. Kogut and L. Susskind, Cornell Univ. preprint CLNS-251 ; K.G. Wilson, Cornell Univ. preprint CLNS-262; A. Chodos, R. Jaffe, K. Johnson, C. Thorn and V. Weisskopf, M.I.T. preprint ~ 387. [2] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 17 (1945) 157; 21 (1949) 425; [3] M. Froissart, Suppl. Nuovo Cimento 14 (1959) 197. [4] S. Blaha, Cornell Univ. preprint.
[5] H.J. Lipkin, Physics Reports (1974); R. Feynman, M. Kislinger, F. Ravndal, Phys. Rev. D3 (1971)
2706. 375