Quasi-confinement of quarks

Volume 65B, number 3

PHYSICS LETTERS

22 November 1976

QUASI-CONFINEMENT OF QUARKS* William I. WEISBERGER Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA

Received 19 September 1976 Renormalization group methods indicate that quarks and color can be completely confined if color gauge fields couple to masslessfermions. For quarks with small mass parameters, they are effectively confined at energies on the scale of color singlet baryon masses.

While the quark model has had many successes in describing the properties of physical hadrons, no states with net quark number (Nq - Net ) ¢ 0 (mod 3) have been observed. This has led to models where the quarks possess a hidden SU(3) symmetry, now called color, but physical states are singlets of the SU(3) color group. Subsequent developments in non-abelian gauge theories have aroused the hope that they may provide a dynamical basis for this confinement hypothesis. In the conventional picture quarks come in three colors and the strong interactions are mediated by a YangMills field gauging the group SU(3). The colored triplets, transforming according the fundamental representation of SU(3), come in several varieties, often called "flavors". If there are N flavors, the strong interactions are approximately invariant under a global SU(N) symmetry. Non-abelian gauge theories with not too many fermions are unique among renormalizable field theories in possessing the property of "asymptotic freedom" [ 1 - 3 ] . The effective coupling constant approaches zero at small distances. The conjectured converse to this behavior is that the effective coupling constant grows indefinitely at large distances so that an infinite amount of energy is required to separate particles with color. The result is supposed to be confinement; only particles belonging to the singlet representation of the color gauge group can be created as physical states [1,4]. However, detailed calculations in perturbative field theory have not supported this picture [5, 6]. In perturbation theory, the infrared behavior of non-abelian gauge theories with massive fermions is tamed by the same mech. anisms that work for abelian theories -- the Kinoshita-Lee-Nauenberg theorem is obeyed [7, 8]. While it can be hoped that the perturbative result has a zero radius of convergence and that color is confined for any finite value of the gauge coupling constant, the field theory models in two-dimensional space time which exhibit confinement do so at the lowest order of perturbation theory via a linear potential. I will take the point of view here that the perturbation investigations do indicate the true solution of the theory. In four dimensions with massive quarks, at least for sufficiently small coupling constant, non-singlet representations of the color gauge group are realized in the physical spectrum. The main point of this paper is to present nonperturbative arguments based on renormalization group methods showing that an infrared catastrophe can still occur when the quark mass parameter in the Lagrangian (the quark bare mass) is equal to zero. In this limit the physical quarks may become infinitely heavy (be confined) and only color singlet particles have finite mass. In this situation, there should be N 2 - 1 massless pseudoscalar particles (Goldstone bosons) associated with the dynamical breakdown of global SU(N) X SU(N) chiral flavor symmetry. In the physical world, the global flavor symmetry is broken in a way which seems to be attributable solely to mass splitting among the quarks of different flavors. Therefore, non-zero mass parameters should be associated * Research supported in part by the National Science Foundation Grant PHY-76-15328. 287

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with the quarks, these masses originate from a mechanism extrinsic to the strong interaction color gauge theory. If these mass parameters are sufficiently small, the physical mass spectrum should be close to that described above. The physical quarks are not completely confined but are much heavier than the low lying color singlet states 4:1 . The pseudoscalar mesons acquire masses. Assume we have a Lagrangian field theory o f gauge fields coupled to a multiplet of spin i particles. The theory is renormalized at Euclidean m o m e n t a specified by a mass IX. We follow Weinberg's procedure [9] o f first renorrealizing and defining the coupling constant g in the massless fermion theory. A quark mass parameter m is introduced by specifying the value of the m ~ vertex in the massless theory at zero momentum transfer and (pq)2 = _IX2. Let M(g, m, IX) denote the physical quark mass, or any other well-defined physical mass. Since M is independent of the choice o f Euclidean renormalization point or o f the choice o f gauge, we have the renormalization group equation

(IX 813IX + 13(g) OlOg - ~(g)m BlOm)M(g, m, U) = 0.

(1)

Also by ordinary dimensional analysis

(Ix 3/3Ix + m B/Bm)M(g, m, Ix) = M(g, m, Ix).

(2)

Assume that M is finite if m 4: 0, at least for sufficiently small g. To study what happens at m = 0 we replace m by Xm and continue from X = 1 to X = 04-2. If we just set m = 0 and calculate in perturbation theory, we get M = 0 to any finite order because of the global 3'5 invariance. A solution of (1) and (2) giving different behavior as X ~ 0, will be interpreted as the true solution o f the theory showing a nonperturbative dynamical breakdown of chiral symmetry. To proceed we combine (1) and (2) to obtain

[(1 + 5,(g))X o/ox - ~(g) OlOg - l lM(g, Xm, Ix) = O.

(3)

If Weinberg's power counting theorem [ 10] for mass insertions holds when we add up all orders of perturbation theory, ~(g) > - 1. Assuming this to be true, we can divide by the positive factor 1 + ~(g) to obtain

IX OlOX-TSOg) OlOg

-

~'(g)]

M(g, Xm, Ix) = 0

(4)

w i t h 7 : 1 3 ( 1 + 7 ) - 1 , 7 = (1 + ~ ) - 1 . The modified ~" function has the same fixed points as the usual t3 function. Define a running coupling constant by 0~(X) _ )t - - - ~ - g@()t)),

g(1) =g.

(5)

The standard solution to (4) is

M(g, Xm, Ix) = exp

~'(~(X')) dX'/X' M(~(X), m, Ix).

(6)

Nothing particular about gauge theories has been used to obtain (6). Therefore, we can use this result to consider the X -~ 0 behavior o f any renormalizable field theory. (The extension to several coupling constants and mass parameters is trivial.) If the theory has no non-abelian gauge fields, there is an infrared stable fixed point at

,1 The gauge bosons remain massless in this situation and can be radiated in groups coupled as a net color singlet. Calculations indicate, however, that in scattering at energies on the scale of the color singlet baryons, such radiation is suppressed by powers of the ratio of singlet baryon mass to physical quark mass. ,2 Though rn, g and ~, are gauge dependent, the definition of zero bare mass by m = 0 is gauge invariant. It is convenient to work in the Landau gauge since this gauge is fixed under renormalization group scaling. 288

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g = 0 [3]. I f g is the domain of attraction of 0, lim ~(~(X)) = 0 h--*0 and we obtain lim M(g, Xm, t-O cx XM(O, m, Ix) ~ O. ~.--,0 In contrast, the origin is infrared unstable for a non-abelian gauge theory. Assume as is standard for all confinement ideas, that ~(g) remains negative for all g v~ 0 so that ~(X) diverges as X ~ 0. There is no upper bound on ~ for strong coupling. If, in fact lim ~(~(X)) diverges faster than (lnX) l+c, the exponential integral in (6) converges ~3 and x~0 1 lira M(g, X m , # ) = C lira M(~,rn, la), lnC=- f "q(~(X'))dX'/X'. (7) x-->O ~---+-0 0 If the fermions transform according to the fundamental representation of the group, their self-energy is repulsive and it is reasonable to expect the quark mass to diverge as~ ~ . For a composite color singlet state the attractive force between its constituents grows stronger as ~ increases so that this state may remain at a finite mass. If the observed breaking of the global flavor symmetry is ascribed to mass splitting among the different quark triplets, we cannot set the quark mass parameters completely to zero. According to this picture, small quark mass parameters in the Lagrangian should imply physical quark masses much greater than those of the low lying color singlets. In (7) X would be small and ~(X) very large but finite. From (7) we see that the mass spectrum of a theory with small quark mass parameters and given coupling constant differs only by a scale factor C from the spectrum of a similar theory with large mass parameters and a larger coupling constant ~ . This suggests studying the color confinement problem by looking at the strong coupling limit. The Feynman path integral formalism is convenient for this purpose and gives formal results which support this picture. In the strong coupling limit it is convenient to replace the gauge field A u by B u = gA u. The generating functional for Green's functions can be written in a general covariant gauge

w(s,n,n)=

ft l [d~] [dr] [act] [dC] e x p ( i fd4x[.E'eff(x)+ J ( x ) B U ( x ) + ~(x)~(x) + "~(x)rl(x)), (8)

with

-

~

"/~eff - -- 4g 2

[Fv(B)'FUV(B) - ~(OUB(x)) 2 ] + ~(x) (iO + B - m ) qJ(x) + OuCt (x).(O u + B X)C(x),

~ = a a(x) - O~(x) + [a(x),a~(x)l. C~ (x), C(x) represent the scalar ghosts. Ju' 4, r~ are external sources for Bu, ~ and 5, respectively. Greens functions are generated by functional differentiation with respect to the appropriate sources. To obtain a strong coupling expansion integrate over the terms of the exponent linear in B

W(J'r~'~)=exP.4gz f

tiSg/

t i ~x I

,3 This is the behavior if we use the lowest order perturbation results for 1~and % though the use of these forms for the strong coupling limit is purely illustrative. It is well known [ 1, 2] that to lowest order/3 = -bg 3, b > 0. A simple calculation gives ~ = ag2 where a = (3g2/87r2)C2(F) > 0, and 6'2(/7) is the quadratic Casimir operator for the fermion representation. Then lim g(~.)

h -b/a.

h--.o

,4 The ratios of bare quark masses are preserved under Weinberg's renormalization procedure and renormalization group scaling. Such ideas as approximate SU(2) X SU(2) chiral flavor symmetry retain their significance. 289

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PHYSICS LETTERS

X I-I 6(4)(~(x)Tu'catO(x)+ifabcOuCtb(X)Cc(X)+J~(x))exp(ifd4x a,x,u 7a is an SU(3) matrix for the fermions, fabc is a structure constant.

22 November 1976

(~(i~b-m)$*3

u

C?.3uC+qJ/*~rl)) (9) "

This may not be a useful form for explicit computation and the formal manipulations are open to suspicion, but it does display certain interesting features of the theory: (1) For g = o% we have a theory of free massive quarks and massless scalar ghost particles subject to the constraint that the local quark plus ghost currents vanish everywhere +5 . (2) Expanding to any finite order in l/g2, we still obtain an expression to be evaluated at ~TuTa~k + ifabc3uC?bCc =0. The zeroth order solution in the strong coupling expansion is realized in a space of states where color is confined ~ . Expansion to a finite order in 1]g 2 does move us out of this colorless space, though colored states may actually exist for arbitrarily large but finite g. Therefore, the lack of evidence for confinement in weak coupling expansions and the demonstration of confinement in the strong coupling limit do not necessarily imply that there is a critical value of the coupling constant at which confinement sets in %7. Strong coupling expansions with large quark mass parameters may prove useful for investigating the colorsinglet spectrum via eq. (7). It may be possible also to study the assumptions made here about the/3 and "~ functions using methods from Euclidean field theory. Were both programs to be implemented successfully it still seems, unfortunately, that we would not have a useful way of estimating the mass scale of the suggested colored states relative to that of the observed color singlet hadrons. ¢5 This is the reverse of an argument by Amati and Testa [ 11 ] that confinement of quarks implies a gauge theory with an infinite coupling constant. ,6 This conclusion agrees with results from strong coupling approximations in lattice gauge theories though the mathematical applicability of these formal manipulations to those calculations is not clear (e.g. [ 12, 13 ] ). ,7 If such a phase transition takes place, there is a divergence of MCg(h),m, ~u)at g(h) = gc(m, la) and of M(g, hm, #) at g = gc(hm, tt). The phase transition surface is readily seen to be given by m = ~f(g) where df/dg = -f(g)/'~(g).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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D. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; Phys. Rev. D10 (1973) 3633. H. Politzer, Phys. Rev. Lett. 30 (1973) 1346. S. Coleman and D. Gross, Phys. Rev. Lett. 31 (1973) 851. S. Weinberg, Phys. Rev. Lett. 31 (1973) 494. Y-P. Yao, Phys. Rev. Lett. 36 (1976) 653. T. Applequist, J. Carrazone, H. Kluberg-Stern and M. Roth, Phys. Rev. Lett. 36 (1976) 768, 1161(E). T. Kinoshita, Journ. Math. Phys. 3 (1962) 650. T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) B1549. S. Weinberg, Phys. Rev. D8 (1973) 3497. S. Weinberg, Phys. Rev. 118 (1960) 838. D. Amati and M. Testa, Phys. Lett. 48B (1974) 227. K. Wilson, Phys. Rev. D10 (1974) 2445. J. Kogut and L. Susskind, Phys. Rev. D l l (1975) 395.