Unitary relations and the number of flavors

Volume 108B, number 6

PHYSICS LETTERS

4 February 1982

UNITARITY RELATIONS AND THE NUMBER OF FLAVORS Reinhard OEHME The Enrico Fermi lnstitute and the Department o f Physics, The University o f Chicago, Chicago, IL, USA and

Wolfhart ZIMMERMANN 1 The Institute for Advanced Study, Princeton, NJ, USA Received 26 May 1981 Revised manuscript received 16 November 1981

Within the framework of QCD, unitarity relations for a gauge invariant, color-singlet correlation function are considered. On the basis of the behavior of the exact gluon propagator in the weak coupling limit, an inconsistency is obtained unless the number of fermions exceeds a lower bound.

In previous work we have encountered interesting features of the exact transverse gluon propagator in quantum chromodynamics [1,2]. Unless the number of fermion fields is sufficiently large, the absorptive part of the propagator is dominated by negative norm states in the weak coupling region. These ghost states are non-perturbative and enforced by superconvergence properties of the propagator function. We also find that the projected propagator, where negative norm states are omitted, diverges for g2 ~ +0 like (g2)-'roo/t~o for "YO0//30 > 0. The divergence is present independent of the method of projection. [Here we denote the anomalous dimension of the gluon field by 7(g) = 70(a)g 2 + .... 3'0(a) = 3%0 + o~'Y01,and we write the Callan-Symanzik function as/3(g) =/30g4 + .... /30 < 0.] These results have been derived in the Landau gauge and in related covariant gauges. They are obtained using renormalization group methods for massless QCD (no intrinsic quark masses), together with analytic properties obtained from Lorentz covariance and the basic spectral conditions. Besides the usual assumptions of non-abelian gauge theories, we require that certain exact Green's functions (Wightman functions) have a weak coupling limit which is given by the formal perturbation expansion of the lagrangian formulation, at least up to the first few terms. It is the purpose of this note to discuss the implications for the conservation of probability of the features of the gluon propagator described above * 1. In a gauge invariant setting, we show that the divergence of the projected propagator for g2 ~ +0 and 700//30 >" 0 is not compatible with unitarity relations. On the other hand there is no problem for 700//30 < O. We find it convenient to consider the correlation function --iGB(X - x ' ) = ( B ( x ) B ( x ' ) )

(1)

of the renormalized, composite operator B(x), which denotes the gauge invariant form corresponding to the expression [4] F•v(x)Fa.v(x) ,

with 1 On leave from the Max-Planck-Institut f'tir Physik und Astrophysik, Munich, West Germany. ,1 A report about our results was presented at the Workshop on Gauge theories (Crete, Greece, 1980) [3]. 416

(2)

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e('TabcAbAc F L =A L +o_

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4 February1982

a = auAa _ avA~ ,

(3)

In the weak coupling limit, the operator B is given by the Wick ordered product

Bo(X) = :AaOuv (x)A~UV(x) : .

(4)

If we denote the Fourier transform o f GB b y

fib = -27riO(ko)OB(k2) ,

(5)

it follows that f o r g 2 -+ +0

PB(k 2 ) -+ PBo (k 2) > 0

(6)

ifk 2 >0. We consider now the unitarity relations for the correlation function G B. Only color-singlet intermediate states contribute, and in view o f the gauge invariance o f B, we expect that no negative norm states make a contribution. We write

(B(x )B(x')) = (B(x ) pB(x')) ,

(7)

where p denotes the projection into a subspace of positive semi-definite metric ,2 For the purpose o f this note, we are only interested in the g2 ~ +0 limit of the exact unitarity relations. Therefore we restrict ourselves here to a few remarks about the construction of these relations for the case o f confined quarks and gluons * 3. In the standard form o f the unitarity equations, the momenta o f all particles are on the mass shells. With confinement, there should be no contributions with vertex functions involving on-shell gluon and quark momenta. To this end, cut self-energy parts are summed up leading to off-shell projected correlation functions which are to be inserted for cut gluon and quark lines. Double counting is avoided by introducing the appropriate irreducibilities for the vertex functions. Our construction o f the unitarity relations, which is independent of perturbation theory, becomes well defined if we introduce a priori an infrared cut-off in the form o f an auxiliary gluon mass *4. This auxiliary mass parameter, which may be intrinsic or generated by a Higgs mechanism, is then taken to zero with appropriate caution [7]. In the weak coupling limit, the leading term o f our unitarity relation for the correlation function (1) is then given by

(B(x)B(x')) = -

x c;

dx1 dx2 dXl' dx2'

if

2.2x2 (x2 - x )P "'x'"2x2(x',

~ , x 1, x 21 uavlplhl(x1 - X l )'

+ ....

(8)

where here and in the following equations color indices are being suppressed. Because o f its simpler transformation properties with respect to the gauge group, we use the projected correlation function o f the field strength F u r : •

+

t

--

t

-]GuvoX(x - x ) - (Fuv(x) pFox(x ) ) .

(9)

*2 For a possible definition of a suitable subspace see ref. [5]. *3 We will discuss the problems of unitarity relations for QCD with confinement, including the role of the composite states, in greater detail elsewhere. Aspects of the exact unitarity relation, which vanish exponentially for g2 .~ +0, are not important for the results obtained in this paper• *4 The use of the gluon mass as an infrared cut-off has been discussed in the literature, mainly within the framework of perturbation theory. Curci and Ferrari [6] introduce the gluon mass by adding an explicit mass term to the lagranglan which preserves the Becchi-Rouet-Stora invarlance. Unitatity is then violated already in the one-loop approximation, but restored in the massless limit for the perturbation expansion. This method should be distinguished from the approach taken by van Dam and Veltman [8 ], where unitarity holds in the one-loop approximation with finite gluon mass, but is violated in the limit. Perturbative unitarity is completely preserved ff the gluon mass is introduced by a Higgs mechanism. This method has been used by De Rfijula et al. [9]. 417

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The vertex functions are correspondingly defined by

(TB(x)F#I

V1

(Xl )Fu2v2 (x2))

' ' ' ' F u2v2(x2)). = f d x ' 1 dx'2 I'Ul'v',~2'v'2(X,Xl,X2)(TFulvi(Xl)Fuavl(Xl))(TFu~v~(x2)

(10)

They involve the exact, time ordered two-point Green's functions of Fur. Taking the Fourier transform of eq. (8), we obtain

OB(k 2) = O(B2)(k2) + .... with

1

1

2 (2~r)4 f dq ~uavlu2v2 (k, q, k - q)

27ro(B2)(k2) -

X G+ ~tlYlPl~.l (q)G;2v202~.2(k - q)FO,XlO2X2(k, q, k - q)

(11)

The structure functions of G+ are given by

Guvox(k ) =-27ri(kukogvx

kukxgvo + kvkxg~o - kvkpgux)O(ko)[P+(k 2) + k-2a+(k2)] (12)

+ 27ri(guogvx - g~,og#x ) 0 (ko)o+(k2), where p+(k 2) t> 0 and o+(k 2)/> O. We now choose renormalization group invariant limits K2(g) = IK2I u(gi, g),

i = 1,2,

2

gi u(gi,g)=exp(f

dx/3-1(x))~exp(1/~og2)

for g2 -->+ 0 ,

(13)

in order to obtain a lower bound for p(B2):

K~

p(B2)(k2) ~>

f

K~ dv2 p+(u2)p+(o2)I(k2,u2,o2),

du2 f

K~

(14)

K~

with

I(k2' u2' 02) -

1

1

2 (2704 f dq

~,ulvlu2v2 (k,

q,

k

-

q)

X Ptqvap]M(q, u2)eu2v2o2x2(k -- q, v2)-PPlx'P2h2(k, q, k - q), Pu~'o x(q,/2) = _21ri(kuk p gt, x -T-...)0 (qo)6 (q2 _ 12).

(15) (16)

We introduce variables g' and g" by U2 = IKZlu(g',g),

02 = IKZlu(g",g),

(17)

where the functions u have been defined in eqs. (13), and K2 < 0 is the renormalization point. With

X+(k2 /g 2 ,g) =-k2p+(k 2 ' g, t~2), and using the renormalization group equations for X+, we obtain 418

(18)

Volume 108B, number 6

p(B2)(k2) ) e x p ( 2 / 0 a

PHYSICS LETTERS

dx 3'F(X)/3-X(x))f_g~ ..dg'2_

, g2

~ dg"2

+

4 February 1982 '

+

U "

g~ f l - ~ X (--u(g,go),go)X ( - (g ,go),go)

X I(k2,lr21u(g',g), Ir 2 [u(g ,g);g, t¢2).

(19)

Here we have applied the renormalization group transformation to the function X÷ in analogy to the methods used in re fs. [ 1,2] for the gluon propagator with A uv in the Landau gauge; the anomalous dimension of Fuv is denoted by 3'F. We obtain the relation

gg

X (-u(g ,g),g) = exp +

(f

dx 3'F(X)fl-l(x)

\g2

)x (-u(g, go), go), +

'

(20)

which should be considered in the sense of distributions; it is also valid for the unprojected Fur-propagation function. Of course, in contrast to Auv , the operator Fur is composite, and we must arrange the renormalizatlon and the related mixing so that we have Fur -+ X ~ F Fur, where Zff 1is given by the exponential factor in eq. (20). In view of our assumptions concerning the limit g2 _+ + 0 of Wightman functions, we have 3"F(g) -+ 3'00g2 in this limit. Let us now take the limit g2 _, +0 in eq. (19). With u(g',g) -+ O, u(g",g) -+ 0, we see from eqs. (10) and (15) that

I ~ I ( k 2, 0, 0; 0, t¢2) = 27rPBo(k2),

(21)

with PBO > 0 for k 2 > 0 according to eq. (6) *s. Evaluating the factor, we obtain finally

PB(k 2) ~ fg2)-2-roo/eo

(22)

in the weak coupling limit. For 3'00/fl0 < 0 (10 ~ N F ~< 16 in the standard model, or 5 ~ 0 (N F ~< 9 in the standard model, or N F ~< 4 with supersymmetry) is in contradiction to the finite weak coupling limit PB -+PBO. We see that the assumptions we have outlined in the introduction, and in refs. [ 1,2], are consistent with our unitarity relations provided there is a sufficient number of fermion fields so that 3'00/180 < 0. But for 3'00//30 > 0, we obtain an inconsistency in a color-singlet channel of a gauge invariant correlation function. As we have pointed out, the divergence is associated with the transition to the weak coupling limit of the exact unitarity relations. A more detailed discussion of our results will be published elsewhere. This work has been supported in part by grants from the National Science Foundation and the Federal Republic of Germany. *s For vanishing mass variables, the function/becomes infrared divergent in the second order of perturbation theory [10]. By methods independent of perturbation theory, we have verified that nevertheless relation (21) holds in the limit g2 _. +0 for the g-dependent mass values considered here.

References [1] R. Oehme and W. Zimmermann, Phys. Rev. D21 (1980) 471; Phys. Lett. 79B (1978) 93. [2] R. Oehme and W. Zunmermann, Phys. Rev. D21 (1980) 1661. [3] R. Oehme and W. Zimmermann, Gauge field propagator and the number of fermions, talk presented at the Workshop on Gauge theories and thek phenomenological implications (Crete, Greece,1980). [4] H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D12 (1975) 467; W.S. Dean and J.A. Dixon, Phys. Rev. D18 (1978) 1113.

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[5] G. Curci and R. Ferrari, Nuovo Cimento 35A (1976) 273; Phys. Lett. 63B (1976) 91; T. Kugo and I. Ojima, Phys. Lett. 73B (1978) 459; Prog. Theor. Phys. 60 (1978) 1869; 61 (1979) 294; Suppl. 66 (1979). [6] G. Curci and R. Ferrari, Nuovo Cimento 32A (1976) 151; I. Ojima, to be published. [7] M. Veltman, Physica 29 (1963) 186; A.M. Polyakov, Zh. Eksp. Teor. Flz. 59 (1970) 542. [Sov. Phys. JETP 32 (1971) 296]. [8] H. van Dam and M. Veltman, Nucl. Phys. B22 (1970) 397. [9] A. De Rfijula, R.C. Giles and R.L. Jaffe, Phys. Rev. D17 (1978) 285. [10] A.H. Mueller, Phys. Rep. 73 (1981) 237; V. Sudakov, Soy. Phys. JETP 3 (1965) 65. [ 11 ] R. Oehme and W. Zimmermann, to be published.

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