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Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986 MASS-SHELL PROPERTIES OF THE DYNAMICAL QUARK MASS L.J. REINDERS and K. STAM Physikalisches Institut der Universiti~t Bonn, Nussallee 12, D-5300 Bonn 1, Fed. Rep. Germany Received 22 July 1986 We discuss the running dynamical quark mass in the framework of the operator product expansion. It is shown that for Ip 21 > m 2 the quark-condensate part of the quark serf-energy has no contributions of order m 2 or higher, and is frozen to its mass-shell value for smaller Ip 21. It was first noted by Politzer [1] that using the operator product expansion (OPE) for the product $(x) 5(0) in the quark propagator a quark may be as- signed a running dynamical mass which is proportion- al to the chiral symmetry breaking condensate (~). This connects the (lagrangian) current quark mass with the constituent quark mass commonly used in potential models. The OPE for the quark propagator reads S(p) = f d4x eipx(oI T(ff (x)ff(O)lO) = C I + C~-qj (01~ffl0) + ... (1) where to first order in a s CT¢, the coefficient of the chiral condensate, is given by the self-energy diagram of fig. 1, and C I contains the ordinary perturbative contributions to the propagator. The dots in (1) stand for terms with operators of higher dimension. Writing S(p) = ilia- m - ~(p)] i i 1 - + Z (p) V_--7- ~ +... (2) pr-m /¢-m p-k k k Fig. 1. The lowest order quark-condensate contribution to the quark propagator. The crosses denote that the quark fields condense. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) the calculation of fig. 1 gives for Z(p) [2] ~_,(p) = (4rCOts/3N)p-2 [(3 + a) - am],C/p2](~qJ), (3) where a is the gauge parameter in a general covariant gauge and m is conventionally identified with the cur- rent quark mass. The application of this result has been hampered by the fact that (3) is gauge dependent. However, it has been argued by Elias and Scadron [3] that al- though the quark propagator is in general gauge de- pendent, it is still possible to define a dynamical quark mass which is gauge invariant on the mass shell. This requires the introduction of a modified equation of motion for soft quark fields which are affected by vacuum fluctuations (like the condensing quarks in fig. 1): $NP(x) = --im dyn ~NP(x) • (4) Consequently, in this picture the parameter m in (3) has to be reinterpreted as the dynamical quark mass mdyn, and atfl = mdyn the gauge-dependent terms in (3) cancel, leading to a gauge-invariant on-shell quark mass: mdyn(P 2 = m2yn) = -- ~ nots( t~)/m2yn . (5) Substituting the common PCAC value (~) ~ -(250 MeV) 3, this yields mdyn ~ 300 MeV in good agree- ment with the constituent quark mass value for u and d quarks. In the following we shall always mean mdyn by rn unless specifically mentioned. In this note we make two observations. Eq. (3) has 125

Mass-shell properties of the dynamical quark mass

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Page 1: Mass-shell properties of the dynamical quark mass

Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

M A S S - S H E L L P R O P E R T I E S OF T H E DYNAMICAL QUARK MASS

L.J. REINDERS and K. STAM

Physikalisches Institut der Universiti~t Bonn, Nussallee 12, D-5300 Bonn 1, Fed. Rep. Germany

Received 22 July 1986

We discuss the running dynamical quark mass in the f ramework of the operator product expansion. It is shown that for Ip 21 > m 2 the quark-condensate part o f the quark serf-energy has no contr ibut ions o f order m 2 or higher, and is frozen to its mass-shell value for smaller Ip 21.

It was first noted by Politzer [1] that using the operator product expansion (OPE) for the product $(x) 5(0) in the quark propagator a quark may be as- signed a running dynamical mass which is proportion- al to the chiral symmetry breaking condensate ( ~ ) . This connects the (lagrangian) current quark mass with the constituent quark mass commonly used in potential models. The OPE for the quark propagator reads

S ( p ) = f d4x eipx(oI T(ff (x)ff(O)lO)

= C I + C~-qj (01~ffl0) + ... (1)

where to first order in a s CT¢, the coefficient of the chiral condensate, is given by the self-energy diagram of fig. 1, and C I contains the ordinary perturbative contributions to the propagator. The dots in (1) stand for terms with operators of higher dimension. Writing

S ( p ) = i l i a - m - ~(p)]

i i 1 - + Z (p) V_--7- ~ +... (2) p r - m / ¢ - m

p -k

k k

Fig. 1. The lowest order quark-condensate contr ibut ion to the quark propagator. The crosses denote that the quark fields condense.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the calculation of fig. 1 gives for Z(p) [2]

~_,(p) = (4rCOts/3N)p-2 [(3 + a) - am],C/p2](~qJ), (3)

where a is the gauge parameter in a general covariant gauge and m is conventionally identified with the cur- rent quark mass.

The application of this result has been hampered by the fact that (3) is gauge dependent. However, it has been argued by Elias and Scadron [3] that al- though the quark propagator is in general gauge de- pendent, it is still possible to define a dynamical quark mass which is gauge invariant on the mass shell. This requires the introduction of a modified equation of motion for soft quark fields which are affected by vacuum fluctuations (like the condensing quarks in fig. 1):

$NP(x) = --im dyn ~ N P ( x ) • (4)

Consequently, in this picture the parameter m in (3) has to be reinterpreted as the dynamical quark mass mdyn, and atf l = mdy n the gauge-dependent terms in (3) cancel, leading to a gauge-invariant on-shell quark mass:

mdyn(P 2 = m2yn) = -- ~ nots( t~)/m2yn . (5)

Substituting the common PCAC value ( ~ ) ~ - (250 MeV) 3, this yields mdy n ~ 300 MeV in good agree- ment with the constituent quark mass value for u and d quarks. In the following we shall always mean mdy n by rn unless specifically mentioned.

In this note we make two observations. Eq. (3) has

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Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

only been proven to be correct to first order in m by expanding in the mass of the condensing quark. Below we shall show that in fact eq. (3) is exact for Ip2i > m 2, i.e. all contributions of higher order in m vanish! The second and most remarkable observation is that at the mass shellp 2 = m 2 the self-energy (and consequently the quark mass) is frozen to its (dyna- mical) mass-shell value.

Nowadays, the Wilson coefficients of quark-con- densate operators in the OPE of a two-point function like (1) are usually calculated by expanding ~(x) into a Taylor series around x = 0 (see for instance ref. [4]) and employing the f'lxed-point gauge [5] which makes it possible to replace all ordinary derivatives by covariant derivatives. For our purposes this meth- od is not very convenient, since it does not reveal the analyticity properties of X(p) as a function o f p 2 ,1 Here we shall use the so-called plane wave method which was the common method in the early days of QCD sum rule calculations [7] (for a review see ref. [8]). This exploits the fact that the OPE is an opera- tor identity and consequently one can single out a particular operator by sandwiching the T-ordered pro- duct of fields in (1) by appropriately chosen states. In our case we sandwich by single quark states with momentum k to calculate the coefficient C~-~0. To retrieve the contribution to the quark propagator the momentum k has to be soft, subject to the condition k 2 = m2yn in view of the equation of motion (4), i.e., we have to average over the momentum k taking ac- count of (4). To first order in g2 C~qj is again given by fig. 1 with momentum k assigned to the condens- ing quark lines,

f d4x e ipx (k lT(~(x) ~(O))lk)

= g2SF(P) 7, ½ X a O(k) ~(k) ~ XbTvSF(P)Dabv( p - k). (6)

In a general covariant gauge

ab _isab Duv(p ) = [g~V -- (1 -- a)p~pv/pE]p -2 , (7)

and the SF(P) in (6) are the free quark propagators corresponding to the external legs in fig. 1. Using

~ka(k) f b ( k ) = (1 /2N) [(K+m)/2mli/6 ab , (8)

,1 However, as has been shown recently [6], this method can be used for proving that order by order the contributions of higher powers in m vanish.

we obtain

2 C ~ ( p , k ) = _i ~ _ ~ C 2 ( R ) S F ( p ) ( - 2 ~ 4 m 1

(p - k ) 2

. 4 ~ + m , . 1 - ( 1 -a )O¢- m)---m---IlJ- m) ~ ] SF(P).

(9) It can easily be verified that to first order in m with the conditions stated above this reproduces the result (3) for Y.(p). To extend the evaluation to all orders in rn the expression (9) has to be averaged over k to all orders. This can be done directly by using the relations

+ 1_1_. O(m2 p2) ( ~ k ~ 2 ) ~ O(p2-m2) m2 - , (lOa)

( ~ _ k ) 4 ) ~ 1 O(P 2 - m 2 ) (p2 _ m2)p2

+ m2(m lz_ p2 i O(m 2 _ p2) , (10b)

( p - k)2/ _ pa P am20(p2 m2)+---.~O(m2-p2),

2P 4 2mZ (10c)

_~ m2p ~ O(p2 - m 2) p4(p2 _ m 2)

+ .... P~ 0 (m 2 - p 2 ) . (10d) m2(m 2 _ p2)

The relations (10) can be derived by expanding in k, averaging order by order and resununing. We fred for the contribution to l~(p) (i.e. dropping the propaga- tors SF(P) in (9)),

4rr°ts 1 E ( p ) - 3N p2 [(3+a)-amkr/P2]<~)O(p2-m2)

4rr% 1 + 3---N- m---2 [(3 + a) - aft/m] (~)O(m 2 - p2) . (11)

This expression is exact to all orders in m. Although strictly speaking the OPE is valid only in the deep euclidean region, eq. (11) is the result of analytic con- tinuation to lower values o f p 2 leading to the freezing of the quark self-energy at its threshold value for [p2 I < m 2 (fig. 2).

Our result also clarifies the discussion in the litera- ture regarding the use of the so-called "regular" or

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Page 3: Mass-shell properties of the dynamical quark mass

Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

zt m

m2 p2"- Fig. 2. The behaviour of ~ as a function p2 according to eq. (11) in the Landau gauge,a = 0.

"irregular" solutions of the chiral symmetry breaking part of the quark self-energy, as obtained from the quark propagator Dyson-Schwinger equation [9]. Asymptotically, the regular solution, i.e., the lip 2 be- haviour is consistent with the OPE. Our eq. (i 1) shows that by continuing the OPE result to p2 < m 2 the ir- regular solution takes over. Thus the OPE justifies the variational Ans~itze to the solution of the quark Dyson-Schwinger equation, as employed in ref. [10], for the total momentum range.

In a completely different context it also implies that the constituent quark mass used in potential models (defined ~ la Politzer, ref. [1]) does not blow up forp 2 ~ 0, but for Ip2l < m 2 is fixed at its on- shell value (5). This may explain why the quark mass can be taken constant in these models.

Finally, let us make a few remarks concerning the gauge dependence ofmdy n. Its introduction via the equation of motion (4) for nonperturbative quark fields was motivated by the wish to cancel the gauge dependence in (3) or (11) at ~¢ = m. But eq. ( 11 ) includes only contributions to first order in a s . Sec- ond order corrections have been calculated in ref. [2], but only to zeroth order in m. The dependence on the gauge parameter a becomes more complicated and it would be very intriguing indeed if on the dynamical mass shell all gauge dependent terms again cancel.

Also, we have cut off the OPE series at the quark condensate operator. Obviously, it is important that the behaviour of N(p) be analyzed with the inclusion of higher-dimensional operators. Results concerning the mixed operator tkouvGUU ~ with a similar cancel- lation of the gauge dependence have recently been reported [11].

Regarding the coefficient of the four-dimensional O a a gluon condensate perator Gu~,Gu~, conflicting results

have appeared in the literature [12,13]. The correct

k I Lk p

Fig. 3. Gluon-eondensate contribution to the quark propaga- tor. The crosses denote that the gluons condense. All quark propagators are free propagators.

contribution to the self-energy in the fixed-point gauge reads [4,8] (fig. 3)

rt 2 m (p2 _ m/0 ((%/70 G 2 > (12) Z<G2>(P) = -3- (p2 - m2)3

In this expression m should be interpreted as the cur- rent quark mass since there are no condensing quarks. Consequently, for massless quarks there is no gluon condensate contribution to the quark self-energy (in the fixed-point gauge). This fact has been known for a long time [14]. However, it is gauge dependent. The same quantity calculated in a covariant gauge is

4rr2 p21C- 3m2],C + 4rn3 ~(G2>(P)= 9 ( p 2 - m 2 ) 3 <(as/rr)G2>, (13)

(again m is the current quark mass). Here we have sandwiched by one-gluon states with momentum k and polarizations a,/3 to select the operator GauvGau~, in the OPE and averaged over the gluon four-momentum and polarizations. The two procedures do not yield the same result. This phenomenon has also been re- cognized at intermediate stages in the computation of current correlation functions, but in that case the final gauge invariant result is the same for both meth- ods [14]. Here it may signal the breakdown of the de- fruition of the gauge-invariant dynamical quark mass when gluon-condensate operators in the OPE are taken into account.

One of us, L.J.R., thanks the Max Planck-Institut in Munich for its hospitality.

References

[1] H.D. Politzer, Nucl. Phys. B 117 (1976) 397. [2] P. Pasqual and R. Tarrach, Z. Phys. C 12 (1982) 127. [3] V. Elias and M. Scadron, Phys. Rev. D 30 (1984) 647. [4] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and

V.I. Zakharov, Fortschr. Phys. 32 (1984) 585.

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Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

[5] V.A. Fock, Soy. Phys. 12 (1937) 404; J. Schwinger, Particles and fields (Addison-Wesley, New York 1970) ; C. Cronstrom, Phys. Lett. B 90 (1980) 267; M.S. Dubovikov and A.V. Smilga, Nucl. Phys. B 185 (1981) 109; M.A. Shifman, Nucl. Phys. B 173 (1980) 13.

[6] V. Elias, T. Steele, M.D. Scadron and R. Tarrach, University of Western Ontario preprint (1986).

[7] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nuel. Phys. B 147 (1979) 385.

[8] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Rep. 127 (1985) 1.

[9] K. Stam, Phys. Lett. B 152 (1985) 238; V.A. Miransky, Phys. Lett. B 165 (1985) 401.

[10] R. Casalbuoni, S. de Curtis, D. Dominici and R. Gatto, Phys. Lett. B 140 (1984) 357; P. Castorina and S.Y. Pi, Phys. Rev. D 31 (1985) 411.

[11] V. Elias, M. Scadron and R. Tarrach, Phys. Lett. B 173 (1986) 184.

[12] T.I. Larsson, Phys. Rev. D 32 (1985) 956. [13] V. Elias, M. Scadron and R. Tarrach, Phys. Lett. B 162

(1985) 176. [14] A.V. Smilga, Soy. J. Nucl. Phys. 35 (1982) 271.

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