ON SPECTRAL FUNCTIONS SUM RULES

Transcription

1 IC/68/61 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON SPECTRAL FUNCTIONS SUM RULES C. G. BOLLINI AND J. J. GIAMBIAGI 1968 MIRAMARE - TRIESTE

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3 IC/68/01 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON SPECTRAL FUNCTIONS SUM RULES* C.G. Bollini ** and J. J. Giambiagi ** * MIRAMARE - TRIESTE July 1968 To be submitted for publication. ** On leave of absence from the Consejo Nacional de Investigaciones Cientificas y Te'cnicas and Fundacion Bariloche, Argentina. ^ On leave of absence from Fundacion Bariloche, Argentina.

4 ABSTRACT Sum rules for the spectral functions of vector and axial fields are obtained from hypotheses on the behaviour of vacuum values of the commutators on the cone. The consequences of vector dominance are examined. A discussion is given on the relation between our hypothesis and that of the equality of the unrenormalized masses and coupling constants. The sources of the fields are taken to be the weak vector and axial vector currents. -1-

5 ON SPECTRAL FUNCTIONS SUM RULES If one starts by assuming the algebra of currents, it is possible to deduce sum rules for the spectral functions. By a similar procedure it is possible to deduce the same sum rules starting from the algebra of fields if the currents are identified with p-times the fields. We shall deduce sum rules for the spectral functions of the fields using hypotheses about the behaviour of the vacuum expectation values of commutators on the light cone 3). We shall identify the currents with the sources of the vector fields and deduce the relation between the corresponding spectral functions. 1. RELATIONS BETWEEN FIELDS AND CURRENTS We shall consider a vector field <f> and an axial field A. The equations of motion are (2) We shall take J to be the vector current and K the transverse part of the axial current; in other words, both J^ and K^ are conserved currents. More explicitly (see Ref. 4), W*»*1l)+." (3) (4) where besides the usual nucleonic terms we have added those derived 4) from the Lagrangian -2-

6 Any conserved vector field satisfies the Lehmann representation whore At>A~ (V) Analogously, for the conserved vector current. we have and,due to the equations of motion (1), (2)> ** ' (8) We also note that q =J>x Q'f-J*X <f>. 01) This formula means that the algebra of charges is related to the m 2 algebra of fields with the factor. 2. SUM RULES In order to deduce the sum rules we shall make assumptions on the behaviour of vacuum expectation values of commutators on the cone and we shall first show that they are equivalent to assumptions on equal time commutators, -3-

7 Eq. (6) implies, for equal times, i, e. t Mj n sty and^of course, the behaviour of equal time commutators fixes the values of the integrals of the spectral functions. On the other hand, for x -» cone, (x) -» t (x_) 6 (x ) so that so that the behaviour of the commutator on the cone also fixes the values of the integrals Now, we shall make the explicit assumption that on the cone the vector ($ ) and axial vector (A ) fields satisfy r* p* (13) A O The factors m /g are suggested by a«(h) in order to obtain the results of the algebra of charges. (See,however,the discussion.) Replacing (13) in each member, -4-

8 r Using now 1 A I J I ^ I (17) where the caret refers to unrenormalized quantities and Z is the wave function renormalization constant. We deduce from (16) Taking this relation into account, (14) implies < 1L i (x), I «&]/»> t*' < / #, I (19) (This relation implies that the bare masses are equal. ) Of course, it is also possible to start first from eq. (19) and then take into account the relation Z^ = i 2 equality of the unrenormalized quantities (g eq. (14) follows immediately. given in Ref. 7. If one accepts the m-g = g, m. = my), 3. CONSEQUENCES OF VECTOR DOMINANCE If we now take (20) and neglect the terms p' in eq. (15) 8) we get (21) -5-

9 or i ft **»( ~ " (22) With this relation, eq. (18) gives 2 z. m. A _ A z v " 2 ' Now we want to find the relation between a and p under the hypothesis of vector dominance. The relevant formulae are eqs. (9) and (10), but we must be careful, as a strict use of formulae (20) would give us cr 0 which is not correct. We must keep in mind that the use of a 6-function is only an approximation to represent a resonance curve for p (p ot p ). In this particular case it is reasonable to make the following approximation: The mean value (/u - m ) defines a width through the formula so that ni) - 4m^F (23) Now we can replace p res by a 6-function (24} This equation is usually written as which leads to the identification _- (26) -6-

10 2 m 4 This result should be compared with G = rj- obtained when the 2, g current is identified with 4>. g T H If we take into account (26) into (21) we get St.? which differs from Weinberg's second sum rule (G. = G.). 4. APPLICATION TO SU(3) Eqs. (1) and (2) can be extended to octets of SU(3). We shall start by assuming a relation analogous to (14), namely 3? where i and j are SU(3) indices. As before, (28) implies or ft J ^ / (29) (30) Analogously to eq. (18),the'use of this relation in (28) allows us to write, for the unrenormalized fields, (implying the equality of the bare masses). Besides (29), eq. (28) implies also (31)

11 J * < 32 > We shall take i = 3, j = 8, with the hypothesis of vector dominance, J ^ (33) Replacing in (32) we obtain (34) > ~ from which *V -4^-^v ' <36) This formula, with A = 1,coincides (except for a positive factor coming from a different definition for the mixing angle) with eq, (7) of SAKURAI 9) which was deduced from Weinberg's sum rules. As pointed out by Sakurai, his eq. (7) is contradictory as m«, > m > nip showing that Weinberg's sum rules are inconsistent. Instead, our formula (36) deduced from the sum rule(32) depends on A. We shall find its value now.. (32) gives (for those spectral functions which are dominated by only one vector meson) -8-

12 l.l..;.! A.... Extending this equality 10 aii me members of the octet we have L- = constant, independent of i. So, we get Replacing in (36), we get (38) and as m (J) > m > m > m p, this formula does not exhibit any contradiction. Eq. (37) when the indices refer to p and K can be used to find the ratio giving (39) which is in good agreement with experiments. 5. DISCUSSION The well-known properties of the unrenormalized spectral functions for vector particles f dy p - \ t f dju^ p/m*' - \ ' m -9- f (

13 2 i 2 f 2 >"vj 2 lead immediately to the equality / d^ P//u = / dju />//u for particles which are assumed to have equal unrenormalized mass. This equality and the trivial one / dju p = / dn p' (= 1) are expressed compactly by the equation (40) Furthermore, when both the unrenormalized and renormalized currents are fixed {in scale) by the requirement that the space 7) integral of the fourth component gives the same "charge" } then the renormalized coupling constant is found to be g = ~^~2" i F r m zi this reason the expression of (41) in terms of renormalized fields, namely (41) when supplemented with the assumption of equal unrenormalized coupling constants, takes the form which is the hypothesis we started with (eqs. (14) and (28)) to deduce the sum rules. Of course, these sum rules coincide with those deduced from the algebra of currents, if one identifies these currents with times the. corresponding fields. However, g we have chosen to identify the weak currents with the actual, renormalized sources of the vector fields. <«, ACKNOWLEDGMENTS The authors are indebted to Professors Abdus Salam and P. Budini and the IAEA for the kind hospitality extended to them at the International Centre for Theoretical Physics, Trieste. Partial support from the Fundacion Saubera'n is also acknowledged

14 REFERENCES AND FOOTNOTES 1) S. WEINBERG,. Phys. Rev. Letters IS, 507(1967); S.L. GLASHOW, H. J. SCHNITZER and S. WEINBERG, Phys. Rev. Letters 19_, 139 (1967); T. DAS, V,S. MATHUR and S. OKUBO, Phys. Rev. Letters 18_, 761 (1967). 2) T.D. LEE, S. WEINBERG and B. ZUMINO, Phys. Rev. 18_, 1029 (1967). 3) S. OKUBO, University of Rochester Report UR , (1967), 4) J. SCHWINGER, Phys. Rev. 165_, 1714 (1968). 5) The other term coming from (x, m) could also give additional sum rules if one were able to make predictions on the behaviour in the vicinity of the light cone. 6) K. JOHNSON, Nucl. Phys. 2^, 435 (1961). 7) N..M. KROLL, T.D. LEE and B. ZUMINO, Phys. Rev. 157, 1376(1967). 8) The influence of p 1 is less important in (13) than in (14) due to 2 the extra factor M in the denominator. 9) J. J. SAKURAI, Phys. Rev. Letters 19_, 803 (1967).

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