IC/68/54 0^ 19. JUL1B6b) 5^1 2 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS DISPERSION CALCULATION OF THE JT- JT SCATTERING LENGTHS N. F. NASRALLAH 1968 MIRAMARE - TRIESTE
1C/68/54 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS DISPERSION CALCULATION OF THE 7T - TT SCATTERING LENGTHS * N.F. Nasrallah I RAM ARE - TRIESTE July 1968 * To be submitted to "Physics Letters",
ABSTRACT Collinear dispersion relations ^J are used to express the physical TC-TC scattering amplitude at threshold in terms of the soft pion results obtained from the current algebra. The scattering lengths are calculated and agree with those obtained by SCEWITCER 2 )'^). m " "» -
DISPERSION CALCULATION OF TTTE ir-ir SCATTERING LENGTHS Current alfeora was firs-it applied to calculating 4) the TT-TC scattering lengths t>y WEISTBERG ' and later by several other authors. They differ in the method of extrapolation from the soft pion limit to the macs-shell. In this note, we use oncesubtracted collinear dispersion relations, introduced, "by FUBINI and FUIiLAN to obtain in a direct way the physical scattering amplitude T(K + 1C,> Tr + TT ) at threshold. We do not assume any definite functional dependance for the off-mass-shell scattering amplitude "but take its proper threshold "behaviour into account. to explain the method. We start from the following expression: We proceed i m it iqy, 2 a b where A ' denote the axial vector currents, m the pion mass and <0[ 9 A i U 1 >= f m 2 with f - - * G tfn e work in the forward direction and in the rest frame of the target pion and set (2) The generalized V, r ard-takaliashi identity 5 ' gives wnere -2-
I rex TT c i d (y)d ( 0 ) [ O - < T O < 0 T D a w ' t> (y)d (0) 0> ] (4.a) V = - IT T/ (4. b) 8 ab S cd S== - 7TTE.1 dy e iqy 6(y 0 ) (mv) 2 b rci ir d / T Q >< 0 [Djy), A^{0)] I 0> ] (4c) and where D Current algebra determines V and eq,(3) "becomes P T U(x) - ^ - x(l-x 2 ) 2 (6 6,, - 5,fi, ) + 6, 6 S. ^ 2 ac bd ad be ab cd ir (5) As cj( x ) has no singularity at x «0, we obtain lira U(x) - -6 - «, S _. n ab cd * 0 (6a) and 3
lim -? - (6 6-6 6 \ ->0 dx 2 ^ ab cd ao bd J ' ± It is now convenient to perform the isospin decomposition of U in the following form: U(x). A(x) 6. 6, + B(x) (6,6. +6 6..) v ' K ' ab cd v ' v ad be ac bd ; + z C(x) (6,6. - 6 6, ) (7) w v ad tc ac bd y v ' ' where A,B and C are even under crossing, i.e., A(x) =. A(-x), B(x) = B(-x), C(x) - C(-x). (8) Also from eq.(3) U(x - 1) - T th (9) where T^, denotes the transition matrix element for the physical tjq,, n n Pi. O process K + K => 7i + R at threshold, Boae symmetry imposes the condition A{x. 1) = B(x - 1) + C(x - 1). (10) Comparing eqs.(7) and (8) we get A(0) r. S (lla) B(0) => 0 (lib) amplitudes C(o) - ~ ii (lie) 7l Using the technique of BJOHKETT ' ' the asymptotic limit of the A,B,C can be seen to be proportional to the equal time -4-
commutators [ I, D, ] and \~D, D, ~, We shall L a Me.t. L a b Je.t. assume these limits to be negligible, as suggested by a single field theoretical model where D ~' TC and by the success of the analogous procedure applied to 7T-1T scattering. We are now able to write for A,B,C the following dispersion relations: 2 C A(x - 1) - -S +4; \ "Z Abs A(x) (12a) 1) dz. Abs (12b) C(x - 1) ', 2 + K xlx 1 Abs C(x) (12c) Starting with eq. (4a) and using the reduction technique, the contributions of the disconnected graphs can be separated. and Abs U A Abs U T -1M decomposed into three parts n a wb (13a) m m b (13b) Abs U III JL a *-»b (13c) -5-
where a bar denotes the connected part of a matrix element and where If, = (m + d ) D ' «(J a y D We recognise that Abs XL. contains the usual singularities due to the exchange of physical particles in the s,s channels, Abs IT contains the mass singularities associated with the vertioes S, 0 j D m _}> and Abs U TTT corresponds finally to the Z graphs of intermediate states identically. \Jt, Tr j Ji X The contribution of the one ~K state to Abs U_ T vanishes The contribution of higher states can be neglected Treiman relation. The least massive state contributing to U TTT is the \A = 2TT^> state, SO that this term contributes only for x > 3. The large value of the denominator appearing in eq.(l2) allows us to neglect the contribution of Abs U TTT* T ^e a r ^hus left with the connected parts of the amplitudes. The integrals in eq,(l2) start at x = 1 which corresponds to physical threshold. We note that an advantage of working in the rest frame is that only s-waves contribute to the dispersion integrals. It is now convenient to take linear combinations of eqa.(l2b) and (12c) in the following form: Soo *(* 2 -D * = t (14a) 2m 2. 2 \ dx Ats (14b) where 2(B(x) + C(x)) x) - 5B(x)-C(x) and U ' 2 (x-1) - T t 2 the EUp«rscripts denoting the isospin. in accordance with the Epirit of the derivation of the Goldberger- -6-
Using now the relations and m(0,2), /-t- \ T» 32 7Tma_ - tlpa; lim Ate U (Of^( Z ) - lira T (0 ' 2 )<x) 32ttn 2 a 2 V x 2-1 (where a n o denote the scattering lengths), we isolate the contributions of the divergent parts of the integrands to the integrals in eqs. (14) (these contributions are of course finite) and write ^ ^ 1 Abs [W (2) (x) - W (2) f 2 2 "" J l x(x 2 -l) 2 (16a) 2 noo 32;rma = ^ _ + 3 2 7 r m + ^ ^ / dx Abs[W (0) (x)-w (0) ( 0 r ^ Ji x( X -IF ff (16b) where U ( We now note that the integrands appearing in eq,(l6) vanish at threshold and are strongly damped. It seems therefore safe to neglect them. If we do, we can solve eqs.(l6) for the scattering lengths and obtain and raa 0..05 (a) or ma Q =.95 ft) (17) ma 2 = -.09 (a) or ma 2 = 1.10 ft) (18) Solution (l8"b) is ruled out experimentally and solution (l7"b) seems too large to "be consistent with our approximations. We keep solutions (17a) and (l8a). Ke note that these solutions agree with -7-
those obtained "by SCHWINGEB. ' oh the basis of chiral dynamics. those of WEIEBERG Our solutions do not coincide with, who assumed a definite functional dependence for the off-eaas-sholl scattering amplitude and who used the 9) additional PCAC self-consistency condition of AJDLER J J. We finally note that this method determines the so-called cr-term, i.e., from eqs. (12a) and (l8a) 7 After completing the present calculation I was informed \>y Professor G. Purlan that be had done a similar one in collahoration with G. CARBOITE, S. DOITIM and 5, SCIUTO 1 K These authors also investigate the interesting possibility of the existence of a low energy s-wave 7T-7r resonance. ACKNOWLEDGMENTS I wish to thank Professors Abdus Salam and P. Budini and the IAEA for hospitality at the International Centre for Theoretical Physics, Trieste. My thanks are also due to Professors A.P. Balachandran and G. Furlan- for commenting on the manuscript.
REFERENCES AND FOOTNOTES l) S. FUBINI and G, FURLAN, to "be published in Ann. Phys. 2) J. SCHiaKGER, Phys. Letters.24J3, 473 (1967). 3) A.P. BALAGHAKDRM, M.G. GUNDZIK and F. NICODHHI, Nuol. Phys, (in press). 4) S. raihberg, Phys. Rev. Letters 1, 616 (1966). 5) K. RAMAN, E.C.G. SUDARSHAN, Phys. Rev. l$± f 1499 (1967) 6) Here we make the usual assumption that the or-term is an isoecalar (see for example Ref.4). 7) J.D. BJORKEF, Phys. Rev. 14J3, 1467 (1966). 8) Unless of course there exists a strong low-energy s-wave 7T-X resonance. 9) S.L. ABLER, Phys. Rev. 137t B1O22 139, B1638 (1965). 10) G. CARBOUE, E. BOHIHI, G. FURLAN and S. SCIOTO, Torino preprint (1968). -9-
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