On Factorization of Coupled Channel Scattering S Matrices

Transcription

1 Commun. Theor. Phys. Beijing, China pp c International Academic Publishers Vol. 48, No. 5, November 5, 007 On Factoriation of Coupled Channel Scattering S Matrices FANG Ke-Jie Department of Physics, Peking University, Beijing 0087, China Received October 30, 006 Abstract We investigate the problem on how to factorie a coupled channel scattering S matrix into a product of simple S matrices. Simple S matrix solutions are found, respecting unitarity, analyticity and being real analytic. The phase shift and its physical meaning produced by these simple S matrices are discussed. PACS numbers:.39.fe,.5.pg,.55.bq, 4.40.Cs Key words: ππ scattering, S matrix Introduction In discussing quantum scattering problems, [] it is meaningful to set up general parametriation forms of the scattering S matrices, based on correct use of analyticity properties established from rigorous quantum field theory results. A famous example is the Dalit Tuan parametriation form, which factories a general S matrix into a product of simpler S matrices, denoted as S i, where the superscript i means the i-th resonance appearing in the scattering process. In the recent few years there appeared a new parametriation form named PKU parametriation form for the elastic scattering amplitude, [ 4] which has been proven to be useful in revealing the resonance structure of low energy ππ scattering amplitudes. The PKU parametriation form is the following: For a partial wave elastic scattering in a given channel, physical S matrix can in general be factoried as S phy = S i S cut, i where S i are the simplest S matrices characteriing the isolated singularities of S phy. [] The expressions for S i can be found in Refs. [] and [3]. The S cut contains only the cuts that can be parameteried in the following simple form: S cut = e iρfs, fs = s Im L fs π L s s s ds + s Im R fs π R s s s ds, 3 where L is the dynamical cut [,3] and R denotes unitarity cuts at higher energies other than the elastic cut. It is worth while noticing that in Eq. the dispersion integral is free of subtraction constants, both for equal [3] and un-equal [4] mass scatterings. The parametriation form, Eqs. 3, has the advantage that S i does not contribute to the discontinuity of f, [] which means the following relation. The discontinuity of f on each cut obeys the following formula: { } discfs = disc i ρs log[sphy s]. 4 Having briefly reviewed the PKU representation form, it is natural then to ask whether it is possible to generalie the above results, at least partially, to multichannel scattering S matrices. Although equation works also in the inelasticity region, the complete expression for a coupled channel S matrix parametriation form is still missing. This paper is therefore devoted to the study on such a problem. It is easy to realie that such a problem is difficult to attack. From Eq. we realie that it is impossible to write a similar expression in the coupled channel case. Because the products of matrices depends on the order of the products, and there is no apparent rule to arrange the order of the products. This paper is not able to solve this problem either. But some limited success is achieved through the study of this paper, on coupled channel scatterings. Preliminaries The coupled channel scattering problem was discussed in a previous unpublished paper, [5] but that paper only focused on the cut structure of equal mass scattering amplitudes and did not touch the unitariation problem. The starting point of our strategy in this paper is to firstly find the most simple S matrices. Here the word most simple means a unitary real analytic matrix, containing the least number of poles on its Riemann sheets. For a coupled channel S matrix: S = + i ρ T I, S = + i ρ T I, S = i ρ ρ T I. 5 In the physical region above the second threshold, the coupled channel unitary S matrix can be written as η e iδπ, i η e iδπ+δ K S = i. 6 η e iδπ+δk, η e iδ K Notice that in above the phase shift δ π is defined in the coupled channel physical region. Below the second threshold, δ π is defined as S = e iδπ. The two δ π s are not the same analytic function. According to Ref. [5] we can obtain the expressions of S matrix on sheets, I, and, S = is S S is det S S S, S I = S det S S det S S det S, S dets The project supported by Hui-Chun Chin and Tsung-Dao Lee Chinese Undergraduate Research Endowment CURE and partly by National Natural Science Foundation of China under Grant Nos and 04503

2 90 FANG Ke-Jie Vol. 48 dets is S S = S is S S. 7 From the coupled channel S matrix we can construct functions which either contain no right hand cut, Ψ SI + S + S I + S, Ψ SI + S + S I + S, 8 or only simple cut dependence Ψ ρ, Ψ ρ, Ψ S S i S det S + det S, S Ψ S S i S det S + det S. 9 S Furthermore, a coupled channel unitarity leads to constraints among functions Ψ and Ψ, Ψ + Ψ = Ψ + Ψ. 0 3 Simple Solutions of Coupled Channel S Matrix When searching for simple S matrix solutions, as inspired by the single-channel discussions, one attempts to firstly neglect all cut integrals appeared in functions Ψ and Ψ and these functions become rational real analytic functions of complex variable. As will be seen below, this step greatly simplifies the solutions. 3. A Solution with a Pair of Poles on Second Sheet In this simple situation, the functions Ψ and Ψ can be written as Ψ = ρ iρ S iρ S + α, Ψ = S + S + β, det S Ψ = ρ i ρ S + det S i ρ S + α, 3 dets Ψ = S + where α and β are subtraction constants. Let a = iρ S, b = iρ S, a = dets S + β, 4 dets iρ S, b = det S iρ S, then a Ψ = ρ + b + α, 5 Ψ = iρ a iρ b + β, 6 a Ψ = ρ + b + α, 7 Ψ = iρ a + iρ b + β. 8 Since expressions Ψ = Ψ, Ψ = Ψ, 9 Ψ = Ψ, Ψ = Ψ, 0 hold on the whole complex plane, so b = a, b = a, α = α, α = α, β = β, and β = β. Let A = Ψ + Ψ, then η = A A 4. Because η is real on the real axis above the second threshold as well as below the first thresholds, and is pure phase between the first and second thresholds, in the absence of the left cut [ infty, 4M K m π], function A should take the following values in different regions: A 4 when 4 and 4t, 3 and 0 A 4 when 4 4t. 4 Notice that in the above conditions we require that when 4, A 4 A = 4 when Z = 4. This condition can be rigorously proved, following Ref. [5]. Using Eqs. 5, 6, and, one can get f A =, 5 where f is a quadratic function of. From the analysis above, A can be parameteried as A = 4 + c 4 4t, 6 where c is a positive real number. Moreover, in order that S = A A 4 Ψ + iψ 7 A has no more cut other than the known left hand and right hand cut, it is convenient and reasonable to set 4 + c d A =, 8 where d is a real number, and 4 d 4t. Using Eqs. 6 and 8, one can get the solutions of c and d for the given and, Let M π =, M K /M π = t, then the first and the second thresholds are located at = 4 and = 4t respectively.

3 No. 5 On Factoriation of Coupled Channel Scattering S Matrices 903 c = ± 64 t 8 t, 9 d = c + t 4 + c, 30 for = t 64t 4. However, c can only take the upper sign, since c > 0 and is imaginary. To get the solutions of a, α, and β, one can put Eqs. 5, 6, 6, and 8 together to get the following equation: [ a ] ρ + a + α + iρ a iρ a c 4 4t + β = 4 +, 3 where b = a has been used. Let = 0, 4, d, respectively, one can get the following equations: α a a = 0, 3 iρ a + i ρ a β = ±, 33 a + a d d + α = 0, 34 iρ a + iρ a d d + β = From these equations, one can easily get the solutions for a, α, and β, a = α = β = [ + d ] Re d 4 + d ρ, 36 ±Im d d Re d 4 + d ρ, 37 ±Re ρ d + d Re d 4 + d ρ. 38 We define A = 4 + c d/ along the real axis, so A < 0 and η = when = 4, contributing ±π, depending on the definition of A 4, [ a ] ρ + a iρ a + α + + iρ a but the conditions when = 0, 4t, d respectively are to the phase of S matrix. In this situation, we define A 4 = 4 Ai between the two thresholds. Physically, the total phase shift equals ero at elastic threshold, so the phase shift contributed by the exponent, e iδπ = Ψ + iψ / A, should be ±π/. Thus, the numerator should be when = 4. All the result should take the upper sign accordingly. After having obtained a, α, β one can easily get the explicit expression of S : S = d c c+4 ρ ρ β ξ + iα ρ, 39 where ξ = + d + [i ρ a + i ρ a ]/β. It is worth while pointing out that the factor on the denominator does not contribute poles to S. On the contrary, they are the roots of S, as has been designed, for the reason that iα ρ + β ξ and the numerator both have roots =,, which is not obvious due to the reformation of the expression. But one can find these from the beginning of the programm with Eqs. 5, 6, 7, and 8. When it comes to obtain the expression of S, the situation is a little different from that of S, because ρ d is a pure imaginary. Similarly, + β = 4 + c 4 4t, 40 α a a = 0, 4 iρ a + iρ a β = ± 4 + 6t c w, 4 iρ a 4t + iρ a 4t + β = ±, 43 a iρ d + a iρ a d d + α = ± + iρ a d d + β. 44 In order that = d is not a pole of S, the right side of Eq. 44 should be negative. We define δ K = 0 when = 4t, so the right side of Eq. 43 should be positive with A being positive on the second threshold. After all, the solution is a = w + g + ih w + f ρ d fh f hρ d + gh + g, h 45 α = a + a, 46 where β = iρ a 4t iρ a 4t, 47 f = +, 48 d g = ρ d 4t, 49 h = i ρ + 4t. 50

4 904 FANG Ke-Jie Vol. 48 Since w has two possible values, S has two distinct solutions. But due to the definition A = 4 + c d/, and since w is the square root of A4, w should be negative coherently. Thus we choose the lower symbol solution. Fig. Phase shift and inelasticity parameter generated by a narrow second sheet pole below the second threshold. Fig. Phase shifts and inelasticity parameter generated by a narrow second sheet pole above the second threshold. To get the explicit expression of S, a tedious calculation is performed to reduce the factor d. Thus S = [ c ] c + 4 d ρ ρ iψ + Ψ, 5 dets = 4 + c d iα ρ + β ξiψ + Ψ. 5 In Fig. we plot phase shift and inelasticity parameter generated by a narrow second sheet pole below the second threshold. In Fig. we plot phse shift and inelasticity parameter generated by a narrow second sheet pole above the second threshold. 3. A Solution of a Pair of Poles on Third Sheet In this situation, one writes S I Ψ = ρ iρ I det S I I S I iρ Idet S I I + α, 53 S I Ψ = det S I I + S I det S I I + β, 54 S I Ψ = ρ i ρ I dets I I S I i ρ IdetS I I + α, 55 S I Ψ = det S I I + S I dets I I + β. 56 For simplicity, the above equations are rewritten as a Ψ = ρ + b I + α, 57 I Ψ = iρ I a iρ Ib I + β, 58 I a Ψ = ρ + b I + α, 59 I Ψ = iρ I a iρ Ib I + β. 60 I The parametriation of Ψ and Ψ is the same as in the Numerical analysis also reveals that such a choice gives a tiny phase shift δ K. On the contrary the solution corresponding to the upper sign has a siable δ K and is un-physical.

5 No. 5 On Factoriation of Coupled Channel Scattering S Matrices 905 situation with a pair of poles on the second sheet, so the solution of S does not change. What is different is that we define A 4 = 4 A i between the two thresholds. Following the similar approach discussed before, one can get the solution for S in this situation, where a = w + g + ih + w + f ρ d fh f hρ d gh g h, 6 α = a + a I, 6 I β = + iρ I a 4t + iρ Ia I 4t, 63 I f = +, 64 I d I g = ρ I d I 4t, 65 I h = i ρ I + I 4t. 66 I phase shift generated by a narrow third sheet pole above the second threshold. The inelasticity parameter is not shown. According to our solution it is the same as the one generated by a second sheet with the same value of pole location. The third sheet pole above second threshold contribution to the phase shift looks odd and seems in contradiction to experiments. But we argue that in physical situation the third sheet pole usually is accompanied by a nearby second sheet pole. In our solutions if we add the second sheet pole and the third sheet pole s contribution to δ π together, the total phase shift looks then reasonable. See Fig. 5. Fig. 4 Phase shift generated by a narrow third sheet pole above the second threshold. Fig. 3 Phase shifts generated by a narrow third sheet pole below the second threshold. In Fig. 3 we plot phase shift generated by a narrow third sheet pole below the second threshold. The inelasticity parameter is not shown. According to our solution it is the same as the one generated by a second sheet with the same value of pole location. In Fig. 4 we plot Fig. 5 Phase shift generated by a pair of narrow second and third sheet poles with the same location above the second threshold. 3.3 A Solution of a Pair of Poles on the Fourth Sheet In this situation, one can write Ψ = ρ det S i ρ S + det S iρ S + α, 67 det S Ψ = S + det S S + β, 68 Ψ = ρ i ρ S i ρ S + α, 69 Ψ = S + S + β. 70

6 906 FANG Ke-Jie Vol. 48 For simplicity we again use the following notations: a Ψ = ρ + b + α, 7 Ψ = i ρ a + i ρ b + β, 7 a Ψ = ρ + b + α, 73 Ψ = i ρ a i ρ b + β. 74 Following the similar approach discussed before, one can get the solution for S in this situation, a = α = β = + d Re d 4 + d ρ, 75 Im d d Re d 4 + d ρ, 76 Re ρ d + d Re d 4 + d ρ. 77 Fig. 6 Phase shifts generated by a narrow 4th sheet pole above the second threshold. In this situation, we define A 4 = 4 A i between the two thresholds. The parameteriation of Ψ and Ψ is the same as in the situation with a pair of poles on the third sheet, so the solution of S does not change. In Fig. 6 we plot phase shift generated by a narrow fourth sheet pole above the second threshold. The inelasticity parameter is not shown since it is the same as the one generated by a second sheet with the same value of pole location. In Fig. 7 we plot phase shifts generated by a narrow fourth sheet pole below the second threshold. Again the inelasticity parameter is not shown due to the reason mentioned before. Fig. 7 Phase shifts generated by a narrow four-sheet pole below the second threshold. 4 Discussions We have exhibited in details the simplest unitary coupled channel S matrix depicting one resonance. Here simplest means that all dynamical cuts and further right-hand cuts are neglected. We get reasonable phase shift and inelasticity diagrams, which provides a crosscheck, demonstrating that we do indeed find the correct solutions. However, as already addressed before, with these simple S matrix solutions at hand, we are still not able to establish expressions for coupled channels, in analog to Eq.. For example the second sheet pole and fourth sheet pole do not obey simple factoriation properties. For example, the product of two S matrices parameteriing two different second sheet poles do not necessarily contain the two poles. Despite of such difficulty remains un-resolved, we think that the current investigation and the expressions we presented may still be helpful in understanding in general the coupled channel scattering problems. Moreover, there does exist factoriation property for third sheet poles. For a given S matrix, containing arbitrary number of poles on different sheets, we may write down S = Si I S R. 78 i Here superscript R means what remains and Si I denote simple S matrices parameteriing different third sheet poles. Take the determinant of the above equation we get det S = det Si I det S R. 79 i A third sheet pole is a ero point of the determinant. From above it is easy to realie that in dets R no ero remains. Therefore above the second threshold the sum δ π + δ K is additive, i.e., it can be represented by the summation of different third sheet pole contributions plus proper background. Therefore our discussions and results are at least

7 No. 5 On Factoriation of Coupled Channel Scattering S Matrices 907 useful, in principle, for setting up a proper dispersion representation for δ π +δ K. Further discussions along this line is needed. Finally we would like to point out that the discussions made for Fig. 5 is interesting, does our result suggest that in reality a narrow third sheet pole is always accompanied by a close second sheet pole? This also deserves further investigations. Acknowledgments The author is grateful to Prof. Han-Qing Zheng for helpful discussions and a careful reading to the manuscript. References [] J. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions, Wiley, New York 97. [] H.Q. Zheng, et al., Nucl. Phys. A [3] Z.Y. Zhou, et al., JHEP [4] Z.Y. Zhou and H.Q. Zheng, Nucl. Phys. A [5] Z.G. Xiao and H.Q. Zheng, hep-ph/00304, unpublished.