Resonance signals in hadron scattering

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(1)RESONANCE SIGNALS IN HADRON SCATTERING. Martha Carolina Cáceres Rodrı́guez. UNIVERSIDAD DE LOS ANDES Science Faculty Physics Department Bogotá D.C. May 2009.

(2) RESONANCE SIGNALS IN HADRON SCATTERING. A thesis submitted for the degree of physicist in the Faculty of Science Universidad de Los Andes. Martha Carolina Cáceres Rodrı́guez. Advisor: Neelima G. Kelkar PhD.. UNIVERSIDAD DE LOS ANDES Science Faculty Physics Department Bogotá D.C. May 2009.

(3) Acknowledgments I would like express my sincere thank you to Dr. Neelima Kelkar for her unconditional assistance, for her directions, patience and all her dedication and support, and most important for the trust she deposited on me. I also would like to present my gratitude to many people, who helped me and lend a hand during the development of my thesis. A special thank you to my Mom, for all her support, her guidance, for being the role model she is to me. To my dad, for his support. To my friends; Mauricio, Lucia, Elizabeth, Ingrid, Marie, Mafe, Raul, Carlos M, they were always there when I need it something, for the long nights of research and study, for the pen, book they lend, for being always there..

(4) Objectives General Objective To study methods of resonance parameter extraction: The extraction of basic resonance parameters from reaction data is one of the important tasks in hadron physics. Ideally one performs complete measurement of independent observables such as differential cross sections and polarization in an experiment. Then one extracts partial wave amplitudes from the data and tries to extract the resonance parameters from these amplitudes. This stage relies on different definitions of what exactly corresponds to a resonance signal. Study and evaluate some methods. The most commonly used method searches for poles of the amplitudes. Other methods include cross sections humps, Argand diagrams, K-matrix poles and time delay methods.. Specific Objectives • The present study aims at discussing the pros and cons involved in the usage of each of these methods. • After having studied the relevant quantum collision theory, the formalism for each of the above mentioned methods will be studied. • A literature survey around the controversies in resonance extraction will be made and an opinion regarding the necessary and sufficient signals for confirming the existence of a resonance will be formed. iv.

(5) Contents. 1 A brief primer on quantum scattering. 1. 1.1. Partial Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.3. Scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 1.4. S-matrix and its connection to the scattering amplitude and phase shift. 8. 2 Standard resonance signals. 10. 2.1. Phase Shift Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.2. Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2.3. S-matrix pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 2.4. T-matrix and K-matrix poles . . . . . . . . . . . . . . . . . . . . . . 18. 2.5. Cross section humps . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. 2.6. Argand Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 3 Characterization of resonances and related controversies. 24. 3.1. Paper 1: Antibound ’States’ and Resonances . . . . . . . . . . . . . . 24. 3.2. Paper 2a, 2b and 2c: Correspondence between Poles and S-Matrix in resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1. Paper 2a: Correspondence between Unstable Particles and Poles in S-Matrix Theory . . . . . . . . . . . . . . . . . . . . 31 v.

(6) vi. 4. CONTENTS 3.2.2. Paper 2b: Correspondence between Unstable Particles and Poles in S-Matrix Theory: The Exponential Decay Law . . . . 38. 3.2.3. Paper 2c: No-Pole S-Matrix Fit to the ∆(1236) Resonance . . 39. 3.3. Paper 3: Model-independent resonance parameter extraction from the trace of K and T matrices . . . . . . . . . . . . . . . . . . . . . . 42. 3.4. Paper 4: Resonance parameters from K-matrix and T matrix poles . 48. Summary. Bibliography. 53 55.

(7) Chapter 1 A brief primer on quantum scattering In this chapter I shall explain the basic concepts which are relevant for characterizing resonance in hadron scattering. In scattering analysis there are many important definitions, such as the cross section and the scattering amplitude. To find the scattering amplitude is indispensable to obtain the cross section. The scattering amplitude can be found through the partial wave expansion method. This method will be explained ahead. Another vital concept is the phase shift, when it appears, and how to calculate it. And finally, it is necessary to make clear the scattering amplitude concept, and the calculus it implies. All these concepts will be clarified below. Let Ψ be the total wave function in a scattering process, where the incoming wave is a plane wave in the z direction eikz , thus. · ¸ eikr ikz Ψ = A e + f (θ, φ) r. (1.1). where f (θ, φ) is the scattering amplitude and eikr /r represents the spherically outgoing wave. The differential cross section can be determined using (1), given by [14] 1.

(8) 2. CHAPTER 1. A BRIEF PRIMER ON QUANTUM SCATTERING. dσ = |f (θ, φ)|2 = f (θ, φ)f ∗ (θ, φ) dΩ then, the cross section is the integral over the solid angle Z Z dσ σ= dΩ = f (θ, φ)f ∗ (θ, φ)dΩ dΩ. 1.1. (1.2). (1.3). Partial Wave Expansion. The partial wave expansion is a method (PWEM) which lets one to calculate the scattering amplitude. This method starts from the fact that the incoming plane wave is a compound of states of all different possible angular momentum. Then the PWEM assumes that each angular momentum component scatters separately given that the potential conserves this quantity. It also defines the potential localized; it means that there is a central potential. The PWEM consists of a connection between the radial solution of the Schrödinger equation and the asymptotic form of the stationary scattering wave function. Then, let us begin with the radial part of the Schrödinger equation: · ¸ ~2 l(l + 1) ~2 d2 u + V (r) + u = Eu − 2m dr2 2mr2. (1.4). To solve the Schrödinger equation let us write a wave function in terms of l and m Ψ(r, θ, φ) =. X. Cl,m Rl (r)Ylm (θ, φ). (1.5). l,m. where R(r) = u(r)/r. So, when there is no potential, V=0, hence ~2 d2 u ~2 l(l + 1) + u = Eu 2m dr2 2mr2. (1.6). · ¸ l(l + 1) d2 u 2 u=0 − 2 + k − dr r2. (1.7). − Now, let k 2 =. 2m E, ~2. then.

(9) 3. 1.1. PARTIAL WAVE EXPANSION. Solving the equation, one finds out that it can be expressed as the superposition of Bessel and Neumann functions, or either as Hankel functions u(r) = A0l rjl (kr) + Bl0 rnl (kr) (1). (2). u(r) = Al rhl (kr) + Bl rhl (kr). (1.8) (1.9). where jl (kr) and nl (kr) are the Bessel and the Neumann functions respectively, and (1) (2) hl (kr) and hl (kr) are Hankel functions. When r → ∞ all these functions behave asymptotically, sin(x − lπ/2) r→∞ x cos(x − lπ/2) nl (x) ≈ r→∞ x jl (x) ≈. (1.10) (1.11). However, if there are only outgoing waves, then Bl = 0. The asymptotic behavior of the Hankel function is given by (1) hl (kr) ≈ r→∞. eikr −i π (l+1) (−i)l+1 eikr e 2 ≈ r kr. (1.12). Now, the total scattered wave is given asymptotically as, Ψscattered =. X. Al Cl,m (−i)l+1. l,m. eikr eikr m Yl (θ, φ) = f (θ, φ) kr r. (1.13). where the coefficients Cl have a very important physical meaning. These coefficients depend on the phase shift, which will be explained later. The different phase shifts also depend on the angular momentum. Then, f (θ, φ) =. 1X Cl,m (−i)l+1 Ylm (θ, φ) k l,m. (1.14). As long as there is azimuthal symmetry because of the central potential definition, then m = 0 r 2l + 1 Pl (cos θ) Ylm=0 = (1.15) 4π Now, it is possible to rewrite f (θ, φ) in terms of angular momentum and k r 2l + 1 1X Pl (cos θ) Cl (−i)l+1 (1.16) f (θ, k) = k l 4π.

(10) 4. CHAPTER 1. A BRIEF PRIMER ON QUANTUM SCATTERING. Replacing f in equation (1.1) " Ψ = Al. 1X Cl (−i)l+1 eikz + k l. r. eikr 2l + 1 Pl (cos θ) 4π r. #. (1.17). It is easy to notice that eikz can be expand using Legendre polynomial such as eikz = eikr cos θ ∞ X = il (2l + 1)jl (kr)Pl (cos θ). (1.18) (1.19). l=0. So as f (θ), f (θ) =. X. (2l + 1)al (k)Pl (cos θ). (1.20). l. where al (k) is the lth wave amplitude, which will be obtained later, and jl is the spherical Bessel function, then, replacing jl (1.10) in (1.19), we get ∞ X. sin(kr − lπ/2) Pl (cos θ) kr l=0 µ i(kr−lπ/2) ¶ ∞ e 1 X l e−i(kr−lπ/2) i (2l + 1) = Pl (cos θ) − 2ik l=0 r r. ikz. e. =. il (2l + 1). (1.21) (1.22). using il = eilπ/2 , ikz. e. ¶ µ ikr ∞ e−(ikr−lπ) 1 X e Pl (cos θ) − = (2l + 1) 2ik l=0 r r. (1.23). This equation shows that the plane wave can be expressed as a sum of partial waves depending on the angular momentum, so, this is the partial wave expansion for a plane wave.. 1.2. Phase Shift. In scattering process there is a superposition of spherical waves which have a different phase as compared to incoming one, and then there is a phase shift related to each one, which depends on angular momentum. This phase shift is the change in the phase of the outgoing waves due to a potential. When the potential is turned on a.

(11) 5. 1.2. PHASE SHIFT. phase shift (δl ) appears. In case of central potentials (with m = 0 in Ylm (θφ)) the total wave function can be expressed as. Ψ(~r) =. X. r. 2l + 1 Pl (cos θ)Rl (r) 4π. (1.24). Since the phase shift appears as a result of the potential, then when r goes to infinity, the radial wave function behaves as,. Rl (kr) −−−→ Al r→∞. sin(kr − lπ/2 + δl ) kr. (1.25). Thus,. Ψ(~r) −−−→ r→∞. ∞ X l=0. Al. ei(kr−lπ/2+δl ) − e−i(kr−lπ/2+δl ) ) Pl (Cosθ) r. (1.26). where Al is a constant As the potential only affects the outgoing wave, then, it is necessary to use the first part of equation (1.26) to find Al . And, using the second part of the equation (1.23), which is also the outgoing wave, is possible to find that. −. Al −i(kr−lπ/2+δl ) (2l + 1) −i(kr−lπ) e e =− r 2ikr. (1.27). Then. Al =. (2l + 1) i(lπ/2+δl ) e 2ik. (1.28).

(12) 6. CHAPTER 1. A BRIEF PRIMER ON QUANTUM SCATTERING Replacing on Ψ (1.26) µ. ei(kr−lπ/2+δl ) − e−i(kr−lπ/2+δl ) ) Ψ(~r) = e 2ik r l µ ¶ X (2l + 1) eikr e−i(kr−lπ) = Pl (Cosθ) e2iδl − 2ik r r l X (2l + 1) µ eikr e−i(kr−lπ) ¶ = Pl (Cosθ) − 2ik r r l {z } | X (2l + 1). i(kr−lπ/2). ¶. Pl (Cosθ). (1.30). (1.31). eikz. +. X (2l + 1) l. = eikz +. X (2l + 1). |l. 2ik. (e2iδl − 1)Pl (Cosθ) {z. eikr r. eikr r. }. f (θ). = eikz + f (θ). 2ik. (e2iδl − 1)Pl (Cosθ). (1.29). ikr. e. r (1.32). (1.33). Then, comparing equations (1.20) and (1.32)the wave amplitude can be written as e2iδl − 1 2ik · ¸ eiδl eiδl − e−iδl = k 2i iδl e sin δl = k. al (k) =. (1.34) (1.35) (1.36). However, it is not complete yet, now one has to calculate the phase shift. It is possible solving the equations (1.4) and (1.8) and keeping in mind the asymptotically behavior of the wave as r goes to infinity. Since the phase shift appears as a result of the potential. It is necessary to consider the case inside the potential region, outside and at the boundary. Let us say that the boundary is at r = r0 , then the ul (r = r0 ) and the derivative dul (r)/dr|r=r0 must be continuous at r0 ; as well as the logarithmic derivatives at the same point. The solution for the exterior region may be expressed as Rl (k, r) = Al (k) [jl (kr) − tan δl nl (kr)]r=r0. (1.37).

(13) 1.3. SCATTERING AMPLITUDE Let be γ the logarithmic derivative, ¸ · 1 dRl γl (k) = Rl dr. 7. (1.38). Now, if γ is evaluated at r = r0 , then it can be written as γl (k) =. k[jl0 (kr0 )/ tan δl (k)n0l (kr0 )] jl (kr0 )/ tan δl (k)nl (kr0 ). (1.39). where jl0 (kr0 ) and n0l (kr0 ) are the derivatives with respect to x evaluated at r = r0 . Solving for δl we get, kj 0 (kr0 ) − γl jl (kr0 ) tan δl (k) = l0 0 (1.40) knl (kr ) − γl nl (kr0 ) Using this expression is possible to calculate the phase shift due to a potential for the exterior region, and for the inside one has to change the evaluating point, getting a very similar equation. Knowing this, is possible to make the next step, calculate the scattering amplitude and the cross section.. 1.3. Scattering amplitude. The scattering amplitude can be obtain from the result above, thus f (θ) =. 1X (2l + 1)eiδl sin δl Pl (Cosθ) k l. (1.41). It is important to see that the scattering amplitude depends on the partial wave amplitude, which depends on δl , the phase shift. According to equation (1.3), the total cross section can be obtain from the scattering amplitude, and replacing f (θ) we get ∞ 4π X σ= 2 (2l + 1) sin2 δl k l=0. (1.42). As the cross section depends on the angular momentum, is possible to write it as a sum of partial cross sections.

(14) 8. CHAPTER 1. A BRIEF PRIMER ON QUANTUM SCATTERING. σ=. ∞ X. σl. (1.43). l=0. where the partial cross section is σl =. 1.4. 4π (2l + 1) sin2 δl k2. (1.44). S-matrix and its connection to the scattering amplitude and phase shift. When the potential is switched on, the plane wave is phase shifted and as r → ∞ then, ¶ ¶ µ µ lπ lπ (1.45) → sin kr − + δl (k) sin kr − 2 2 This is what we would find on integrating the Schrödinger equation out from nonsingular behavior at the origin. Thus, µ ¶ µ ¶ i e−i(kr−lπ/2) ei(kr−lπ/2) i e−i(kr−lπ/2) Sl (k)ei(kr−lπ/2) → (1.46) − − 2k r r 2k r r This expression introduces the scattering matrix Sl (k) = e2iδl (k). (1.47). which must lie on the unit circle |S| = 1 to conserve probability the outgoing current must equal the ingoing current. If there is no scattering, that is, zero phase shift, the scattering matrix is unity. Thus, Sl (k)ei(kr−lπ/2) ei(kr−lπ/2) → r r. (1.48). The complete wave function in the far region (including the incoming plane wave) is therefore: Ã ! X eikr (S (k) − 1) l Pl (cos θ) (1.49) Ψ(r, θ, φ) = eikr cos θ + (2l + 1) 2ik r l.

(15) 9. S-MATRIX. The il factor canceled the e−ilπ/2 . The −1 in (Sl (k) − 1) is there because zero scattering means S = 1. Alternatively, it could be regarded as subtracting off the outgoing waves already present in the plane wave, as discussed above. There is no φdependence since with the potential being spherically-symmetric the whole problem is azimuthally-symmetric about the direction of the incoming wave. It’s easy to prove the optical theorem for a spherically-symmetric potential: just take the imaginary part of each side of the equation f (θ, k) =. 1X (2l + 1)eiδl (k) sin δl (k)Pl (cos θ) k l. (1.50). at θ = 0, using Pl (1) = 1, Imf (θ = 0, k) =. 1X (2l + 1) sin2 δl (k) k l. (1.51). from which the optical theorem Imf (0, k) = kσ/4π follows immediately. It’s also worth noting what the unitarity of the lth partial wave scattering matrix Sl† Sl = 1 implies for the partial wave amplitude al (k) = k1 eiδl (k) sin δl (k) . Since Sl (k) = e2iδl (k) it follows that Sl (k) = 1 + 2ikal (k). (1.52).

(16) Chapter 2 Standard resonance signals This chapter gives the reader the descriptions of the different methods used in literature for characterizing resonances.. 2.1. Phase Shift Jump. The phase shift jump is a method that lets one know if there is a resonance in a scattering process. This jump is linked to the phase shift because it has been found that for the energy values that cross section has a rapid increase, the phase shift has a sudden jump by π. These energy values are the energy (or central mass) of the resonance. Then, it is possible to state that when there is a resonance process, the phase shift will experience an abrupt jump. Additionally, this method provides the expressions to obtain the resonance energy and the width. For elastic resonance scattering without background there is the Breit-Wigner . Here ER corresponds to formula. Consider a resonance occurring at E = ER − iΓ 2 the central mass and Γ is the width of the resonance. As the central phase shift jumps through π/2 at E = ER , there appears a peak in the distribution dδl /dE centered around ER (see figure (2.1)). The definition for a quasistationary state is 10.

(17) 11. 2.1. PHASE SHIFT JUMP given then by these three conditions[1]: ¯ d2 δl (E) ¯¯ =0 dE 2 ¯E=ER ¯ d3 δl (E) ¯¯ <0 dE 3 ¯E=ER ¯ ¯ ¯ ¯ ¯ dδl (E) ¯ 1 ¯¯ dδl (E) ¯¯ ¯ ¯ ¿¯ ER2 ¯ dE ¯ER dE ¯ER. The first step is to define the function µ ¶−1 dδl (E) f (E) = dE. (2.1) (2.2) (2.3). (2.4). the derivative of f at E = ER must be zero, then the second derivative of δl respect to E is zero, as is shown µ ¶−1 d dδl (E) 0 (2.5) f (E) = dE dE µ ¶−2 2 dδl (E) d δl (E) = − (2.6) dE dE 2 µ ¶−2 2 dδl (E) d δl (E) 0 =0 (2.7) f (E)E=ER = − dE dE 2 And the second derivative of f is ¶2 µ ¶µ ¶−3 µ 3 ¶2 µ 2 dδl (E) dδl (E) d δl (E) d δl (E) 00 f (E) = 2 − dE 2 dE dE 3 dE For E = ER the first terms vanish, so: ¶2 µ 3 ¶µ d δl (E) dδl (E) 00 f (E) = − dE 3 dE Then, according to equations (2.2) and (2.3), v s u (ER ) u −2 E22 dδldE 2f t R = ¿1 d3 δl (ER ) ER2 f 00 3. (2.8). (2.9). (2.10). dE. Using Taylor series, f (E) can be written as f (E) = f +. f 000 f (n) f 00 (E − ER )2 + (E − ER )3 + · · · + (E − ER )n 2 3! n!. (2.11).

(18) 12. CHAPTER 2. STANDARD RESONANCE SIGNALS. n+1 , then the terms of Taylor series converge if ∀n > N such as f (n+1) < f (n) (E−E R) third order and bigger are consider negligibly. Now from equation (2.4), one knows that dE (2.12) d(δl (E)) = f (E). Then, δl (R) is given by, δl (E) =. Z. f+. dE − ER )2. f 00 (E 2. (2.13). Doing the integral, Ã ! E − ER 2 −1 p + α10 tan δl (E) = 00 00 ff 2f /f Ãp ! r 2f /f 00 2 −1 tan = − + α1 f f 00 E − ER r. (2.14) (2.15). p Let us say that α1 − α10 = 2f /f 00 (π/2), and let us define two new parameters p p r = 2f /f 00 and Γ = 2f /f 00 . Calculating the parameters we get, s (dδl (E)/dE)3 (2.16) r = −2 3 d δl (E)/dE 3 s (dδl (E)/dE) (2.17) Γ = −2 3 (d δl (E)/dE 3 ) Notice that r is dimensionless and Γ has energy dimension. From (2.10) one gets that Γ ¿ 2ER . Now, replacing (2.17) and (2.16) in (2.15) we obtain that δl = r tan−1. Γ/2 + α1 ER − E. (2.18). Γ/2 ER − E. (2.19). As long as δl (E) = δlR (E) + δlbg , then δlR = r tan−1 Taking r = 1, δlR can be expressed as tan−1. Γ/2 i = [ln(E − ER + iΓ/2) − ln(E − ER − iΓ/2)] ER − E 2. (2.20).

(19) 13. 2.2. TIME DELAY. There it has an infinite order branch point at E = ER + iΓ/2 and E = ER − iΓ/2. E is always in the analyticity domain, which means that E can moves from the real axis around the branch point and return to the initial point on the same axis, then by π. Thus, the partial S-matrix, ln(E − ER − iΓ/2) raises by 2πi, and tan−1 EΓ/2 R −E which has analytic domain, will change as Sl (E) → Sl (E)e2πir. (2.21). For positive integer values of r. Hence, for r = 1 ER − E + iΓ/2 2iδbg e l ER − E − iΓ/2 µ ¶ bg −iΓ = 1+ e2iδl ER − E + iΓ/2 = 1 + 2ikal (E). Sl (E) =. = e. 2iδlbg. (1 + 2ikaR l (E)). (2.22) (2.23) (2.24) (2.25). Then the resonance partial wave amplitude is obtained, aR = l. 1/2Γ 1 k ER − E − 12 iΓ. (2.26) (2.27). Using the expression for the cross section from the first chapter, then σlR (E) =. 4π (Γ/2)2 (2l + 1) k2 (ER − E)2 + (Γ/2)2. (2.28). Then, plotting the resonance phase shift an the resonance cross section, as in figure (2.1), it easy to notice that the resonance energy is in the top of the cross section distribution, and the width Γ corresponds to the distance between the inflection points of the cross section, and also these two parameters are related to the phase shift jump, where the ER is in the middle point and the width match the jump by π.[1]. 2.2. Time Delay. We haven’t discussed yet about the time dependant equations. There is a method that implies the time on resonant scattering process analysis. The time delay method.

(20) 14. CHAPTER 2. STANDARD RESONANCE SIGNALS. Figure 2.1: Breit-Wigner cross section and its relation to the resonance phase shift [1] is simple and very clear to find out if in a scattering process has been resonance or not. The principle of this method is kind of obvious. If there is resonance, then the scattering process will take longer than when there is no resonance. Then, this method consist on find and expression that can be plotted, and according to the plot of the time delay, as a function of energy, it allows one to know where is a resonance and for what values of E. Thus, it is possible to obtain the resonance energy. Let us begin with the asymptotic form of the wave packet 1 Ψ(r, t) ' √ [ei(kr+ωt) cos(r∆k + t∆ω) r i(kr+ωt+2δ). −e. (2.29). cos(r∆k − t∆ω + 2∆δ)]. Now, to find a peak, one knows that the derivative has to be zero, then, for the first part ei(kr+ωt) cos(r∆k+t∆ω), the condition is sin(r∆k+t∆ω) = 0, thus r∆k+t∆ω = 0, so, rp = −t. dω dk. (2.30).

(21) 15. 2.2. TIME DELAY. Where dω/dk is the group velocity. Similarly, for the second term sin(r∆k − t∆ω + 2∆δ) = 0, then rp = −t. dδ dω −2 dk dk. (2.31). Comparing the two rp and subtracting them, one gets µ. dω ∆t = 2 dk dδ = 2~ dE. ¶−1. dδ dk. (2.32) (2.33). It is important to notice that this last equation shows a connection between time delay and phase shift. The derivative of the phase shift must be zero for local minimums and maximums, and reaches the maximum at inflection points. According to this, it is easy to notice that the time delay should get a peak for the middle point of the phase shift jump, where the resonance energy produce an inflection point. The analysis using a wave packet started in the early fifties ([D.Bohm], [Eisenbud] and [Wigner])[11], where they obtained an expression for the time delay ∆t in binary collisions. In the case of elastic scattering, they derived ∆t in terms of the energy derivative of the scattering phase shift as follows: ∆t = 2~. dδ dE. (2.34). The wave packet analysis of time delay was extended by Eisenbud to inelastic collisions [Eisenbud]. He defined the delay time matrix ∆t, such that an element ∆tij of this matrix, corresponded to the time interval between the outgoing wave in channel j and the ingoing wave in channel i . This time delay, ∆tij , is related to the S-matrix as follows: · ¸ −1 dSij ∆tij = Re −i~(Sij ) dE. (2.35). we see that for purely elastic scattering (j = i) and using S = e2iδ , we get back dS from the above equation [11]. ∆tij = 2~ dE.

(22) 16. 2.3. CHAPTER 2. STANDARD RESONANCE SIGNALS. S-matrix pole. For scattering processes is necessary to have a positive energy because the mass of the resonance has to be real and positive, and as a consequence, the imaginary momentum has to be negative. To show this, let us start with energy expression E=. P2 2m. (2.36). The energy can be also written as a sum of real and complex part, E = ER + iEI. (2.37). P 2 = (PR + iPI )2 = PR2 − PI2 +i 2PR PI | {z } | {z } real part. Then,. (2.38) (2.39). Complex part. ER = PR2 − PI2 EI = 2iPR PI. (2.40) (2.41). then, as long as we need a real positive energy, because it is the mass of the resonance, PI2 < PR2. (2.42). Since the resonance pole occurs at ER − iΓ/2, PI is negative; then we are on the unphysical plane and then is where resonance takes place. The minus sign in ER − iΓ/2 is essential to give rise to an exponential decay of the resonance of the type e−Γt . Let us start writing ul in terms of the S matrix and the Hankel functions, as i ul (kr) −−−→ (h− (kr) − Sl h+ l l (kr)) r→∞ 2. (2.43). Now, one should use the Jost function to obtain the regular solution for the radial Schrödinger equation, i ∗ + φl (k, r) −−−→ (Fl (k)h− l (kr) − Fl (k)hl (kr)) r→∞ 2. (2.44).

(23) 17. 2.3. S-MATRIX POLE Then, Sl is defined by Sl =. F∗l (k) Fl (−k) = Fl (k) Fl (k). (2.45). where Fl is the definition of a Jost function [15]. Thus, φl (k, r) = Fl (k)ul (kr). (2.46). Let us say that there is a point (p̄) that makes the Jost function vanished, which also means that Sl has a pole. The resonance pole, p̄ = pR − ipI. (2.47). occurs on the unphysical sheet of the Riemann surface as shown in figure(2.2). The physical sheet is define as ImP > 0 and unphysical sheet as ImP < 0. Where pR and pI are the real and the imaginary components of p, and then we have to re-defined pI > 0.. Figure 2.2: Complex planes of p and E Let us expand Fl by taylor (n). Fl (p) = F0l (p − p̄) +. F00l F (p − p̄)2 + · · · + l (p − p̄)n 2 n!. (2.48). As Fl (p̄) has a simple zero, then Fl (p) ≈. µ. dFl dp. ¶. p̄. (p − p̄). (2.49). which give as the function for a real interval centered on pR , that works as long as pR is close to the axis. To find the phase shift in the interval, we can say that p = p0 and is real, as can be seen in the figure(2.3). [15].

(24) 18. CHAPTER 2. STANDARD RESONANCE SIGNALS. Figure 2.3: resonant part of the phase shift [15] Then, tan δ (R) = δ (R) = tan−1. 2.4. pI pR − p0. pI = −Arg(p − p̄) pR − p0. (2.50) (2.51). T-matrix and K-matrix poles. The K-matrix formalism was introduced by S. N. Gupta around 1950 in a paper published in 1951 [19]. Then, the formalism was included in his book Quantum electrodynamics [20]. In 1979, Cutkosky used T and K matrices for pion-nucleon partial-wave analysis, [21]. After that, in 1993 S. N. Gupta and collaborators used it to analyze the W, Z and the Higgs-boson scattering at SSC energies, [22]. And more recently there are some controversial papers that will be discussed in the next chapter in sections 3.3 and 3.4, there will be explained the formalism of T and K-matrix poles.. 2.5. Cross section humps. This method consist on building and checking the plot of the cross section and find a peak, wherever the peak reaches its maximum value, the energy is equal to the resonance energy. And the width is defined as the thickness of cross section plot, at the middle of the height. Remembering from chapter 1 that Ψ = eikz + ikr d2 ul (kr) when r goes to infinity the Schrödinger equations is given by − + fh (θ) e r , and dr2 i k2 −. l(l+1) r2. ul (kr) = 0, then, the behavior of the asymptotic solution is ul (kr) −−−→ r→∞.

(25) 19. 2.5. CROSS SECTION HUMPS. sin(kr − lπ/2 + δl (k)). This can be compared to the Bessel function jl (x) ≈. r→∞. sin(x − lπ/2), where x = kr. From there we got f (θ, k), and then we found out that σ=. ∞ X. σl. (2.52). l=0. where the partial cross section is σl =. 4π (2l + 1) sin2 δl k2. (2.53). Now, ploting σl , one gets the bell shape, where the parameters can be found.. Figure 2.4: It shows the total cross section for π + p and π − p elastic scattering up to a beam energy of the π + or π − of 2 GeV. Then, the humps are the resonances occuring in pion nucleon elastic scattering. These can either be nucleon resonances (N*) or Deltas. Nucleon have spin and isospin 1/2 and Deltas have both spin and isospin 3/2. The first one around 300 MeV of pion beam energy, is the Delta resonance with a mass around 1232 MeV. After partial wave analysis, this one appears in the P33 partial wave (P means L = 1, 3 is for 2s, s=spin= 3/2 and the next 3 is for 2I (I=isospin=3/2S)).[17] However, it does not mean that every peak in the cross section plot corresponds to a resonance process because this kind of process occurs in a single partial wave, and the cross section is the superposition of all waves, then it is important to verify this for single waves..

(26) 20. 2.6. CHAPTER 2. STANDARD RESONANCE SIGNALS. Argand Diagrams. The Argand diagrams are useful to obtain resonance parameters such as the energy and width of the resonance from experimental data. These diagrams can be made drawing z, which is given by, (2.54). z = 2kal As we have seen before, for the Breit-Wigner resonance, = aR l. 1/2Γ 1 k ER − E − 12 iΓ. (2.55). ER − E + 12 iΓ 1/2Γ 1 = k ER − E − 12 iΓ ER − E + 12 iΓ 1 Γ/2 = ((ER − E) +i (Γ/2)) k (ER − E)2 + 14 Γ2 | {z } | {z } Real. (2.56) (2.57). Im. Then, it must be drawn in the complex plane. For convenience, let us write z as, z = −i(e2iδl − 1). (2.58). = −i(cos 2δl + i sin 2δl − 1). = sin 2δl +i (1 − cos 2δl ) | {z } | {z } Real. (2.59) (2.60). Im. Then, it can be plotted as a unitary circle centered at i, as is shown, in figure [2.5]. It is not difficult to notice that the top of the circle corresponds only to the imaginary part, then the real part is zero. This means that at that point E = ER , δl (ER ) = π/2. As energy increases, z moves in a circular path in an anti-clockwise sense. To consider the background phase shift, and similarly to equation [1.36], it is necessary to write the background amplitude in terms of the phase shift δlbg (background δlbg here means the non-resonant part) bg. abg l. eiδl sin δlbg −i i2δbg = = (e l − 1) k 2k. (2.61). And the total wave amplitude can be express as bg al = a R l + al. (2.62).

(27) 2.6. ARGAND DIAGRAMS. 21. Figure 2.5: Argand diagram for the resonant elastic scattering amplitude - BreitWigner resonance -[1] Then, for elastic resonance with background, the resonant amplitude will be rotated by 2δlbg as can be seen in the next figure. In this case the resonance energy will not be at the top, it will rotate too, then it will be at the diametrically opposite point of the z bg point. All this is valid if the background remains constant with energy changes, however, if it does not, the point of resonance will not be well defined. When there is a resonant inelastic scattering, the circle gets smaller and distorted [1]. To illustrate the Argand diagrams, here is an example of πN scattering plot from experimental data as shown in figure (2.7):.

(28) 22. CHAPTER 2. STANDARD RESONANCE SIGNALS. Figure 2.6: Argand diagram for the resonant elastic scattering amplitude in the presence of background [1].

(29) 2.6. ARGAND DIAGRAMS. 23. Figure 2.7: Argand diagram for the resonant elastic scattering amplitude of pionnucleon. The amplitude is plotted in the real and imaginary axes versus the total energy, and the Argand diagram shows three changes of curvature, where resonance took place. [18].

(30) Chapter 3 Characterization of resonances and related controversies In this chapter I will present and discuss some, old as well as recent, papers which I found, in connection with the methods of resonance extraction. These works use some of the methods described in the previous chapter to characterize resonance from hadron scattering data.. 3.1. Paper 1: Antibound ’States’ and Resonances. The first paper to discuss was published in 1973 in the American Journal of Physics [13]. Hans Ohanian and Carl Ginsburg, who are part of the physics department of Rensselaer Polytechnic Institute, discussed a simple example of elastic scattering in one dimension. They used four different methods of resonance extraction. Those have been called in the previous chapter as: 1. Phase shift jump, 2. Cross section hump, 3. Time delay and 4. Internal wave amplitude. The last one will be explained ahead. They started their paper with the simplest case for one dimension scattering process. A free particle potential, whose wave function as a stationary state can be written as 24.

(31) 3.1. PAPER 1: ANTIBOUND ’STATES’ AND RESONANCES. φ(x) =. 1 ikx (e − e−ikx ) = sin kx 2i. 25. (3.1). where k 2 = 2mE . Now, they said that there is a potential V 6= 0 for x > 0, then the ~2 solution for the exterior region will have a phase shift due to the potential, such as φ0 (x) =. 1 ikx+2iδ (e − e−ikx ) 2i. (3.2). Now, the scattered wave, or outgoing wave, is defined as ψscattered (x) = φ0 (x) − φ(x) 1 2iδ = (e − 1)eikx 2i. (3.3) (3.4). using equation (1.33), the scattering amplitude can be written as f (θ) =. e2iδ − 1 2i. (3.5). Then, what they called the scattering probability or strength of scattering, can be expressed as 1 2iδ |e − 1|2 = sin2 δ 4. (3.6). which is proportional to the cross section (1.44), then, let us call this method as the cross section hump method in order to relate this to chapter 2. Another method, which also has been discussed in chapter 2, is the time delay that is nothing but the derivative of the phase shift with respect to the energy. It can also be expressed as the specific time delay by 1 dδ R dk. (3.7). They developed an example to discuss the antibound state. An antibound state is what is more commonly known in textbooks and literature as a ”virtual ” state. Such a state is not a true physical state since it gives rises to a growing exponential wave at large times (and hence cannot be normalized). However, its theoretical existence (for example as a pole in the amplitude at negative energies) does have an effect on cross section. An example is the singlet state of the deuteron..

(32) 26. CHAPTER 3. CHARACTERIZATION OF RESONANCES. The bound state of the neutron and proton at −2.22MeV is the ground state of the deuteron which has about 96% probability be in the L = 0, S = 1 (spin-triplet state) 3 S1 and 4% in the L = 2, S = 1, 3 D1 state. The antibound or virtual state is found with the neutron-proton in singlet state, L = 0, S = 0, i.e. 1 S1 , at about -100KeV. Consider an infinite potential well by the right side and a step by the left side, leaving in the middle a well of depth V0 and width a, as is shown in figure (3.1).. Figure 3.1: The square well[13] Then, for this well, the general solution for the interior part is given by 0. 0. ψ(x) = A0 eik x + B 0 e−ik x. (3.8). where k 02 = 2m(E + V0 )/~ and A and B arbitrary constants. Let us say that A0 = −B 0 , and let A be equal to 2iA0 , then ψ(x) = A sin k 0 x. (3.9). for x < A. Now, for the exterior region we have that for x < 0, Ψ(x) = 0, and for x > a the solution is the same that have been found above (3.2) ψ(x) =. 1 ikx+2iδ − e−ikx ) (e 2i. Then, using boundary conditions for x = a, one can obtain that µ 0 ¶ k cot k 0 a + ik 2iδ −2ika e =e k 0 cot k 0 a − ik. (3.10). (3.11). The paper shows an example of neutron-proton interaction with two different values of V0 , for triplet state and for singlet state, as is shown in figures (3.2) and (3.3) respectively. In the figures is represented the extraction methods explained above; (a) Phase shift, (b) Cross section hump, (c) Specific time delay, and (d) Interior wave amplitude |A|..

(33) 3.1. PAPER 1: ANTIBOUND ’STATES’ AND RESONANCES. 27. Figure 3.2: Scattering in a square well, V0 = 3.4~2 /(2ma2 ). This potential describes the interaction between the proton and neutron in the triple state. [13]. Figure 3.3: Scattering in a square well, V0 = 2.1~2 /(2ma2 ). This potential describes the interaction between the proton and neutron in the singlet state.[13].

(34) 28. CHAPTER 3. CHARACTERIZATION OF RESONANCES. Comparing the figures is possible noticed that δ(0) = π for triplet state, and δ(0) = 0 for singlet state. Then, for singlet state dδ/dk is large, which implies that E ≈ 0, hence, this is what is called as an antibound state. This can be explained with Levinson’s theorem expressed as δ(0) = N π. (3.12). where N is the number of bound states, then, if there are no bound states δ(0) = 0, as in singlet state, is the antibound state. Generalizing for bound states with E < 0, the equation (3.11) becomes k 02 =. 2m(V0 − |E|) ~2. (3.13). k2 =. i2m|E| = iκ ~2. (3.14). and. where |E| = −E is the binding energy. Then, applying boundary conditions at x = a one gets ¶ µ 0 k cot k 0 a − κ 2iδ 2κa (3.15) e =e k 0 cot k 0 a + κ and ψ(x) =. 1 2iδ −κx (e e − eκx ) 2i. (3.16). which has the wrong behavior at x → ∞, then one has to get it right, hence, e−2iδ = 0. (3.17). In this case, the interaction neutron-proton has only one solution, which corresponds to the binding energy of the deuteron, |E| = 2.22MeV. An antibound state can be obtained by eliminating the decreasing exponential from equation (3.16), then an antibound state can be defined as e2iδ = 0. (3.18).

(35) 3.1. PAPER 1: ANTIBOUND ’STATES’ AND RESONANCES. 29. hence, k 0 cot k 0 a − κ = 0. (3.19). However, this is unphysical because the wave function increases exponentially at large distances. Then, for δ with E small and negative, we get k002 = 2mV0 /~ and µ ¶ 2ik 2iδ 0 cot k0 a + O(k 2 ) (3.20) e = 1 − 2ika + k00 therefore δ ≈ nπ + k. ·µ. ¶ ¸ 1 0 cot k0 a − a + O(k 2 ) k00. (3.21). and then ¯ dδ ¯¯ = (k00 cot k00 a)−1 − a dk ¯k=0. (3.22). now, if the energy is near zero then κ will be small κ ≈ k00 cot k00 a. (3.23). and ¯ 1 dδ ¯¯ ≈ −a ¯ dk k=0 κ. (3.24). This shows that for a small value of κ, the specific time delay is large positive near zero energy; in contrast, the specific time delay at bound states is large negative. The cross section, which is proportional to 1/k 2 will be given by sin2 δ. α. 1 κ2. (3.25). Finally, the authors show an example of scattering on a square well with bar~2 ~2 rier, where V0 = 1.0 2ma 2 and V1 = 5.0 2ma2 . There they plotted the four different methods discussed along the paper, as is shown in the figure (3.4), and based on the plot they conclude..

(36) 30. CHAPTER 3. CHARACTERIZATION OF RESONANCES. 2. 2. ~ ~ Figure 3.4: Scattering in a square well with barrier, V0 = 1.0 2ma 2 and V1 = 5.0 2ma2 . (a) phase shift, δ; (b) Cross section, sin2 δ ; (c) specific time delay, a−1 dδ/dk ; (d) interior wave amplitude, |A|. [13]. According to the last plot and the analysis above, they find out some conclusion about the different methods of resonance extraction. For instance, phase shift jump can be used as a concluding method if the following conditions take place: 1. There is a sudden increase by π going through π/2 and 2. This increase is given as a consequence of an increase on energy. While the internal wave amplitude cannot be used as an ultimate method; the only thing that can be taken for sure is that the amplitude of oscillation of the system is larger at resonances, however, if the amplitude rises, one is not necessary a resonance. On the other hand, they found that cross section hump is not sufficient condition for the existence of a resonance because the cross section reaches a maximum when the phase shift δ goes through π/2, but not always there is an increase of energy when δ crosses π/2. In other words, if δ moves across π/2 it does not mean that there was a phase shift jump. Nevertheless, the authors stated that it is possible to know the existence of a resonance if the peak is very narrow and sharp, since it means that |dδ/dk| is large and then this behavior corresponds to a resonant state. Finally, they settled that a sharp maximum in the time delay is the most clear an accurate method to affirm that resonance took.

(37) PAPER 2A, 2B AND 2C: S-MATRIX POLES. 31. place in certain scattering process. Time delay works so good because the minimum and maximum points at phase shift represent minimums in time delay, while the inflection point of phase shift is a sharp maximum in time delay plot, then, the peak can be found only at resonant energy.. 3.2. Paper 2a, 2b and 2c: Correspondence between Poles and S-Matrix in resonance. In this section will be discussed three papers that are all connected. The first paper Correspondence between Unstable Particles and Poles in S-Matrix Theory presents a discussion about the construction of a S-Matrix satisfying all requirements of analyticity, unitarity, and threshold and asymptotic behavior in energy, with the characteristics associated with resonance, such as cross section hump and phase shift jump, and consequently a large cross section and a time delay of 2~/Γ, without a pole in the unphysical sheet. In this paper the authors recognized at the end that the phase shift δ could not ensure the exponential decay law of the unstable particle, then two of the authors published another paper, Correspondence between Unstable Particles and Poles in S-Matrix Theory: The Exponential Decay Law, where they obtain a S-matrix with the same characteristics in the first paper, and possessing the phase shift jump resonance with the exponential decay law, without an associated pole. They also proved the existence of a local potential which yields the S-matrix. The two papers mentioned above are based on an artificial mathematical example. Hence, the third paper, by the same authors, shows a real example of a resonance with no pole S-matrix.. 3.2.1. Paper 2a: Correspondence between Unstable Particles and Poles in S-Matrix Theory. The authors of this paper built three different S matrices for three different cases where the S-matrix satisfies the requirements of analyticity, unitarity, and threshold and the characteristic asymptotic behavior in energy. The analyticity properties of S matrix are relevant to the physical interpretation of dynamical properties of.

(38) 32. CHAPTER 3. CHARACTERIZATION OF RESONANCES. interacting elementary particles. Analyticity gives rise to a branch cut stretching on the positive real energy axes, for elastic scattering process, while for any new inelastic process there correspond a superposed branch cut; and bound states give rise to simple poles for real negative energies. In the same way, it is taken for granted that any unstable particle has a corresponding pole of the S-matrix on the unphysical sheet of the energy Riemann surface, as was explained in the previous chapters, see figures (2.2) and (3.5). The paper recognizes that the characteristics that are associated with an unstable particle must be the cross section humps and the phase shift jump, which correspond to some experimental consequences as large cross section and time delay, respectively.. Figure 3.5: Physical sheet of the Riemann surface corresponds to Imp > 0 and unphysical sheet corresponds to Im p < 0.. Before the examples it is necessary to show some mathematical preliminaries. To build up an S-matrix without pole, they had to assign a phase shift on the positive axis of energy. Real for real positive energies that vanishes at infinity like E l+1/2 for the lth wave scattering process in elastics scattering processes. Then, the phase shift will be a step function that jumps as 2mπ → (2m + 1)π. What it is important is to build up that step function in terms of simple analytic functions. For this purpose, the authors used a theorem that allows to approximate continuous functions, in a finite interval, by a polynomial to any desired degree of accuracy. Then, they defined next functions:.

(39) 33. PAPER 2A, 2B AND 2C: S-MATRIX POLES. 2 n. gl,n (x) = (1 − x ). µ. x2 1− 2 η. ¶l. ,. (3.26). for 0 < η < 1, then, 1 hl,n (x) = Kl,n. Z. x. gl,n (t)dt. (3.27). η. where Kl,n =. Z. 0. gl,n (t)dt. (3.28). η. These functions have the following properties:. hl,n (0) = 1,. hl,n (η) = 2. hl,n (x) → O[(x + η)l+1 ]. h0l,n (x) ≥ 0. −1 h0l,n (0) = Kl,n. for |x| < η. h0l,n (0) → +∞. (3.29) when x → −η. when n → +∞. (3.30) (3.31) (3.32) (3.33) (3.34). Using these functions and its properties it is possible to build up S matrices having cross section humps and phase shift jump without any associated pole on the unphysical sheet. S matrix with essential singularities and no poles It is shown how one can construct a no poles S matrix for elastic scattering process of two spinless particles. They also built up a potential which gives rice to the considered phase shift. It starts with an expression for a phase shift like this π ³ a ´1/2 hl,n [φ(E)] (3.35) δl (E) = 2 E where (E − a) (3.36) φ(E) = η (E + a).

(40) 34. CHAPTER 3. CHARACTERIZATION OF RESONANCES. From the equations (3.35) th derivative of δl is given by · ¸ ³ a ´1/2 π ³ a −1/2 −3/2 ´ 0 0 hl,n [φ(E)] + − a E hl,n [φ(E)] δl (E) = 2 2 E from (3.36), we can calculated the derivative, then, φ0 (E) = η , hence φ0 (a) = 2a δl0 (E. 2ηa , (E+a)2. (3.37). and for E = a,. ·µ ¶ ¸ 1 π η 1 − hl,n [0] + = a) = 2 2a 2a Kl,n · ¸ π η 1 1 − δl0 (a) = 2 2a Kl,n 2a. (3.38) (3.39). The derivative shows the correct behavior of the phase shift, as long as there is an inflection point of δl at E = a. For large values of n, the function π2 hl,n [φ(E)] is very small in the interval 0 ≤< a and almost π when E > a, then the phase shift is restricted in such a way that after the phase shift jump, the function vanishes at infinity; and even though the phase shift goes through π/2 again, as can be seen in figure (3.6), the derivative is not positive and we can identify that there is no resonance.. Figure 3.6: Phase shift’s behavior, V0 = 2.1~2 /(2ma2 )[3] The analytic properties of the S matrix are defined by the chosen phase shift δl (z). This phase shift has a pole of the order of 2n+2l +1 at z = −a in the complex energy plane, has a kinematical branch cut and vanishes when z goes to infinity in any direction..

(41) PAPER 2A, 2B AND 2C: S-MATRIX POLES. 35. As a consequence, the S matrix Sl (z) = e2iδl (z) has an essential singularity at z = −a in the real an the complex sheets of the energy Riemann surface. Besides the discussed singularities, the S matrix is analytic and tends to 1 for z → ∞ in any direction of the sheets of the energy Riemann surface. To build up the potential from the phase shift they used the result of another paper - unfortunately it is not available and it is not possible to discuss the procedure-, at the end they give an expression for the potential in momentum representation, hp |V | p0 i = −. 1 Φ(Ep )Φ(Ep0 ) 4π 2 pEp p0 Ep0. (3.40). where Ep = p2 /2µ, and ½ · ¾ ¸ Z ℘ ∞ δ(E 00 )dE 00 sin δ(Ep ) Φ(Ep ) = exp − π 0 E 00 − Ep. (3.41). where ℘ means that the Cauchy principal value of the integral has to be taken. Finally, it is important to point out that with the phase shift defined there will not be bound states. S matrix with no singularities apart from the kinematical and dynamical branch cuts This example gets closer to the real physical condition. Now the S-matrix satisfies three requirements: 1. S(z) is analytic in the complex energy z plane cut from −∞ to −b and from 0 to +∞. The left-hand cut will be chosen to be of the logarithmic type, as suggested by the structure of the S matrix for a superposition of Yukawa potentials; and the right-hand cut is the kinematical and is therefore of the square-root type. 2. S(z) goes to 1 for z → ∞ in any direction in the complex z plane and in any sheet of the energy Riemann surface. 3. There are no other singularities, besides the branch points mentioned above, in the unphysical sheets..

(42) 36. CHAPTER 3. CHARACTERIZATION OF RESONANCES. Then the phase shift can be expressed as ¶1/3 µ π ³ a ´1/2 b + E hl,n [φ(E)] δl (E) = 2 E b+a. (3.42). on the positive real axis, where. φ(E) =. (b + E) η ln(1 + a/b) (b + a). (3.43). This phase shift is similar to the one before, but a cubic-root factor has been inserted so that the analytic continuation of the phase shift does not blow up at the dynamical branch cut. The functions in the equations (3.42) and (3.43) are determined in such a way that the logarithm and the cube root of (b + e)/(b + a) have the branch cut from −∞ to −b, while the square root of (a/E) has a branch cut from 0 to +∞. The derivative of the phase shift at E = a is given by · ¸ 1 1 1 1 1 1 π 0 (3.44) − + δl (a) = 2 Kl,n ln(1 + a/b) a + b 2a 3 b + a then, the phase shift has an inflection point The parameter a controls the position of resonance, n does the same to the width as well as b with the dynamical branch cut; it is relevant to point out that the parameters are independent. The phase shift is analytic in the complex z plane cut between −∞, −b and 0, +∞; and the analytic continuation of the δ vanishes when z goes to infinity in any direction. Then, the S matrix is analytic in all the z plane but the dynamical and kinematical branch lines. Due to the absence of the bound states, the phase shift generates a Jost function Dl (z) · ¸ Z 1 ∞ δl (E 0 )dE 0 Dl (z) = exp − (3.45) π 0 E0 − z This function is analytic in the whole z plane cut between 0, ∞, and never vanishes. It goes to +1 for |z| → +∞. The analytic continuation DII (z) of the equation (3.45) across the cut in the interval 0 ≤ z < +∞ can be expressed as DII (z) = e2iδl (z) Dl (z) ≡ Sl (z)Dl (z). (3.46). This equation possesses a logarithmic branch point at z = −b created by the corresponding branch point of the phase shift δl (z). And DII (z) does not vanish and tends to +1 as |z| → +∞.

(43) PAPER 2A, 2B AND 2C: S-MATRIX POLES. 37. Inclusion of an inelastic process The present case is the closer one to the actual physical situation. This example is similar to the first one that was explain before, but in this case, inelastic scattering process, the phase shift becomes imaginary at E = 0. Then, the complex part of the phase shift in the elastic process has to vanish at E = 0. And the cross section peak will take place at E = a, where the phase shift goes through π/2, while its derivative, time delay, will be large and positive, and its complex part will remain invariant in this energy region.. π δl (E) = 2. µ. ¶µ. ¶1/3 b+E b+a ) (3.47) ¶1/2 · ¸l+1 µ b+E b+E a−c b + E ln hl,n [φ(E)] − × ln ln2 c−E b+c b λ. a 1/2 E (. where a, b, c, λ are positive; φ(E) is the same in equation (3.43), the parameter λ has to ensure that the derivative of the imaginary part of the phase shift is equal to zero at E = a, and the parameter a is chosen larger than c to guarantee that the resonance will take place at the inelastic energy region. The functions in the equation (3.47) are determined in such a way that the logarithms and the cube root (b + E)1/3 have the branch cut from −∞ to −b, while the square roots of (a/E) and (c − E) have their branch cut from 0 to +∞ and from c to +∞, respectively. From the unitary requirement of the S-matrix, the complex part of the phase shift has to be positive. And all the parameters are independent. Finally, all these conditions allow the correct behavior of the phase shift at resonance energy (E = a), since the phase shift goes through π/2 with positive derivative, in the interval c ≤ E ≤ 2a + a2 /b. The analytic structure is the same discussed in the first example, and, as it was wanted, the S-matrix has no pole or any other singularity in any sheet of the energy Riemann surface. Summarizing, it is possible to state that there are, at least, three cases of what they recognize as a resonance -phase shift jump, cross section hump and time delay in a scattering process- where the S-matrix can be built in such a way that there is no pole associated with it. It is important to emphasize that the S-matrix satisfies all the requirements of analyticity, unitarity, and threshold and the characteristic.

(44) 38. CHAPTER 3. CHARACTERIZATION OF RESONANCES. asymptotic behavior in energy. Actually, the S-matrix pole is No sufficient and No necessary condition for a resonance. And as a consequence, there are no reason to establish a correspondence between an unstable particle and a S-matrix pole. The next paper was made to clarify and discus in detail that the phase shift proposed here does not guarantee the exponential decay law of the unstable particle.. 3.2.2. Paper 2b: Correspondence between Unstable Particles and Poles in S-Matrix Theory: The Exponential Decay Law. From the last paper it was clear that the δl built does not ensure the exponential decay law expected for an unstable particle at large time interval. Then, two of the authors made this paper where they have modified the formalism to build up the known S matrix with the usual requirements of analyticity, unitarity and asymptotic behavior, and the expected phase shift jump with the exponential decay and no pole associated to the unphysical sheet. To build up a phase shift that ensures the exponential decay law it is not sufficient to create a phase shift that goes through π/2 at the resonance energy, one have to take care of the details of the energy dependence in the resonance region. Then one obtains the S matrix without an associated pole. The authors compared two phase shift with a small difference in the relevant energy interval. They found that the difference can be made arbitrarily small. Therefore, if the δ1 (E) gives rise to a pure exponential decay law for a certain time interval, δ2 (E) will give rise to the same decay law in the time interval considered. Moreover, the difference shows that the value of the derivative at the resonance point is not of crucial relevance for the time development of state. Considering the phase shift corresponding to the Breit-Wigner resonance, δBW (E) = arctan[−Γ/2(E − a)]. (3.48). is possible to notice that difference between this phase shift and another one does not depend on Γ, then the difference argument is not valid for this case. Starting from the equation (3.48) they notice that δn (E) = π/2 is a polynomial in E of degree 2n−1, approximating the Breit-Wigner phase shift in the energy interval considered..

(45) 39. PAPER 2A, 2B AND 2C: S-MATRIX POLES. But δn (E) cannot be chosen as a physical phase shift due to it does not have the correct asymptotic behavior at zero an infinite energies. Then, an acceptable phase shift can be obtain by multiplying δn (E) by two factors that govern the behavior in zero and infinite energies. Thus, considering the identity µ ¶1/2 µ ¶−1/2 µ ¶1/2 X E E−a E (2k − 1)!! a − E k = (3.49) 1+ 1≡ a a a (2k)!! a k=0 valid for |E − a| < a, then let be µ ¶1/2 X m (2k − 1)!! a − E k E Zm (E) = a (2k)!! a k=0. (3.50). that not only possess the correct zero-energy behavior, but also can be made arbitrarily close to 1, inside |E − a| < ∆ by increasing m. Now, for the infinite-energy behavior, let us use the function F (E, c) = exp{−c ln2 [1 + (E − a)/(a + b)]};. b, c > 0. (3.51). Then, the phase shift can be expressed as δn,m (E, c) = Zm (E)δn (E)F (E, n). (3.52). This function possess the correct behavior for zero and infinite energies, as well as any other required characteristic, as has been seen in the previous paper, and ensures that the S-matrix has no pole associated to a resonance. Finally, the paper proves that there exists a potential yielding the found phase shift, then , it is absolutely possible to obtain an S-matrix that ensures the exponential decay law, with no pole associated to a resonance.. 3.2.3. Paper 2c: No-Pole S-Matrix Fit to the ∆(1236) Resonance. In the two papers before this one, the authors shown that mathematically, it is possible to have a resonance, or in other words, a phase shift jump with no pole on the complex plane. In this paper it is actually an example of a pion-nucleon 3 − 3 scattering, with 14 experimental points of the phase shift jump that were fitted to a.

(46) 40. CHAPTER 3. CHARACTERIZATION OF RESONANCES. 4 parameters s-matrix with no poles associated. The phase shift built has a precision of 1◦ . The data of the experiment was taken from another paper [ref 2] that fitted the points to a S-matrix pole in the unphysical sheet, having errors of the order of 0.1◦ . Five different resonance fits to δ33 that had found a S-matrix pole in the complex energy plane for a value of energy of 1211 − 50iMeV. In this paper was necessary to use 4-parameter formulas to obtain a χ2 of . 1 per point. According to the paper 2a, the phase shift can be expressed as. π δ33 (W ) = 2. µ. ER E. ¶1/2 R X(W ) (1 − t2 )n (1 − t2 /η 2 )dt −η R0 (1 − t2 )n (1 − t2 /η 2 )dt −η. (3.53). where n is a positive integer, η is between 0 and 1 (0 < η < 1), E = W − MN − Mπ , W is the center of mass of energy, and X(W ) is a function of energy that the S matrix obtained from the δ33 has the properties: 1. S(W ) = −1 at E = ER 2. S(W ) is analytic in the complex energy plane cut from −∞ to W0 (dynamical left-hand branch cut) and from MN + Mπ to +∞(physical kinematical cut). 3. S(W ) does not possess any other singularity (in particular there are no poles) in any sheet of the complex energy plane. 4. S(W ) possess the correct threshold behavior (δ33 k3 ) and approaches unitary as W → ∞ along any direction and any sheet According to these properties, the authors proposed two functions: ln[(E + µ)/(ER + µ)] X1 (W ) = −η ln[µ/(ER + µ)] · ¸ η E+µ ER + µ X2 (W ) = ln − ln ln(ER /µ) E + ER2 /µ ER + ER2 /µ. (3.54) (3.55). where µ = MN + Mπ − W0 . Evaluating δ33 with the equations (3.54) and (3.55) and changing the parameters n, η, ER andW0 the error was about 1◦ and 4◦ respectively, As is shown in the figure (3.7). As the authors were able to fit experimental data of the ∆++ to a simple no-pole S-matrix with a high degree of accuracy, they said.

(47) PAPER 3: K AND T MATRICES. 41. Figure 3.7: Theoretical phase shift δ33 calculated using X1 (W ) (continuous line) and X2 (W ) (dashed line). The dots are the experimental points.[5]. that it is possible to do the same for the other particles; nevertheless, thinking in the physics of the problem, as a symmetry model or a dynamical scheme, one might prefer associate a resonance with the Breit-Wigner form, which implies, of course, the S-matrix with an associated pole in the unphysical sheet of the Riemann surface, But, it is important to emphasize that it is not because it is easier or the only way to do it. Even though the built no-pole S-matrix satisfies all the requirements explain above, the underlying physics is unknown. There is also a warning about all those resonance that do not fit in a symmetric scheme and have been questioned. There should be analyze other possibilities to fit the data, as a no-pole S-matrix..

(48) 42. 3.3. CHAPTER 3. CHARACTERIZATION OF RESONANCES. Paper 3: Model-independent resonance parameter extraction from the trace of K and T matrices. This paper published in Physics Letters B in 2008 [4], by S. Ceci, A. Švarc, B. Zauner, D.M. Manely and S. Capstick, presents a model-independent method for the extraction of resonance parameters, which will be explained ahead. In theoretical hadron physics, to make a connection between resonances that are predicted by quark models and experiments has been an issue. This problem is evident because the methods for extracting resonance parameters present different conclusions for the same experimental data. As the authors show, this can be found in the Review of Particle Physics where the resonances have been parametrized in two ways: using Breit-Wigner parameters, and with T-matrix complex pole positions. These two methods produce quite different parameters due to the discrepancy in the analysis details, such as the number and character of the included channels, different parameterization schemes, analyticity constraints for scattering amplitudes, the choice of background models, the method of unitarization of the S matrix, among others. Nevertheless, the differences are present using some other method. Therefore, the authors concluded that the problem is that Breit-Wigner parameters are conventionally model dependent by nature. Consequently, the purpose of this paper is to establish a new model-independent method to find resonance parameters, defined as K-matrix mass and width, exactly at the energies of the K-matrix poles. The method uses the trace of corresponding K and T matrices to eliminate ambiguities caused by the multichannel character of the formalism. Additionally, the resonance parameters extracted are defined at the energy of the K-matrix poles, hence, they are independent of specific parametrization of the K matrix. The authors observed that T -matrix trace simplifies the formalism without loss of generality, and shows resonant behavior more prominently than any T -matrix element does. Finally, making some calculations, they found that the T-matrix trace shows resonant behavior at energies matching those of the K-matrix poles. For the scattering processes, in this paper, is necessary using multichannel scattering, which is nothing but the description of the system by operators acting in an orthonormal wave-function space, then the transition probabilities for physical.

(49) 43. PAPER 3: K AND T MATRICES. processes are given by a matrix elements in some chosen basis. Such as the transition probability for a two body system of channels a and b, where Pa→b = |hb; q|Ŝ q |a; qi| where q are all the quantum numbers conserved in the scattering reaction. In particular, for πN scattering spin and parity are conserved and isospin is almost conserved (charge symmetry is slightly violated). Conservation for probability can be ensured if S matrix is unitary, then it can be expressed as S = e2iδ , where δ is a Hermitian matrix in the channel indices. This δ is defined in an arbitrary basis, but it can be transformed in an orthonormal wave-function basis, as δ = U † δD U where U is an unitary matrix δD is a diagonal matrix. Therefore, S matrix can be diagonalized by the same transformation: S = U † e2iδD U . The K matrix is defined as K = i(I − S)/(I + S), where I is the unit matrix. This matrix can be written in the eigenstate basis as: K = U † tan δD U . It turns out that K is a real matrix, since S is unitary and symmetric, hence U is a real orthogonal matrix. They defined an orthogonal vector basis {E 1 , E 2 , ..., E N }, where  1  0 E1 =   .. .. 0 . 0 . .. . . . .. 0 .. ..  0  . ..  , .. EN. ...,. 0. . 0 . . .  .. . . =  . . 0 ..  . 0 .. ..  . .   0 0 0 1. (3.56). as every diagonal N × N can be spanned in this basis, hence tan δD =. N X. E i tan δ i. (3.57). i=1. where N is the number of channels and δ i is the eigenphase shift. There can be defined a coupling matrices equation that connects the scattering operators in any basis with their diagonal form in the eigenstate basis, χi = U T E i U. (3.58). that can be written as a projector, satisfying N X i=1. χi = I,. χi χj + χj δij. (3.59).

(50) 44. CHAPTER 3. CHARACTERIZATION OF RESONANCES. where δij is the Kronecker δ symbol. Now the K and T matrices can be expressed as N X χj tan δ j (3.60) K= j=1. T =. N X. j. χj eiδ sin δ j. (3.61). T . I + iT. (3.62). j=1. where they are connected by. K=. Later in this paper will be used the trace concept, which is the sum of all the elements on the diagonal of a matrix. There are two relevant properties, (i) The trace of product of matrices is invariant to cyclic permutations of the matrices: T r[ABC] = T r[BCA] = T r[CAB]; and (ii) the trace is distributive with respect to scalars: T r[αA+βB] = αT r[A]+βT r[B]. Notice that the orthogonal transformation (3.58) preserves the trace of a matrix: T rχi = 1.. (3.63). To extract the parameters, the authors present a procedure, but first it is necessary to make some important considerations. The elements of tan δD and χj correspond to the kinematical variable. The rth element of the diagonal matrix tan δD can be described in terms of many functions of energy, hence Γr /2 + tan δBr (3.64) Mr − W where the tan δBr is the whole contribution of background, and the first term is a simple pole, where W is the total energy, and Mr and Γr are the K matrix mass and width, respectively, and at the poles are defined as: tan δ r =. MrR = Mr (MrR ),. R ΓR r = Γr (Mr ).. (3.65). Now, the corresponding K and T matrix are given by: N. X Γ0r /2 + K = χ χj tan δ j Mr − W j6=r. (3.66). X Γ0r /2 j + T = χ χj eiδ sin δ j 0 Mr − W − iΓr /2 j6=r. (3.67). r. N. r.

(51) 45. PAPER 3: K AND T MATRICES. where the second term in both equations represent the contribution of the coupled channel background. And Γ0r /2 = Γr /2 + (Mr − W ) tan δBr . For a K matrix pole at some energy MrR , the matrix element χrab at the same energy gives the coupling strength of the resonance with mass MrR and the decay width ΓR r from channel a r to b. The diagonal element of the matrix χ is the branching fraction xra of a given resonance to the channel a, so xra = χraa. (3.68). The dependence of the resonance parameters on the channel can be reduced by using only the diagonal elements of the K and T matrices. There are two methods to obtain these matrices, by unitary coupled-channel partial wave analysis, or by using part wave T matrices . To remove the channel-resonance mixing, one has to get the traces from K and T matrices in equations (3.69) and (3.70). Then, the traces of the K and T matrices can be written as N. X Γ0r /2 tan δ j T r[K] = + Mr − W j6=r. (3.69). N. T r[T ] =. X j Γ0r /2 eiδ sin δ j + Mr − W − iΓ0r /2 j6=r. (3.70). Now the T matrix trace is a good start to the model-dependent extraction method, but the authors states the following procedure to extract K-matrix pole parameters[4]: 1. The parameter extraction procedure starts when a full T matrix has been obtained from an energy-dependent partial wave analysis of experimental data. 2. Use equation (3.62) to obtain the full K matrix from the known T matrix. 3. Poles of Tr[K] are found in order to obtain the masses of a set of NR resonances M1R , ..., M1N R , defined by Equation (3.65) at the position of the pole. 4. Multiplying both sides of equation (3.69) by (MkR − W ) and setting the energy.

(52) 46. CHAPTER 3. CHARACTERIZATION OF RESONANCES W to the value of the kth resonance mass (MkR ), then " # N 0 X Γ /2 r + tan δ j (MkR − W ) (MkR − W )T r[K] = MkR − W j6=r. (3.71). N. (MkR. Γ0r X + − W )T r[K] = tan δ j (MkR − W ) 2 j6=r. (3.72). solving for the width, ΓR k. = 2 lim. W →MkR. ΓR = 2 lim k. W →MkR. " £. (MkR. − W )T r(K) −. (MkR − W )T r(K). ¤. N X. tan δ. j. (MkR. j6=r. #. − W). (3.73) (3.74). 5. The branching fraction of a resonance to a given channel can be obtained similarly, but this time using the diagonal K-matrix element, Kaa from equation (3.66) # " N 0 X Γ /2 χj tan δ j (MkR − W ) (3.75) + (MkR − W )K = χraa Rr Mk − W j6=r (MkR. − W )Kaa =. Γ0 χraa r 2. +. N X j6=r. χj tan δ j (MkR − W ). (3.76). using equation (3.68) and solving for the branching fraction " # N X 2 xka = lim (MkR − W )Kaa − χj tan δ j (MkR − W ) (3.77) R R Γk W →Mk j6=r xka =. £ R ¤ 2 (M lim − W )K aa k R ΓR k W →Mk. (3.78). 6. Repeat steps (4) and (5) for all resonances found in (3). Finally, the extracting method was illustrated in this paper for some experimental data taken from other paper. They used πN and ηN as the channels in the analysis and π 2 N as an effective two-body channel. Using the proposed method they found, with small calculation, very similar results to the original publication, which used Breit-Wigner formula. This paper also shows a comparation between the.

(53) PAPER 4: K-MATRIX AND T MATRIX POLES. 47. K-matrix poles and the trace of T-matrix. The resonance position found by using K matrix poles directly correspond to the positions of the peaks in the Im(Tr[T ]), and the zeros in Re(Tr[T ]). While the peak of the T matrix elements corresponding to individual channels show a certain deviation from that behavior. This suggest that using individual channels to obtain the resonance parameter induces an error, which does not appear using the trace of T matrix. The analysis of the data with the proposed method works very good, and all the differences in the parameters values can be explained by the original analysis, furthermore, all those problems can be easily solved by adding channels in the partial wave analysis. Summarizing some ideas, the model-independent method is presented as a better method than the extraction of Breit-Wigner parameters from the T matrix, here are some of the advantages, according to the authors: • The T-matrix is model dependent, and the results for pole parameter are also model dependent. In contrast, the presented procedure for extracting the Kmatrix pole parameters from the T-matrix is model independent. • The shown procedure to extract resonance parameters is simple and straightforward once the full T matrix is known, whereas the another method requires a complicated diagonalizing process. • The method removes completely the background at resonance energies by using the trace of the K matrix, while extracting Breit-Wigner parameters from the T matrix, the background is not totally removed and actually makes a substantial contribution to the T matrix, even at the resonance energies. Additionally, this method has some others relevant issues that should be pointed out, in particular, the traces of K and T matrices are used in order to eliminate the problem due to the multi-channel aspect of formalism. Moreover, as long as the method get rid of the contribution of background and channel mixing at K-matrix poles energies, the parameter values obtain at those energies must be compared directly to the predictions of quark model and QCD calculations. To sum up, the model-independent method shown in this paper seems to be very accurate and much easier than some others that have been used, however, a recent paper, which will be discuss below, disagrees with this model..

(54) 48. 3.4. CHAPTER 3. CHARACTERIZATION OF RESONANCES. Paper 4: Resonance parameters from K-matrix and T matrix poles. A very recent paper of 2008 and published few months ago in Physical Review C, presents a discussion of the paper 3 [4]. In this paper, the authors, Workman, Arndt and Paris, tested some experimental data using the model-independent procedure discussed above. The test was made over a set of amplitudes determined from pion-nucleon scattering and eta-nucleon production data. They used the partial wave T matrix, and then they extracted the K-matrix poles positions and residues for real energies. From this analysis the authors found two results that make them to evaluate the convenience of using the position and the residues of the K matrix poles as a modelindependent characterization of resonance structure. The first issue was about using a different form of tan δ r (3.64), in which case, the K matrix pole and residue are not independent of the background, tan δBr , as it was shown in paper 3. And second, they used a T matrix determined in another paper, and then they calculated numerically the related K matrix, where some of the poles of the T matrix had no associated K-matrix poles for real energies. Actually, in the original paper was assumed that K matrix conserves the structure of T matrix, but, it does not. This papers starts showing the relevant equations of paper 3, but as long as those have been shown here above, I am presenting the ones that are different. Additionally, in this paper is used a different notation, for simplicity, I will use the original notation (paper three notation). The authors did again the whole derivation of equations (3.69) and (3.70) in relation to assumption (3.64); then they compared the result with the result obtained from a different assumption for parametrization of the resonant eigenphase, taken from some paper, as follows. The K matrix was defined in terms of T matrix as, T = K(1 − iK)−1. (3.79). Then, they diagonalized K matrix by an orthonormal transformation U, described by KD = U T KU. (3.80).