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The Yukawa Potential: from Quantum Mechanics to Quantum Field Theory

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The Yukawa Potential: from Quantum Mechanics to Quantum Field Theory The Yukawa potential: from Quantum Mechanics to Quantum Field Theory Author: Christian Y. Arce Borro Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Catalonia, Spain. Advisor: Josep Taron Roca Abstract: Quantum Mechanics and Quantum Field Theory treatment for the scattering of two fermions is compared to find out an instantaneous effective Yukawa potential describing the inter- action in the non relativistic limit. The comparison is made between the S matrix element for the scattering of 2 particles in time independent QM and the non relativistic limit of the scattering amplitude of 2 fermions due to a Yukawa interaction in QFT. I. INTRODUCTION This connection will be made introducing an S matrix element as follows [1]: The understanding of interactions between particles is essential for any theory that attempts to explain the fun- damental structure and behaviour of matter, because it's Sfi = h f j ii (3) this understanding which allows to make predictions that can be then observed (or not) in real life, and thus test Manipulating eqs. (1) and (2) in order to work out eq. a theory. (3), we can obtain the following relations between j ai Quantum Mechanics explains that particles are sources of and jφai: potential, and this potential covers some region of space that if another particle were to enter it, an atractive or ± 1 ± j i = jφai + V^ j i (4) repulsive interaction depending on the potential would a (E − H^ ± i) a then occur. a 0 1 Quantum Field Theory describes interactions from the j ±i = jφ i + V^ jφ i (5) a a ^ a perspective of virtual particle exchange between the par- (Ea − H ± i) ticles that are interacting, so instead of an instantaneous potential, there's a deeper understanding on how the Where the ±i have been introduced to avoid the sin- propagation of information is carried. gularities provided that we take the limit ! 0 at the The purpose of this work is to connect these two descrip- end of the calculation. The initial (final) states are then tions in the scattering of 2 fermions, in order to derive in defined as the ones with the + (−) sign for the physical the non relativistic limit one of the first discovered effec- reasons discussed in [1], [2], and so (3) now reads: tive potentials between nucleons, the Yukawa potential. This paper is written in natural units, = c = 1, and ~ S = h −j +i (6) bold letters denote 3-vectors. fi f i If we develop (6) by using the hermitian conjugate of − eq. (5) for the h f j, and then using (4) on the first term II. S MATRIX IN QM + for the j i i, one gets: Given a scattering process in QM due to some poten- tial V^ , we will work with a hamiltonian H^ that can be − + 2i ^ + h f j i i = hφf jφii − 2 2 hφf jV j i i written as H^ = H^0 + V^ , with H^0 being the free (non in- (Ef − Ei) + teracting) hamiltonian, and the potential V^ causing the ^ + = δfi − 2πiδ(Ef − Ei)hφf jV j i i (7) interaction. In this description, we want the solutions for the Schr¨odingerequation of H^ and H^ to have the same 1 0 Where we have used the identity lim = δ(x). eigenvalue, meaning that when we connect the interac- !1 π x2 + 2 tion V^ the energy stays the same as in the free case: In eq. (7) we can substitute (4) as many times as we want, getting every time higher powers of V^ and thus an infinite series for the perturbative expansion of the H^0jφai = Eajφai (1) scattering amplitudes. H^ j ai = Eaj ai (2) The scattering process will be described by how some III. 2 PARTICLE SCATTTERING IN QM initial j ii and final j f i interacting states with their respective free solutions jφii and jφf i relate to each other. To first order in V , the second term in (7) becomes: The Yukawa Potential: From Quantum Mechanics to Quantum Field Theory Christian Y. Arce Borro Where the S subindex refers to the operator in the Schr¨odingerpicture. Staying in the interaction picture, ^ −2πiδ(Ef − Ei)hφf jV jφii (8) states evolve according to the following equation: In orther to compare this result with the scattering of two fermions in QFT, we develop the term hφf jV^ jφii dj (t)iI i = H^I j (t)iI (15) in (8) with a two particle state of plane waves, jφii = dt 0 0 jp1; p2i, jφf i = jp1; p2i and a traslational invariant po- tential: If one tries to write the solution of (15) as j (t)iI = UI (t; t0)j (t0)iI with U(t0; t0) = 1, that would be: Z Z 0 0 t 3 3 i(p −p )x1 i(p −p )x2 Z d x1 d x2V (x1 − x2)e 1 1 e 2 2 (9) UI (t; t0) = 1 + (−i) dt1HI (t1) t0 t t If the masses of the two particles are m1; m2, we can Z Z 1 + (−i)2 dt dt dt H (t )H (t ) + ::: (16) change fx1; x2g for the center of mass and the relative 1 1 2 I 1 I 2 t0 t0 coordinates fxCM ; xRg : Eq.(16) can be written in a compact way introducing m1x1 + m2x2 the time ordering operator T, which puts earlier time xCM = defined operators to the right and later ones to the right: m1 + m2 xR = x1 − x2 (10) R t ^ 0 0 −i HI (t )dt The determinant of the jacobian matrix for this trans- U(t; t0) = T e t0 (17) formation is equal to 1, and so (9) now reads: Where since T is linear, the time ordering of the ex- ponential is defined as the taylor series with each term Z 0 0 time ordered. The S matrix is then defined as [3]: 3 i(p1+p2−p1−p2)xCM d xCM e 0 0 Z m2(p1−p1) m1(p2−p2) 3 i − xR m1+m2 m1+m2 S^ = lim U(t ; t ) (18) × d xRV (xR)e (11) − + t±→±∞ From this change of variables we can then see that mo- In QFT the S matrix is often decomposed separating mentum conservation delta arises naturally, and defining the non interacting term from the interacting as follows: 0 q = p1 − p1 as the momentum transfered from one par- ticle to the other, the integral over xR becomes a Fourier transform of the potential. With this, the S matrix ele- S = 1 + iT (19) ment to first order in V^ for the scattering of 2 particles is: When we calculate scattering amplitudes, the identity term doesn't contribute because we consider processes in which the initial state is different from the final one. 4 0 0 Sfi = δfi−i(2π) δ(Ef −Ei)δ(p1+p2−p1−p2)Ve(q) (12) V. FERMION-FERMION SCATTERING IN QFT IV. S MATRIX IN QFT We want to look at the fermion-fermion scattering due In QFT,we treat scattering from a time evolution per- to a Yukawa interaction: spective, meaning that the scattering amplitudes are cal- Z 3 ¯ culated with the matrix element between the final state HI = d xgφ (20) and the time evolution of the initial one. ^ ^ ^ Given a certain interacting hamiltonian H = H0 + Hint, Where g is the coupling constant of the interacting the- ^ where H0 describes the particle fields (Dirac, Klein Gor- ory, with jgj < 1, and the φ, , ¯ fields are defined in the ^ don, photons etc.) and Hint the interaction between appendix. We will proceed to do this using Wick's the- them, when dealing with scattering it is useful to use the orem for the time ordered operator [3]. The free hamil- interaction picture. In this picture, states and operators tonian will be the sum of real Klein Gordon and Dirac are defined as: hamiltonians, and every operator or state we write is in the interaction picture, thus we will drop the subindex I. Our initial and final states, jii and jfi, will be two iH0t j (t)iI = e j (t)iS (13) fermions with definite momenta and spin with the rela- iH0t −iH0t O^I = e O^Se (14) tivistic normalization: Treball Fi de Grau 2 Barcelona, June 2019 The Yukawa Potential: From Quantum Mechanics to Quantum Field Theory Christian Y. Arce Borro We now use the symetry of the propagator DF (x−y) = p p ry sy DF (y − x) and multiply the first two terms by DF (y − x) jii ≡ jp1; p2i = 2E1 2E2b b j0i (21) p1 p2 and the last two by DF (x − y) so when we integrate x 0 0 s0 r0 p 0 p 0 and y the contributions are the same for (x,y) as for (y,x). hfj ≡ hp1; p2j = h0jbp0 bp0 2E2 2E1 (22) 2 1 Plugging the exponentials from the propagator as men- The scattering amplitude for the process is: tioned and performing the itegrals over x and y, omitting the integral over the momentum q of the propagator, (27) 0 0 −i R d4xφ ¯ becomes: hp1; p2jT e jp1; p2i (23) The zero order term of the exponential series in (23) 0 0 u¯ 0 (p )u (p ) u¯ 0 (p )u (p ) × is clearly contributing to the 1 in (19), and so it is not r 1 s 2 s 2 r 1 8 0 0 relevant for the scattering process. Now if we look at (2π) 2δ(p1 − p2 + q)δ(p2 − p1 − q) the first order, this has only one φ ∼ ay + a, and these 0 0 − u¯r0 (p1)ur(p1) u¯s0 (p2)us(p2) × creation and anihilation operators conmute with the 8 0 0 b0s and c0s because they create and anihilate different (2π) 2δ(p2 − p2 + q)δ(p1 − p1 − q) (28) particles.
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