The Yukawa Potential: from Quantum Mechanics to Quantum Field Theory

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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 interaction 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 = hψf |ψii (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 |ψai Quantum Mechanics explains that particles are sources of and |φai: potential, and this potential covers some region of space that if another particle were to enter it, an atractive or ± 1 ± |ψ i = |φai + Vˆ |ψ i (4) repulsive interaction depending on the potential would a (E − Hˆ ± i) a then occur. a 0 1 Quantum Field Theory describes interactions from the |ψ±i = |φ i + Vˆ |φ i (5) a a ˆ 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 = hψ−|ψ+i (6) bold letters denote 3-vectors. fi f i If we develop (6) by using the hermitian conjugate of − eq. (5) for the hψf |, and then using (4) on the first term II. S MATRIX IN QM + for the |ψi i, one gets: Given a scattering process in QM due to some potential Vˆ , we will work with a hamiltonian Hˆ that can be − + 2i ˆ + hψf |ψi i = hφf |φii − 2 2 hφf |V |ψi i written as Hˆ = Hˆ0 + Vˆ , with Hˆ0 being the free (non in- (Ef − Ei) + teracting) hamiltonian, and the potential Vˆ causing the ˆ + = δfi − 2πiδ(Ef − Ei)hφf |V |ψi i (7) interaction. In this description, we want the solutions for the Schrödingerequation of Hˆ and Hˆ to have the same 1 0 Where we have used the identity lim = δ(x). eigenvalue, meaning that when we connect the interac- →∞ π x2 + 2 tion Vˆ 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 Vˆ and thus an infinite series for the perturbative expansion of the Hˆ0|φai = Ea|φai (1) scattering amplitudes.

Hˆ |ψai = Ea|ψai (2)

The scattering process will be described by how some III. 2 PARTICLE SCATTTERING IN QM initial |ψii and ﬁnal |ψf i interacting states with their respective free solutions |φii and |φf 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, ˆ −2πiδ(Ef − Ei)hφf |V |φii (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 |Vˆ |φii d|ψ(t)iI i = HˆI |ψ(t)iI (15) in (8) with a two particle state of plane waves, |φii = dt 0 0 |p1, p2i, |φf i = |p1, p2i and a traslational invariant potential: If one tries to write the solution of (15) as |ψ(t)iI = UI (t, t0)|ψ(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 {x1, x2} for the center of mass and the relative 1 1 2 I 1 I 2 t0 t0 coordinates {xCM , xR} : 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 ˆ 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 exponential 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 Sˆ = 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 particle 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 Vˆ 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 xgφψψ (20) and the time evolution of the initial one. ˆ ˆ ˆ Given a certain interacting hamiltonian H = H0 + Hint, Where g is the coupling constant of the interacting the- ˆ where H0 describes the particle fields (Dirac, Klein Gor- ory, with |g| < 1, and the φ, ψ, ψ¯ fields are defined in the ˆ 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, |ii and |fi, will be two iH0t |ψ(t)iI = e |ψ(t)iS (13) fermions with definite momenta and spin with the rela- iH0t −iH0t OÎ = e OˆSe (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 r† s† DF (y − x) and multiply the ﬁrst two terms by DF (y − x) |ii ≡ |p1, p2i = 2E1 2E2b b |0i (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). hf| ≡ hp1, p2| = h0|bp0 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 d4xφψψ¯ becomes: hp1, p2|T e |p1, 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 ∼ a† + 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. Since our initial and final states don’t have This 2 factor that appears from our convenient use of any bosons in them, this would end up with either a|0i DF (x − y) cancels the 1/2! from (24). Performing the or h0|a†, giving it zero contribution. integral over q with either of the deltas that appear on each term, the final amplitude is: We see then, the leading contribution is the second order: ig2(2π)4δ(p + p − p0 − p0 )× 2 Z Z 1 2 1 2 (−ig) 4 4 n o d x d y T φψψ¯ φψψ¯ (24) 0 0 2! x y [¯ur0 (p1)us(p2)][¯us0 (p2)ur(p1)] (p0 − p )2 − m2 The only contribution from the time ordered product 2 1 φ to the physical proccess is the one that anihilates two 0 0 ! [¯ur0 (p1)ur(p1)][¯us0 (p2)us(p2)] fermions on the right and two fermions on the left [3], − 0 2 2 (29) (p1 − p ) − m [4]. Plugging it into (23) and omitting the integrals over 1 φ x and y, we get the following expression: This would correspond to the two possible diagrams of two identical fermions that interact exchanging a virtual 0 0 2 ¯ ¯ scalar of mass mφ. The diagram for the first term in (29) hp1, p2|(−ig) : φψψ x φψψ y : |p1, p2i (25) is: The scalar propagator (see (39)) that arises from the p0 p0 contracion of the φ’s won’t be shown in the expressions 1 2 until it is needed. The momenta and spins for fields (KG particles not included) are written as k1, k2, k3, k4 and l1, l2, l3, l4 from left to right. Expression (25), omitting q the integrals and the constant factors, once it’s normal ordered, looks like: p1 p2 i(k1−k2)x i(k3−k4)y e e u¯l (k1)ul (k2) u¯l (k3)ul (k4) 1 2 3 4 And the diagram for the second one is: 0 0 s r l1† l3† l2 l4 r† s† × (−1)h0|b 0 b 0 b b b b b b |0i (26) p2 p1 k1 k3 k2 k4 p1 p2 p0 p0 The minus sign comes from conmuting bl2 and bl3† and 1 2 k2 k3 the ”[ ]” are to denote that the Dirac spinor indexes are contracted inside of it. The minus sign will be left out until the final result. Using the anticonmutation relations q (40), we move the b and b† operators from the fields of † the interaction through the b ’s and b’s of the initial and p1 final state. After performing then the integrals over all p2 the k’s, the expression left is:

VI. OBTENTION OF THE YUKAWA 0 0 0 0 i(p1−p2)x+i(p2−p1)y u¯r0 (p1)us(p2) u¯s0 (p2)ur(p1) e POTENTIAL 0 0 0 0 i(p −p2)x+i(p −p1)y − u¯r0 (p )ur(p1) u¯s0 (p )us(p2) e 2 1 1 2 So far we have computed the amplitude for the +(x ↔ y) (27) scattering of 2 particles in QM, and the amplitude for

Treball Fi de Grau 3 Barcelona, June 2019 The Yukawa Potential: From Quantum Mechanics to Quantum Field Theory Christian Y. Arce Borro the scattering of 2 fermions in QFT formalism. In orther to compare them, a few aspects must be taken into Z d3q eiqr consideration: 2 V (r) = −g 3 2 2 (2π) q + mφ −g2 Z ∞ |q|2 Z 1 Notice how the amplitude in QFT has 2 different = d|q| d(cos θ) sin θei|q|r cos θ terms, and in QM we only have one. In the QM case, (2π)2 2 2 0 |q| + mφ −1 when we fixed |p0 , p0 i and |p , p i as the final and ini- 1 2 1 2 −g2 Z ∞ |q| 1 tial state, we forced the particles to be distinguishable, = d|q| (ei|q|r − e−i|q|r) 2 2 2 0 (2π) 0 |q| + m ir since momenta {p1, p1} were assigned to the coordinate φ x , and {p0 , p } to x . The calculation done in QFT 2 Z ∞ 1 2 2 2 ig |q| i|q|r doesn’t have this restriction, as can be seen diagramati- = d|q| 2 e (34) (2π)2r |q| + m2 cally. −∞ φ If we had distinguishable fermions in QFT, as happens Closing the countour integral in the upper half of the to be the case of QM, the only contribution to the am- complex plane: plitude would be the second term in (29), corresponding to the second diagram, which is the term that from now on we will consider for the comparison between the two g2 V (r) = − e−mφr (35) amplitudes. 4πr The amplitude in QFT must also be taken in the non This is, the Yukawa potential arises as the lowest order relativistic limit, meaning that m p. We shall consider effective potential for the interaction between two distin- only terms to the lowest order in momenta, so p0 = m + guible fermions in the non relativistic limit. p2/2m → m, and then for any 4-momentum p = (m, p). In this limit, we have: VII. CONCLUSIONS

√ ξ We have seen how the scattering of two fermions due u (p) → m λ (30) λ ξ to a short range potential V is treated in time indepen- λ dent Quantum Mechanics, and calculated the S matrix 0 2 0 2 2 (p1 − p1) = 0 − |p1 − p1| ≡ −q (31) element to ﬁrst order in V . QFT formalism for interactions has been introduced, and we have obtained the † scattering amplitude to leading order for 2 fermions that Where ξ are two component spinors that satisfy ξ ξ 0 = λ λ interact exchanging a Klein Gordon scalar. δλλ0 , and q is the 3-momentum carried by the virtual scalar. The scattering amplitude then reads: We have seen that in the non relativistic limit and con- sidering distinguible particles, these two results can be matched if the Yukawa potential is substituted as the potential causing the interaction in QM. ! 2 4 0 0 2 δr0rδs0s −ig (2π) δ(p1 + p2 − p1 − p2)(2m) 2 2 (32) −q − mφ VIII. APPENDIX

A. Klein Gordon and Dirac fields Imposing spin conservation in this last expresion, we can finally connect it to the last term on the r.h.s. of µ 2 (12): The Klein Gordon equation (∂µ∂ + m )φ = 0 describes a field for neutral bosons of spin 0. Its solution is: Z d3p 1 −g2 ˆ √ −ipx † ipx 2 φ(x) = 3 aˆpe +a ˆpe (36) Ve(q) = (2m) 2 2 (33) (2π) 2E q + mφ Wherea ˆ anda ˆ† are the anihilation and creation operators for Klein Gordon bosons, and all of the conmutators 2 The factor (2m) comes from the relativistic normal- between them are zero except: 0 0 ization of our states in QFT: hp|p iQF T = 2Eδ(p − p ), with E = m. We are now comparing it with an equa- † 3 tion that has been computed with the non relativistic aˆ~p, aˆ~q = (2π) δ(~p − ~q) (37) QM normalization hp|p0i = δ(p − p0), so the factor (2m)2 should be dropped. The Dirac equation (i∂/ − m)ψ = 0 and its adjoint The integration on q can now be performed: (i∂/ + m)ψ¯ = 0 describe a field for spin 1/2 fermions,

Treball Fi de Grau 4 Barcelona, June 2019 The Yukawa Potential: From Quantum Mechanics to Quantum Field Theory Christian Y. Arce Borro containing particles and antiparticles. The solutions for Written as a 4-momentum integral, this propagator is: these two equations are:

Z d3p 1 ˆ √ Z d4p ie−ip(x−y) ψ(x) = 3 (2π) 2E φxφy = DF (x − y) = 4 2 2 (41) (2π) p − mφ + i X −ipx ˆλ ipx λ† e uλ(p)bp + e vλ(p)ˆcp (38) λ It is this exchange of a virtual φ particle between the two fermions expressed with this propagator which gives Z d3p 1 ¯ √ rise to the interaction and, in an eﬀective way, to a non ψ(x) = 3 (2π) 2E relativistic instantaneous Yukawa potential, as in (33). X ipx ˆλ† −ipx λ e u¯λ(p)bp + e v¯λ(p)ˆcp (39) λ

Where ˆb (ˆc) and ˆb† (ˆc†) are the anihilation and creation operators for Dirac particles (antiparticles). Since these are fermions, obey anticonmutation relations, and all of the anticonmutators between them are zero except: Acknowledgments

ˆλ ˆλ0† λ λ0† 3 λλ0 bp , bq = cˆp , cˆq = (2π) δ(~p − ~q)δ (40) I would like to thank my advisor Josep Taron for his constancy and patience, and for the great guidance B. Propagator of the Klein Gordon field throughout the overall process this work has been. I’d also like to thank my friends and family, specially my The Wick contraction of two Klein Gordon fields at dif- mother, for always supporting me no matter the situa- ferent times gives the propagator of the KG field (scalar). tion.

[1] L. de la Pe˜na, Introduction to Quantum Mechanics, 3d ed. , Mexico: FCE, UNAM, 2006. [2] M. Gell-Mann and L. Goldberger, ”Formal Theory of Scat- tering”. Phys. Rev. 91, no2 : 398–408 (1953). [3] M. E. Peskin and D. V. Schroeder, An introduction to Quantum Field Theory, USA, Westview Press, 1995. [4] D. Tong, Lecture notes on Quantum Field Theory, Cam- bridge, 2006-2007. [5] J. J. Sakurai, Modern quantum mechanics, Addison- Wesley Pub. Co, 1994

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