Yukawa Potential from QFT

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PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 12

Wednesday, January 22, 2020

Yukawa Potential from QFT

Let us compute the scattering amplitudes in Yukawa theory. A simple application of the rules we derived would be scattering of distinguishable fermions, in the non-relativistic limit. By comparing the amplitude for this process to the Born approximation formula from non-relativistic quantum mechanics, we can determine the potential V (r) created by the Yukawa interaction. If the two interacting particles are distinguishable, only the ﬁrst diagram in Fig. ?? contributes.

Figure 1: Scattering diagram giving rise to the Yukawa potential in non-relativistic limit.

In the non-relativistic limit, we keep terms only in lowest order in the 3-momenta. That is, up to O(p2, p02, ··· ). Noting that M p we have p2 p0 = m + ≈ m (1) 2m PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

For the momenta involved in the scattering process

p = (m, p), k = (m, k), p0 = (m, p0), k0 = (m, k0) (2)

Using these expressions we have

(p0 − p)2 = −|p0 − p|2, (3) and we have relations such as ! √ ξs us(p) = m , (4) ξs where ξs is a two-component constant spinor normalized to

0 0 (ξs )†ξs = δss (5)

The spinor product in the expression for the scattering amplitude

2 0 1 0 iM = (−ig) u¯(p )u(p) 0 2 2 u¯(k )u(k) (6) (p − p) − mφ becomes

0 0 0 u¯s (p0)us(p) = 2m(ξs )†ξs = 2mδss (7) 0 0 0 u¯r (k0)ur(k) = 2m(ξr )†ξr = 2mδrr . (8)

We see that the spin of each particle is separately conserved in this non-relativistic scattering interaction. Putting together the pieces of the scattering amplitude Eq. (6), we ﬁnd

2 ig 2 ss0 rr0 iM = 0 2 2 (2m) δ δ . (9) |p − p| + mφ

This gives the Yukawa interaction,

2 2 −g Ve(q) = (2m) 2 2 . (10) |q| + mφ

We note that the factor (2m)2 comes from the relativistic normalization of our states

hp|p0i = 2Eδ(p − p0), (11) with E = m. We need to compare it with the non-relativistic quantum mechanics normalization

hp|p0i = δ(p − p0). (12)

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Thus we need to drop the factor (2m)2 when comparing to Born approximation expression, which assumes conventional non-relativistic normalization of states. The Born approximation to the scattering amplitude in non-relativistic quantum mechanics, written in terms of the potential function V (x), has the form

0 0 hp |iT |pi = −iVe(q)(2π)δ(Ep0 − Ep), q = p − p. (13)

The additional δ(3)(p0 − p) goes away when we integrate over the momentum target. Inverting the Fourier transform we get

Z 3 2 d q −g iq·x V (x) = 3 2 2 e (2π) |q| + mφ 2 Z ∞ 2 Z 1 −g |q| i|q|r cos θ = 2 d|q| 2 2 d(cos θ) sin θe (2π) 0 |q| + mφ −1 2 Z ∞ iqr −iqr −g 2 e − e 1 = 2 dq q 2 2 4π 0 iqr q + mφ −g2 Z ∞ q eiqr = 2 dq 2 2 . (14) 4π ir −∞ q + mφ

The contour of the integral can be closed above in the complex plane, and we pick up the residue of the simple pole at q = +imφ. Thus, we ﬁnd, g2 1 V (r) = − e−mφr, (15) 4π r which is an attractive “Yukawa potential”, with range

1 = ~ , (16) mφ mφc which is the Compton wavelength of the exchanged boson. Yukawa made this potential for his theory of the nuclear force, and worked backwards from the range of the force, about 1 fermi, to predict the mass (about 200 MeV) of the required boson, the pion.

Fermion-anti-fermion Scattering in Yukawa Theory

We can also consider the process where a fermion and an anti-fermion scatter oﬀ of each other. For the process given in Fig. 2. In this process we need to evaluate, non-relativistically,

! s0 ! 0 0 1 ξ v¯s(k)vs (k0) ≈ m(ξs† − ξs†) 1 0 ξs0 0 = −2mδss . (17)

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Figure 2: A fermion and an anti-fermion scattering oﬀ of each other through Yukawa interaction.

† † In this process also we must work out the fermion minus sign. Upon using |p, ki = apbk|0i and 0 0 hp , k | = h0|bk0 ap0 , we can write the contracted matrix element as the one given in Fig. 3.

Figure 3: Contracted matrix element for the scattering of a fermion and an anti-fermion in Yukawa theory.

To untangle the contractions, we need to perform three operator interchanges. Thus there is an overall factor of −1. This cancels the extra minus sign in Eq. (17). Thus we see that the Yukawa potential between a fermion and an anti-fermion is also attractive, and identical strength to that between two fermions.

Anti-Fermion-anti-fermion Scattering in Yukawa Theory

We can also consider a process in which two anti-fermions scatter oﬀ of each other. We ﬁnd that the potential is still attractive. Thus we conclude that Yukawa potential is universally attractive.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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