PHY646 - Quantum Field Theory and the Standard Model
Even Term 2020 Dr. Anosh Joseph, IISER Mohali
LECTURE 23
Wednesday, February 19, 2020
Topics: UV and IR Divergences in QED.
In the last lecture we obtained an explicit, though complicated, expression for the one-loop vertex correction:
α Z 1 V µ = dx dy dz δ(x + y + z − 1) 2π 0 zΛ2 1 ×u¯(p0) γµ ln + (1 − x)(1 − y)q2 + (1 − 4z + z2)m2 ∆ ∆ iσµνq 1 + ν 2m2z(1 − z) u(p), (1) 2m ∆ where ∆ = −xyq2 + (1 − z)2m2. The bracketed expressions are our desired corrections to the form factors. Before we move on to interpreting this result, let us summarize the calculational methods used. The techniques are common to all loop calculations:
1. Draw the diagram(s) and write down the amplitude.
2. Introduce Feynman parameters to combine the denominators of the propagators.
3. Complete the square in the new denominator by shifting to a new loop momentum variable, l.
4. Write the numerator in terms of l. Drop odd powers of l, and rewrite even powers using identities like Z d4l lµlν Z d4l 1 gµνl2 = 4 . (2) (2π)4 D3 (2π)4 D3
5. Perform the momentum integral by means of a Wick rotation and four-dimensional spherical coordinates. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
The momentum integral in the last step will often be divergent. In that case we must define (or regularize) the integral using Pauli-Villars prescription or some other device.
UV and IR Divergences in QED
Let us consider the coefficient of uγ¯ µu. It has an ultraviolet divergence ∼ ln Λ. It also has an infrared divergence. Let us assume that the photon is on-shell: q2 = 0. Then
2 2 2 ∆µ = m (1 − z) + zµ . (3)
This leads to the term
Z 1 m2(1 − 4z + z2) dx dy dz δ(x + y + z − 1) 2 2 2 . (4) 0 m (1 − z) + zµ
We have
Z 1 Z 1 Z 1 Z 1 Z 1−z dzf(z) dy dx δ(x − (1 − y − z)) = dzf(z) dy 0 0 0 0 0 Z 1 = dzf(z)(1 − z), (5) 0 where the term in the square brackets, on the left-hand side, is zero unless 1 − y − z > 0, That is, y < 1 − z. Thus we have
Z 1 m2(1 − 4z + z2) Z 1 −2 + (1 − z)(3 − z) dz(1 − z) = dz(1 − z) , (6) 2 2 2 µ2 0 m (1 − z) + zµ 0 2 (1 − z) + z m2 which is finite if µ 6= 0. If µ = 0, we have Z 1 −2 dz + (3 − z) , (7) 0 (1 − z) and the first term diverges as z → 1. Thus µ 6= 0 regulates this divergence. Note that in ∆ = −xyq2 +m2(1−z)2, z → 1 implies x = y = 0 and ∆ → 0. The term (l2 −∆)−1 becomes divergent when l2 → 0. That is for the case of low energy (IR). This is also the same as saying that the virtual photon goes on-shell. Now that we have parametrized the UV divergence in Eq. (1) let us try to interpret it. The 2 divergence appears in the worst possible place: It corrects F1(q ) = 0, which should be fixed at the value 1. But this is the only effect of the divergent term. We will therefore adopt a simple but completely ad hoc fix for this difficulty: Subtract from the above expression a term proportional to 0 µ the zeroth-order vertex function (¯u(p )γ u(p)), in such a way as to maintain the condition F1(0) = 1.
2 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
In other words, make the substitution
2 2 δF1(q ) → δF1(q ) − δF1(0), (8) where δF1 denotes the first-order correction to F1. The justification of this procedure involves the minor correction to our S-matrix formula
sum of all connected, amputated 4 (4) X iM· (2π) δ (pA + pB − pf ) = Feynman diagrams with pA, pB . (9) incoming, pf outgoing
In brief, the terms we are subtracting corrects for our omission of the external leg correction diagrams of Fig. 1.
Figure 1: The process of electron scattering from another, very heavy, particle, at one-loop order.
We will provide justification of this statement later in the course.
Expression After Correcting for UV and IR Divergences
With both of these provisional modifications, the form factors are
Z 1 2 α F1(q ) = 1 + dx dy dz δ(x + y + z − 1) 2π 0 m2(1 − z)2 m2(1 − 4z + z2) + q2(1 − x)(1 − y) × ln + m2(1 − z)2 − q2xy m2(1 − z)2 − q2xy + µ2z m2(1 − 4z + z2) − + O(α2). m2(1 − z)2 + µ2z Z 1 2 2 α 2m z(1 − z) 2 F2(q ) = dx dy dz δ(x + y + z − 1) 2 2 2 + O(α ). (10) 2π 0 m (1 − z) − q xy
2 Note that neither the UV nor the IR divergence affects F2(q ).
3 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
We can therefore evaluate unambiguously
Z 1 2 2 α 2m z(1 − z) F2(q = 0) = dx dy dz δ(x + y + z − 1) 2 2 2π 0 m (1 − z) α Z 1 Z 1−z z = dz dy π 0 0 1 − z α = . (11) 2π
Thus we get a correction to the g-factor of the electron
α g = 2 1 + = 2.0023, (12) 2π with the next correction of order α2. We see that the anomalous magnetic moment of the electron is
g − 2 α a ≡ = ≈ 0.0011614. (13) 2 2 2π
This result was first obtained by Schwinger in 1948. Experiments give
ae = 0.0011597. (14)
4 The coefficients of the QED formula doe ae are now known through order α . QED is the most stringently tested - and the most dramatically successful - of all physical theories.
Resolution of IR Divergence
The IR divergence has a straightforward resolution. We have evaluated the diagram given in Fig. 2 and found that a divergence happens when the “loop” photon becomes on-shell.
Figure 2: The process of electron scattering from another, very heavy, particle, at one-loop order.
Now let us consider the “bremsstrahlung” process in which a real photon is present in the final state. See Fig. 3.
4 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
Figure 3: Bremsstrahlung processes.
In the limit that the energy E(p−k) is very small, the final state photon would not be detected in any experiment. We find that if we amputate
2 2 dσ 0 dσ α −q −q 2 (p → p ) = 1 − ln 2 ln 2 + O(α ) , (15) dΩ dΩ 0 π m µ 2 2 dσ 0 dσ α −q −q 2 (p → p + γ) = + ln 2 ln 2 + O(α ) , (16) dΩ dΩ 0 π m µ
dσ where the first factor, dΩ 0, is the tree-level result. The separate cross-sections are divergent. But their sum is independent of µ and therefore finite. That is, the IR divergences exactly cancel. What About UV Divergences? It turns out that the UV divergences have much more important consequences for QFTs. We will delve more into the structure of UV divergences.
References
[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).
[2] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).
5 / 5