PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 26 Tuesday, February 25, 2020 Topics: Dimensional Regularization of φ4 Theory. One-loop Correction to Two-Point Function. Dimensional Regularization of φ4 Theory In this lecture we will discuss a regularization scheme in which we can treat the dimension d of spacetime as a continuous complex variable. This scheme is called dimensional regularization. It is a consistent and well-defined method in perturbation theory. A main advantage of dimensional regularization is that the form of the loop integrals and the method of calculation of integrals are unchanged though the dimensionality has been changed to d dimensions. The divergences of the integrals will reside in the Γ-functions that have simple poles at d = 4; 6; 8; ··· . Other advantages of this scheme are that, in the regularized theory, it preserves Poincaré invariance and gauge symmetry. A disadvantage is that it works only in the perturbative regime of quantum field theory. (Note that another regularization scheme, the lattice regularization, breaks Poincaré symmetry though it respects local gauge symmetry. This scheme works in the nonperturbative regimes of quantum field theories as well.) Let us begin with the Lagrangian of the φ4 theory 1 1 1 L = @ φ∂µφ − m2φ2 − λφ4: (1) 2 µ 2 4! In four spacetime dimensions we have the action S = R d4xL. Since the action is dimensionless we have [φ] = M and [λ] = M 0. Let us look at the theory in d spacetime dimensions. We then have S = R ddxL giving [φ] = d −1 4−d M 2 and [λ] = M . The Lagrangian can be written in d spacetime dimensions as 1 1 1 L = @ φ∂µφ − m2φ2 − λµ4−dφ4; (2) 2 µ 2 4! PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where µ is an arbitrary mass scale (called the unit of mass [’t Hooft 1973]) that we introduced to keep λ dimensionless. In our new notation, the contribution from the φ4 vertex has now changed to −iλ ! −iλµ4−d. One-loop Correction to Two-Point Function Let us look at the one-loop correction to the propagator of the φ4 theory in d spacetme dimensions. See Fig. 1. k p p Figure 1: A one-loop correction to the propagator in φ4 theory. We have i 1 Z ddk i i I = −iλµ4−d : (3) p2 − m2 + i 2 (2π)d k2 − m2 + i p2 − m2 + i We need to evaluate the loop integral Z ddk 1 : (4) (2π)d k2 − m2 + i Let us look at the following integral, in d − 1 space dimensions and 1 time dimension Z ddk 1 I (∆; n) = d (2π)d (k2 − ∆ + i)n 1 @ n−1 Z ddk 1 = : (5) (n − 1)! @∆ (2π)d (k2 − ∆ + i) 0 0 2 2 d d Performing Wick rotation k ! ikE, k ! −kE, and d k ! id kE we have n−1 Z d −i @ d kE 1 Id(∆; n) = d 2 : (6) (n − 1)! @∆ (2π) (kE + ∆ − i) 2 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Let us use the so-called Schwinger proper-time trick or Schwinger representation for propagator 1 Z 1 = dse−sP (7) P 0 to rewrite the propagator 1 1 Z 2 −s(kE +∆−i) 2 = dse : (8) kE + ∆ − i 0 (In the above, s is called the proper time variable due to reasons not relevant for this course.) Thus the integral takes the form n−1 Z d −i @ d kE 1 Id(∆; n) = d 2 (n − 1)! @∆ (2π) (kE + ∆ − i) n−1 Z d Z 1 −i @ d kE 2 −s(kE +∆−i) = d dse (n − 1)! @∆ (2π) 0 n−1 Z 1 Z d −i @ d kE 2 −s(∆−i) −skE = dse d e : (9) (n − 1)! @∆ 0 (2π) Expanding out the d number of Gaussian integrals n−1 Z 1 Z 1 Z 1 −i @ −s(∆−i) dk1 −sk2 dkd −sk2 Id(∆; n) = dse e 1 ··· e d (n − 1)! @∆ 0 −∞ 2π −∞ 2π −i @ n−1 Z 1 1 1 = dse−s(∆−i) : (10) d=2 d=2 (n − 1)! @∆ 0 (4π) s In the above we used 1 rπ 1 rπ 1 rπ d 1 d 1 1 ··· = = p p = : (11) 2π s 2π s 2π s 2 π s (4π)d=2 sd=2 The integral simplifies to, upon using ∆ ≡ ∆ − i, −i Z 1 1 1 I (∆; n) = ds(−s)n−1e−s∆ d d=2 d=2 (n − 1)! 0 (4π) s n−1 Z 1 (−i)(−1) 1 n−1− d −s∆ = dss 2 e d=2 (n − 1)! (4π) 0 n Z 1 i(−1) 1 1 n− d −1 −t = dtt 2 e d=2 n−d=2 (n − 1)! (4π) ∆ 0 n d (−1) 1 Γ(n − 2 ) = i d · d · : (12) 2 n− 2 Γ(n) (4π) ∆ In the above we used the change of variable t = s∆, and the integral representation of the 3 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Gamma function Z 1 Γ(x) = dttx−1e−t: (13) 0 Thus we have the propagator Z ddk 1 (−1)n Γ(n − d ) 1 = i 2 : (14) d 2 n d=2 d (2π) (k − ∆ + i) (4π) Γ(n) n− 2 ∆ A Note on Gamma Functions We have the identity zΓ(z) = Γ(1 + z): (15) For z = , 1 Γ() = Γ(1 + ) 1 Z 1 = dtte−t 0 1 Z 1 Z 1 = dte−t + dte−t ln t + O(2) ; (16) 0 0 giving 1 Γ() = − γ + O(); (17) where Z 1 γ = dte−t ln t ' 0:5772 (18) 0 is the Euler-Mascheroni constant. Now, setting z = −1 in the formula Γ(z) = z−1Γ(z + 1) we get 1 Γ(−1 + ) = Γ() −1 + −1 1 = − γ + O() 1 − 1 = − 1 + + O(2) − γ + O() 1 = − + 1 − γ + O() : (19) Thus we have 1 Γ(−1 + ) = − + γ − 1 + O(): (20) Similarly, we have 1 1 3 Γ(−2 + ) = − γ + + O() : (21) 2 2 4 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 One-loop Correction to Two-Point Function II Let us continue with the loop part of the one-loop correction to the propagator in φ4 theory. We have 1 Z ddk i L = (−iλµ4−d) : (22) 2 2 (2π)d k2 − m2 + i Now, taking d = 4 − 1 (−1) Γ(1 − 2 + ) 1 L = λµ · i · 2 · 2 2− 1−2+ 2 (4π) 2 Γ(1) (m2) 2 m2 = −iλµ (4π)/2Γ(−1 + )(m2)−/2: (23) 32π2 2 Using 1 a(4−d) = e(4−d) ln a = 1 + (4 − d) ln a + (4 − d)2(ln a)2 + ··· (24) 2 we have, for d = 4 − , a = 1 + ln a + ··· : (25) Using this insight, we will make use of the following approximations µ = 1 + ln µ + ··· ; (26) (4π)/2 = 1 + ln(4π) + ··· ; (27) 2 2 Γ(−1 + ) = − − 1 + γ + ··· ; (28) 2 2 − 2 (m ) 2 = 1 − ln m + ··· : (29) 2 The loop integral now becomes m2 2 L = −iλ − − 1 + γ − 2 ln µ − ln(4π) + ln m2 + O() 2 32π2 m2 2 m2 = −iλ − + γ − 1 + ln + O() : (30) 32π2 4πµ2 Rearranging the terms we get L λm2 1 λm2 m2 2 = + 1 − γ − ln + O(): (31) i 16π2 32π2 4πµ2 This renormalization prescription, where the counterterms (the diverging terms) are pure poles at the physical value of d, is called the minimal subtraction (MS) scheme. Note: The MS scheme is the standard method in the context of QCD. Minimal subtraction tends to produce large coefficients in the perturbation expansion due to the term ln(4π) − γ. It has become conventional to work with a modified scheme, called the MS scheme. In this scheme, the µ 5 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 of MS scheme is written as r eγ µ =µ ¯ : (32) 4π In the MS scheme the loop integral takes the form L λm2 1 λm2 m2 2 = + 1 − ln + O(): (33) i 16π2 32π2 µ¯2 The first term blows up as ! 0. The second term is the finite part of the correction. This term is not that important but it depends on the arbitrary mass µ. Note also that this result is independent of the external momentum pµ. Since the unit of mass (or scale) µ is arbitrary, the expression given in Eq. (31) depends on three parameters instead of two. A change of µ amounts to a change of renormalization prescription, so that the change can be compensated, in this case by a change in m. In effect, the minimal subtraction is a one-parameter family of renormalization prescriptions. References [1] L. H. Ryder, Quantum Field Theory, Cambridge University Press (1996). [2] J. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge Monographs on Mathematical Physics, Cam- bridge University Press (1984). 6 / 6.