Lecture 14 the NSVZ Beta-Function Outline Theme: Assorted Comments on QFT

Lecture 14 The NSVZ Beta-Function Outline Theme: assorted comments on QFT

• Superconformal symmetry.

• Renormalization scheme dependence. • The NSVZ beta-function. • Wave function renormalization.

Reading: Terning 7.6-7, 8.2, 8.6. Superconformal Symmetry Motivation: • The IR ﬁxed point of a QFT is scale invariant. • This means it is also conformally invariant.

• Conformal symmetry imposes powerful constraints. • In SUSY QFT the symmetry is superconformal invariance which imposes even more powerful constraints. • Strategy: construct general unitary representations of the superconformal group. The Conformal Algebra Generators:

• Lorentz rotations/boosts: Mµν = −i(xµ∂ν − xν ∂µ).

• Translational generators: Pµ = −i∂µ.

2 α • Special conformal generators Kµ = −i(x ∂µ − 2xµxα∂ ).

α • Dilation operator: D = ixα∂ .

Counting generators: 6 + 4 + 4 + 1 = 15. This is the dimension of the conformal algebra. Algebra: these generators generate SO(4, 2). Radial Quantization An alternative basis for the generators:

0 Mjk = Mjk , 0 1 Mj4 = 2 (Pj − Kj) , 0 i D = − 2 (P0 + K0) , 0 1 Pj = 2 (Pj + Kj) + iMj0 , 0 i P4 = −D − 2 (P0 − K0) , 0 1 Kj = 2 (Pj + Kj) − iMj0 , 0 i K4 = −D + 2 (P0 − K0) . 0 0 0 0 Notation: Mmn,Pn,Kn,D with m, n = 1, 2, 3, 4. Some commutators in the algebra:

0 0 0 [D ,Pm] = −iPm , 0 0 0 0 [Pm,Kn] = −2i(δmnD + Mmn) . Highest Weight Representations Highest weight representations: the highest weight state is annihilated 0 by all Kn. 0 0† Action by Pn = Kn creates descendant states. Scaling dimension d: the Dilation operator is diagonalized so D0 = −id.

The highest weight state: where the scaling dimension takes it lowest (!) value. Scaling dimensions of Descendant states: d increases by 1 for each action 0 by Pn. Unitarity Unitarity: all states have positive norm. Positive norm for some speciﬁc descendant states (with any m 6= n): 0 ˜ 0 ˜ ˜ 0 ˜ Pm|d, (j1, j1)i ± Pn|d, (j2, j2)i ⇒ d ≥ ±hd, (j1, j1)|iMmn|d, (j2, j2)i .

In the space of operators at a given level of scaling dimension we expand:

0 i 0 0 iMmn = 2 (δmαδnβ − δmβδnα)Mαβ = (V · M )mn .

Diagonalization (in the m, n index) through

0 1 0 2 2 02 V · M = 2 [(V + M ) − V − M ] , so that scaling dimension is bounded by

1 0 d ≥ 2 [C2(r) + C2(V ) − C2(r )] , for any representation r0 that appears in the decomposition V + r where V is the vector of SO(4). Quadratic Casimir of SO(4) ' SU(2) × SU(2):

˜ P 2 ~2 ~˜ ˜ ˜ C2(j, j) = m,n Jmn = 2(J + J) = 2[j(j + 1) + j(j + 1)] .

Some examples:

C2(scalar) = C2((0, 0)) = 0 , 1 3 C2(spinor) = C2(( 2 , 0)) = 2 , 1 1 C2(vector) = C2(( 2 , 2 )) = 3 .

Some bounds on scaling dimensions:

1 d ≥ 2 (0 + 3 − 3) = 0 (scalar) , 1 3 3 3 d ≥ 2 ( 2 + 3 − 2 ) = 2 (spinor) , 1 d ≥ 2 (3 + 3 − 0) = 3 (vector) .

Remark: these are for gauge invariant operators so for the vector the operator of lowest dimension is the current J µ. 0 0 Similar arguments applying Pi Pk on a scalar state gives d(d − 1) ≥ 0 .

So unless d = 0 (the identity operator) we must have d ≥ 1 , for a scalar field. Interpretation: a free scalar field has scaling dimension d = 1. At a nontrivial fixed point the scaling dimension cannot any be smaller. Superconformal Symmetry For SUSY theories, conformal symmetry is enhanced to superconformal symmetry.

New features (N = 1 SUSY): the SUSY charge Qα, the R-charge R, the conformal SUSY generator Sα. Heuristic interpretation: superconformal symmetry in D = 4 is SO(4, 2), enhanced with fermion generators. It is analogous to D = 6 SUSY: SO(5, 1), enhanced with fermion generators. Upon dimensional reduction, N = 1 SUSY in D = 6 becomes N = 2 SUSY in D = 4. Component expansion of superconformal symmetry has same structure: there are two supercharges Qα and Sα. The R for some purposes plays the role of a central charge. Unitarity and Superconformality The important anti-commutator:

0 0 i 0 0 3 {Qα,Sβ} = 2 Mmn(ΓmΓn)αβ + iδαβD − 2 (γ5)αβR.

Unitarity imposes non-negative norm of the states: 0 ˜ 0 ˜ aQα|d, R, (j1, j1)i + bQα|d, R, (j2, j2)i .

Computing norm for each α 6= β using S0 = Q0† (in Euclidean space): ˜ i 0 3 ˜ d ≥ ±hd, R, (j1, j1)| 2 MmnΓmΓn − 2 (γ5)αβR|d, R, (j2, j2)i .

Decompose to independent SU(2) components (for spinors this is just 1 5 the chirality P± = 2 (1 ± γ )): ~ ~ 3 ~˜ ~˜ 3 d ≥ P+ 4J · S − 2 R + P− 4J · S − 2 R .

i (Spin operator Smn = 4 [Γm, Γn]). Unitarity and Superconformality

~ ~ 1 The spin j representation has J + S spin j ± 2 so 2J~ · S~ = (J~ + S~)2 − J~2 − S~2 = −j − 1 or j , except for j = 0 where 2J~ · S~ = 0 is only option. Thus for j 6= 0: 3 ˜ 3 ˜ d ≥ dmax = max 2(j + 1) + 2 R, 2(j + 1) − 2 R ≥ 2 + j + j .

For scalars j = 0:

3 d ≥ 2 |R| . 3 Remark: applying the general bound for j = 0 gives d ≥ 2 + 2 |R| ⇒ there is a gap. More precise statement (that takes more work to establish): for j = 0 3 3 either d = 2 |R| or d ≥ 2 + 2 |R|. The saturation is for chiral superfields. The NSVZ β-function Conformal symmetry is at the fixed point. We next reconsider the running of the coupling towards the fixed point. According to holomorphy: the SUSY gauge coupling runs only at one- loop where

g3 P β(g) = − 16π2 3T (Ad) − j T (rj) .

This appears in contradiction with other (true) statements about the running: • the “exact” β function of NSVZ is P 3 3T (Ad)− T (rj )(1−γj ) g j β(g) = − 16π2 1−T (Ad)g2/8π2 .

• one- and two-loop terms in β function are scheme independent. Renormalization Schemes Renormalization condition: defines the coupling in terms of a physical amplitude. 1 4 Example 1: in 4! gφ theory, the four point amplitude (at some definite energy) is exactly g (the definition of the coupling g). This determines the finite parts of the counterterms. Example 2: work in dimensional regularization and subtract just the singular parts as → 0 (minimal subtraction, MS). The predictions of QFT are the values of other physical amplitudes, expressed in terms of the coupling that was defined. The renormalization scheme ambiguity: different computational scheme express physical predictions in terms of different couplings g0 = g + ag3 + O(g5) . Two-loop Universality of the β-fct. The β-function (in some scheme) is

dg 3 5 7 β(g) = d ln µ = b1g + b2g + O(g ) .

In another scheme

0 3 5 7 g = g + a1g + a2g + O(g ) , there is a diﬀerent β-function

0 0 ∂g 03 05 07 β (g ) = β(g) ∂g0 = b1g + b2g + O(g ) .

Remark: the dependence on the ai only appears at higher order! Scheme Dependence of Λ-scale Next: aside on the dynamical scale. The dynamical scale Λ is introduced as an integration constant:

dg bg3 8π2 µ µ dµ = − 16π2 ⇒ g2(µ) = b ln Λ . In asymptotically free theories Λ is the analogue of the coupling constant.

Terminology: dimensional transmutation. is the feature that the “coupling constant” Λ is dimensionful. Question: what is the status of Λ at higher loops? And does it depend on renormalization scheme? Scheme Dependence of Λ-scale At higher order:

2 dg 1 3 5 8π µ 8πb2 µ µ = − 2 b1g + b2g + ... ⇒ 2 = ln + 2 ln ln + ... dµ 16π b1g Λ b1 Λ or the inverse relation:

2 2 − 8π − 16π b2 ln g+... b g2 b2 Λ 1 1 µ = e .

In an alternate renormalization scheme with coupling

0 3 g = g + a1g + ... the dynamical scale is

2 8π2 16π b2 0 2 0 − − ln g +... 16π a1 b g02 b2 +... Λ 1 1 b1 Λ µ = e = e µ . The “dots” are all of higher order so: the relation between Λ in different renormalization schemes depends only on the first order. In practice: determine the relation between Λ in different schemes by comparing results for some reference process evaluated in both schemes one loop order. Then use that relation in all other processes, and to all orders. In SUSY theories: this is relevant when comparing finite coefficients computed in different renormalization schemes. Status We have discussed renormalization scheme dependence of the β-function and the dynamical scale Λ. Conclusion: it is a red herring. The key distinction between the running of the holomorphic coupling (one loop exact) and the NSVZ β-function: only the later is canonically normalized. Derivation of the NSVZ β-function: relate normalizations exactly, using anomalies. Holomorphic vs Canonical Coupling The holomorphic coupling:

1 R 2 a a Lh = 2 d θ W (Vh)W (Vh) + h.c., 4gh where

1 1 θYM τ 2 = g2 − i 8π2 = 4πi , gh a a a Vh = (Aµ, λ ,D ) .

The canonical gauge coupling for canonically normalized ﬁelds: 1 θYM R 2 a a Lc = 2 − i 2 d θ W (gcVc)W (gcVc) + h.c. 4gc 32π

The key observation: these deﬁnitions are not equivalent under Vh = gcVc because of a rescaling anomaly. The rescaling anomaly is completely determined by the axial anomaly. Rescaling Anomaly

Matter ﬁelds Qj have additional rescaling anomaly from:

0 0 1/2 Qj = Zj(µ, µ ) Qj .

Analysis: rewrite the axial anomaly in a manifestly supersymmetric form using the path integral measure as

D(eiαQ)D(e−iαQ) = DQDQ i R 2 T (rj ) a a × exp − 4 d θ 8π2 2iα W W + h.c. .

Identify Z = e2iα (with α complex) so

1/2 1/2 D(Z Qj)D(Z Qj) = DQjDQj j j i R 2 T (rj ) a a × exp − 4 d θ 8π2 ln Zj W W + h.c. . Rescaling Anomaly 2 For the gauge ﬁelds (gauginos) take Zλ = gc so

i R 2 2T (Ad) a a D(gcVc) = DVc × exp − 4 d θ 8π2 ln(gc) W (gcVc)W (gcVc) + h.c. .

So for pure SUSY Yang–Mills:

R i R 2 1 a a Z = DVh exp 4 d θ 2 W (Vh)W (Vh) + h.c. gh R i R 2 1 2T (Ad) a a = DVc exp 4 d θ 2 − 8π2 ln(gc) W (gcVc)W (gcVc) + h.c. . gh Canonical and holomorphic coupling related as

1 1 2T (Ad) g2 = Re 2 − 8π2 ln(gc) . c gh Remark: relation between the two couplings is logarithmic so one cannot be expanded in a Taylor series around zero in the other. (This is unlike renormalization scheme dependence). Rescaling Anomaly Include the matter ﬁelds: 1 1 2T (Ad) P T (rj ) g2 = Re 2 − 8π2 ln(gc) − j 8π2 ln(Zj) . c gh

Differentiate with respect to ln µ, find NSVZ β-function: P 3 3T (Ad)− T (rj )(1−γj ) g j β(g) = − 16π2 1−T (Ad)g2/8π2 . where the anomalous dimension of the field is

dZj γj = − d ln µ .

Summary: the NSVZ β-function gives the exact running of the canonical coupling. It is related to the holomorphic β-function through a nontrivial rescaling that is determined by anomalies. Wavefunction Renormalization

The anomalous dimension γj is 1/2 of the anomalous mass. Compare: we previously found that the mass term in the superpotential does not require renormalization. This may seem like a contradiction. The kinetic terms in the Lagrangian require wave function renormalization (singular rescaling between classical and quantum ﬁelds):

∗ µ µ Lkin. = Z∂µφ ∂ φ + iZψσ ∂µψ .

The renormalization factor Z is a non-holomorphic function of the parameters Z = Z(m, λ, m†, λ†, µ, Λ) .

If we integrate out modes down to µ > m at one-loop order

† Λ2 Z = 1 + cλλ ln µ2 , where c is a constant determined by the perturbative calculation. If we integrate out modes down to scales below m we have

† Λ2 Z = 1 + cλλ ln mm† . Wavefunction renormalization means couplings of canonically normalized ﬁelds run. The running mass and running coupling are related to the holomorphic parameters in the superpotential as

m λ Z , 3 . Z 2