Lecture 14The NSVZ Beta-Function

OutlineTheme: 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 SymmetryMotivation:

• 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 whichimposes even more powerful constraints.

• Strategy: construct general unitary representations of the super-conformal group.

The Conformal AlgebraGenerators:

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

• Translational generators: Pµ = −i∂µ.

• Special conformal generators Kµ = −i(x2∂µ − 2xµxα∂α).

• Dilation operator: D = ixα∂α.

Counting generators: 6 + 4 + 4 + 1 = 15. This is the dimension of theconformal algebra.

Algebra: these generators generate SO(4, 2).

Radial QuantizationAn alternative basis for the generators:

M ′jk = Mjk ,

M ′j4 = 12 (Pj −Kj) ,

D′ = − i2 (P0 +K0) ,

P ′j = 12 (Pj +Kj) + iMj0 ,

P ′4 = −D − i2 (P0 −K0) ,

K ′j = 12 (Pj +Kj)− iMj0 ,

K ′4 = −D + i2 (P0 −K0) .

Notation: M ′mn, P′n,K

′n, D

′ with m,n = 1, 2, 3, 4.

Some commutators in the algebra:

[D′, P ′m] = −iP ′m ,[P ′m,K

′n] = −2i(δmnD′ +M ′mn) .

Highest Weight RepresentationsHighest weight representations: the highest weight state is annihilatedby all K ′n.

Action by P ′n = K ′†n creates descendant states.

Scaling dimension d: the Dilation operator is diagonalized so D′ = −id.

The highest weight state: where the scaling dimension takes it lowest (!)value.

Scaling dimensions of Descendant states: d increases by 1 for each actionby P ′n.

UnitarityUnitarity: all states have positive norm.

Positive norm for some specific descendant states (with any m 6= n):

P ′m|d, (j1, j̃1)〉 ± P ′n|d, (j2, j̃2)〉 ⇒ d ≥ ±〈d, (j1, j̃1)|iM ′mn|d, (j2, j̃2)〉 .

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

iM ′mn = i2 (δmαδnβ − δmβδnα)M ′αβ = (V ·M ′)mn .

Diagonalization (in the m,n index) through

V ·M ′ = 12 [(V +M ′)2 − V 2 −M ′2] ,

so that scaling dimension is bounded by

d ≥ 12 [C2(r) + C2(V )− C2(r′)] ,

for any representation r′ that appears in the decomposition V + r whereV is the vector of SO(4).

Quadratic Casimir of SO(4) ' SU(2)× SU(2):

C2(j, j̃) =∑m,n J

2mn = 2( ~J2 + ~̃J) = 2[j(j + 1) + j̃(j̃ + 1)] .

Some examples:

C2(scalar) = C2((0, 0)) = 0 ,C2(spinor) = C2(( 1

2 , 0)) = 32 ,

C2(vector) = C2(( 12 ,

12 )) = 3 .

Some bounds on scaling dimensions:

d ≥ 12 (0 + 3− 3) = 0 (scalar) ,

d ≥ 12 ( 3

2 + 3− 32 ) = 3

2 (spinor) ,d ≥ 1

2 (3 + 3− 0) = 3 (vector) .

Remark: these are for gauge invariant operators so for the vector theoperator of lowest dimension is the current Jµ.

Similar arguments applying P ′iP′k 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 anontrivial fixed point the scaling dimension cannot any be smaller.

Superconformal SymmetryFor SUSY theories, conformal symmetry is enhanced to superconformalsymmetry.

New features (N = 1 SUSY): the SUSY charge Qα, the R-charge R, theconformal 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 = 2SUSY 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 SuperconformalityThe important anti-commutator:

{Q′α, S′β} = i2M

′mn(ΓmΓn)αβ + iδαβD

′ − 32 (γ5)αβR .

Unitarity imposes non-negative norm of the states:

aQ′α|d,R, (j1, j̃1)〉+ bQ′α|d,R, (j2, j̃2)〉 .

Computing norm for each α 6= β using S′ = Q′† (in Euclidean space):

d ≥ ±〈d,R, (j1, j̃1)| i2M′mnΓmΓn − 3

2 (γ5)αβR|d,R, (j2, j̃2)〉 .

Decompose to independent SU(2) components (for spinors this is justthe chirality P± = 1

2 (1± γ5)):

d ≥ P+

(4 ~J · ~S − 3

2R)

+ P−

(4~̃J · ~̃S − 3

2R).

(Spin operator Smn = i4 [Γm,Γn]).

Unitarity and SuperconformalityThe spin j representation has ~J + ~S spin j ± 1

2 so

2 ~J · ~S = ( ~J + ~S)2 − ~J2 − ~S2 = −j − 1 or j ,

except for j = 0 where 2 ~J · ~S = 0 is only option.

Thus for j 6= 0:

d ≥ dmax = max(2(j + 1) + 3

2R, 2(j̃ + 1)− 32R)≥ 2 + j + j̃ .

For scalars j = 0:

d ≥ 32 |R| .

Remark: applying the general bound for j = 0 gives d ≥ 2 + 32 |R| ⇒

there is a gap.

More precise statement (that takes more work to establish): for j = 0either d = 3

2 |R| or d ≥ 2 + 32 |R|. The saturation is for chiral superfields.

The NSVZ β-functionConformal symmetry is at the fixed point. We next reconsider the run-ning of the coupling towards the fixed point.

According to holomorphy: the SUSY gauge coupling runs only at one-loop where

β(g) = − g3

16π2

(3T (Ad)−

∑j T (rj)

).

This appears in contradiction with other (true) statements about therunning:

• the “exact” β function of NSVZ is

β(g) = − g3

16π2

(3T (Ad)−

∑jT (rj)(1−γj)

)1−T (Ad)g2/8π2 .

• one- and two-loop terms in β function are scheme independent.

Renormalization SchemesRenormalization condition: defines the coupling in terms of a physicalamplitude.

Example 1: in 14!gφ

4 theory, the four point amplitude (at some definiteenergy) is exactly g (the definition of the coupling g). This determinesthe finite parts of the counterterms.

Example 2: work in dimensional regularization and subtract just thesingular parts as ε→ 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 schemeexpress physical predictions in terms of different couplings

g′ = g + ag3 +O(g5) .

Two-loop Universality of the β-fct.The β-function (in some scheme) is

β(g) = dgd lnµ = b1g

3 + b2g5 +O(g7) .

In another scheme

g′ = g + a1g3 + a2g

5 +O(g7) ,

there is a different β-function

β′(g′) = β(g) ∂g∂g′ = b1g′3 + b2g

′5 +O(g′7) .

Remark: the dependence on the ai only appears at higher order!

Scheme Dependence of Λ-scaleNext: aside on the dynamical scale.

The dynamical scale Λ is introduced as an integration constant:

µ dgdµ = − bg3

16π2 ⇒ 8π2

g2(µ) = b ln µΛ .

In asymptotically free theories Λ is the analogue of the coupling constant.

Terminology: dimensional transmutation. is the feature that the “cou-pling constant” Λ is dimensionful.

Question: what is the status of Λ at higher loops? And does it dependon renormalization scheme?

Scheme Dependence of Λ-scaleAt higher order:

µ dgdµ = − 116π2

(b1g

3 + b2g5 + . . .

)⇒ 8π2

b1g2= ln µ

Λ + 8πb2b21

ln ln µΛ + . . .

or the inverse relation:

Λµ = e

− 8π2

b1g2− 16π2b2

b21

ln g+....

In an alternate renormalization scheme with coupling

g′ = g + a1g3 + . . .

the dynamical scale is

Λ′

µ = e− 8π2

b1g′2− 16π2b2

b21

ln g′+...= e

16π2a1b1

+... Λµ .

The “dots” are all of higher order so: the relation between Λ in differentrenormalization schemes depends only on the first order.

In practice: determine the relation between Λ in different schemes bycomparing results for some reference process evaluated in both schemesone loop order.

Then use that relation in all other processes, and to all orders.

In SUSY theories: this is relevant when comparing finite coefficientscomputed in different renormalization schemes.

StatusWe have discussed renormalization scheme dependence of the β-functionand 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 canonicallynormalized.

Derivation of the NSVZ β-function: relate normalizations exactly, usinganomalies.

Holomorphic vs Canonical CouplingThe holomorphic coupling:

Lh = 14g2h

∫d2θ W a(Vh)W a(Vh) + h.c.,

where1g2h

= 1g2 − i

θYM8π2 = τ

4πi ,

Vh = (Aaµ, λa, Da) .

The canonical gauge coupling for canonically normalized fields:

Lc =(

14g2c− i θYM

32π2

) ∫d2θ W a(gcVc)W a(gcVc) + h.c.

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 AnomalyMatter fields Qj have additional rescaling anomaly from:

Q′j = Zj(µ, µ′)1/2Qj .

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

D(eiαQ)D(e−iαQ) = DQDQ× exp

(− i

4

∫d2θ

(T (rj)8π2 2iα

)W aW a + h.c.

).

Identify Z = e2iα (with α complex) so

D(Z1/2j Qj)D(Z1/2

j Qj) = DQjDQj× exp

(− i

4

∫d2θ

(T (rj)8π2 lnZj

)W aW a + h.c.

).

Rescaling AnomalyFor the gauge fields (gauginos) take Zλ = g2

c so

D(gcVc) = DVc × exp(− i

4

∫d2θ

(2T (Ad)

8π2 ln(gc))W a(gcVc)W a(gcVc) + h.c.

).

So for pure SUSY Yang–Mills:

Z =∫DVh exp

(i4

∫d2θ 1

g2h

W a(Vh)W a(Vh) + h.c.)

=∫DVc exp

(i4

∫d2θ

(1g2h

− 2T (Ad)8π2 ln(gc)

)W a(gcVc)W a(gcVc) + h.c.

).

Canonical and holomorphic coupling related as

1g2c

= Re(

1g2h

)− 2T (Ad)

8π2 ln(gc) .

Remark: relation between the two couplings is logarithmic so one cannotbe expanded in a Taylor series around zero in the other. (This is unlikerenormalization scheme dependence).

Rescaling AnomalyInclude the matter fields:

1g2c

= Re(

1g2h

)− 2T (Ad)

8π2 ln(gc)−∑jT (rj)8π2 ln(Zj) .

Differentiate with respect to lnµ, find NSVZ β-function:

β(g) = − g3

16π2

3T (Ad)−∑

jT (rj)(1−γj)

1−T (Ad)g2/8π2 .

where the anomalous dimension of the field is

γj = − dZjd lnµ .

Summary: the NSVZ β-function gives the exact running of the canonicalcoupling. It is related to the holomorphic β-function through a non-trivial rescaling that is determined by anomalies.

Wavefunction RenormalizationThe anomalous dimension γj is 1/2 of the anomalous mass.

Compare: we previously found that the mass term in the superpotentialdoes not require renormalization. This may seem like a contradiction.

The kinetic terms in the Lagrangian require wave function renormaliza-tion (singular rescaling between classical and quantum fields):

Lkin. = Z∂µφ∗∂µφ+ iZψσµ∂µψ .

The renormalization factor Z is a non-holomorphic function of the pa-rameters

Z = Z(m,λ,m†, λ†, µ,Λ) .

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

Z = 1 + cλλ† ln(

Λ2

µ2

),

where c is a constant determined by the perturbative calculation.

If we integrate out modes down to scales below m we have

Z = 1 + cλλ† ln(

Λ2

mm†

).

Wavefunction renormalization means couplings of canonically normalizedfields run.

The running mass and running coupling are related to the holomorphicparameters in the superpotential as

mZ ,

λ

Z32.