Lecture 14 The NSVZ Beta-Functionlarsenf/Lecture14.pdf · Renormalization Schemes Renormalization...
Transcript of Lecture 14 The NSVZ Beta-Functionlarsenf/Lecture14.pdf · Renormalization Schemes Renormalization...
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.