Treatment of 5 in Dimensionally ... - physics.ntua.gr

Treatment of γ5 in Dimensionally-RegularizedChiral Yang-Mills Theory with Scalar Fields

Hermès Bélusca-Maï[email protected]

Department of Physics, Faculty of Sciences,University of Zagreb, Croatia

Workshop on Connecting Insights in Fundamental Physics:Standard Model & Beyond, Corfu Summer Institute

August 31 September 11, 2019

[Work In Progress], with Amon Ilakovac, Marija Maor-Boºinovi¢ (PMF Zagreb),

and Dominik Stöckinger (IKT, TU Dresden)

DReg The R-Model BRST symmetry Calculations @ 1-loop Towards RGE (WIP) Summary

Context & Motivation

Realistic models in 4D contain chiral fermions (e.g. Standard Model andextensions; SUSY models, ...).

Dimensional Regularization (DReg) with MS(bar) subtraction scheme widelyused in calculations, in the literature, in automated codes, etc., because itdoesn't break (explicit) gauge and Lorentz symmetries.

∃ numerous schemes in DReg for treating γ5 matrix (see e.g. the reviews

[Gnendiger...2017,Bruque...2018], and [Larin1993,Trueman1995,Jegerlehner2000]

non-exhaustive) → perform consistent loop calculations, reproduce known results(ABJ anomaly, etc.).

However it is not clear what happens at all loop orders. In certain situations, at> 1-loop level some naive γ5 treatment is used, supplemented with manualrestoration of symmetries (using e.g. Ward IDs, etc.). Other schemes[Kreimer1990,'94] even abandon the cyclicity property of the Trace operation...

Hermès Bélusca-Maïto (PMF Zagreb) γ5 in DReg χ YM Theory w/ Scalars SM & Beyond, Corfu 2019 1 / 34


Our aims

In this talk:

I Propose using algebraic renormalization techniques in DimensionalRegularization (DReg) with the 't Hooft-Veltman-Breitenlohner-Maisonscheme (BMHV), following [Martin,Sanchez-Ruiz1999];

I Obtain a treatment of γ5 matrix and other related intrinsically 4-dimensionalLorentz objects that is consistent by construction at any loop order.

Revisit the renormalization of a generic Chiral Yang-Mills Theory supplementedwith real scalar elds; example on an explicit calculation at 1-loop.

Restore BRST (and possible other) symmetries; recover known resultsconcerning gauge anomalies.

Work In Progress One of our aims is to obtain RGEs and compare themagainst the usual ones, i.e. method of [Machacek,Vaughn1983,'84,'85].

These results are also needed for 2+ - loop calculations.



Additionally...

Note: A similar procedure may need to be followed in principle inDimensional-Reduction scheme for SUSY theories... (open question)

Disclaimer: We do NOT claim that the other schemes are inconsistent;however a more solid proof of these in the context of algebraicrenormalization would be welcome so as to have more condence in them.Exhibiting relations between these schemes?

Our status wrt. the work [Martin,Sanchez-Ruiz1999] (and related ones):

No scalars With scalars

Abelian [Sanchez-Ruiz2002]

Non-abelian YM [Martin,Sanchez-Ruiz1999]w/o VEV: Our work

w/ VEV: ?

N-ab. YM inBkgd-eld gauge

· · · Extension of · · · · · · previous works · · ·



Outline

1 Dimensional regularization (DReg)

2 The R-ModelDening action S0; extension to d dimensionsCharge conjugation

3 BRST symmetryEective action and BRST invarianceCompleted R-Model in d dimensionsBRST invariance of R-Model @ tree-levelNormal Products; BRST invariance of R-Model @ loop-level

4 Calculations @ 1-loop1-loop singular counterterm actionBonneau Identities1-loop nite counterterm actionL-Model?

5 Towards an RGE Work In Progress!

6 Summary


Outline






6 Summary



Dimensional regularization (DReg) ['t Hooft,Veltman1972] (1/3)

DReg introduced by 't Hooft and Veltman (1972) as away to elegantly regularize loop amplitudes in(non-SUSY) Yang-Mills theories.

Gauge and Lorentz symmetries explicitly preserved(6= in e.g. Pauli-Villars or other cut-os regs.).

Observations:

I Lorentz structures considered as whole objects without any mention ofexplicit values for their Lorentz indices.

I d-dimensional loop integrals are formal and made to obey the usual rules oflinearity, translation, scaling, etc.

(See also [Collins1986].)




Formally extend 4D space-time into d = 4− 2ε-dimensions: Md ⊃ M4 withMd = M4 ⊕ M−2ε. Small ε > 0 regularizes UV divergences (ε < 0 for IR divs.).

Metrics: gµν = gµν + gµν . One can construct an inverse metric gµν

(mathematically via dual space). Similarly for the other subspaces.The · quantities are the so-called evanescent objects.

Properties:

gµνgνρ = δρµ (d-dimensions) , gµν g

νρ = gµν gνρ = δρµ (4-dimensions) ,

gµν gνρ = gµν g

νρ = δρµ (−2ε-dimensions) , gµν gνρ = 0 (orthogonality) ,

andgµνg

νµ = d , gµν gνµ = 4 , gµν g

νµ = −2ε .

Similar behaviour with other tensorial quantities, e.g.:

gµνkν = kµ , gµνk

ν = kµ , gµν kν = 0 .




Q1? DimReg OK for bosonic elds and also 4-component fermions, but whatabout Dirac γ matrices? Usual γµ can be constructed (see [Collins1986]).Convention for Tr1 (spinor space): Tr1 = 4 so as to make results simpler.

Q2? What about intrinsically 4-dimensional objects: γ5 and εµνρσ?(Inconsistency if γ5 anticommute:Tr(γ5γµ1

· · · γµ4) ∝ (d− 4) Tr(γ5γµ1

· · · γµ4) = 0 when d→ 4

instead of 4iεµ1µ2µ3µ4 .)Properties for Dirac matrices:γ5 = (i/4!)εµνρσγ

µγνγργσ , γµ, γν = 2gµν , γ5, γµ = 0 , but

[γ5, γµ] = 0 , and: γ5, γ

µ = γ5, γµ , [γ5, γ

µ] = [γ5, γµ] .

Consistency: Scheme proved to be axiomatically consistent at all orders byBreitenlohner and Maison [Breitenlohner,Maison1975, Breitenlohner,Maison1977].⇒ 't Hooft - Veltman - Breitenlohner - Maison scheme (BMHV scheme).Together with MS(bar) subtraction (subtracting the ε poles, possibly with somenite part) ⇒ Dimensional renormalization (DimRen).


Outline






6 Summary



The R-Model dening action S0

Model with generic gauge group G (usually SU(N); can be something else...)with right-handed (RH) fermions in right (R) rep. of G and scalars in S rep. ofG, both coupling to gauge bosons.Originally dened in 4 dimensions, using either Weyl, or Dirac fermions withprojectors PR/L = (1± γ5)/2.

S(4D)0 =

∫d4 x (L(4D)

YM + L(4D)Ψ + L(4D)

Φ + L(4D)Yuk + L(4D)

gh + L(4D)g-x ) ,

with:

L(4D)YM =

−1

4F aµνF

a µν , L(4D)Φ =

1

2(DµΦm)2 −

λmnopH

4!ΦmΦnΦoΦp ,

L(4D)Ψ = iΨi /∂PRΨi + gSTR

aijΨi /G

aPRΨj ≡ iΨi /D

ijRΨj ,

L(4D)Yuk = −(YR)mijΦmΨ

C

i PRΨj + h.c. ,

L(4D)gh = ∂µca ·Dab µcb , L(4D)

g-x =ξ

2BaBa +Ba∂µGaµ .

Note the Yukawa interaction with charge-conjugated fermion ( 6= Dirac model where left

component couples to right component).



The R-Model action S0 extended to d dimensions

Formally extend S0 to d dimensions such that propagators retain their form ind-D. Trivially done for bosonic elds. RH fermions introduce two problems:

1 Kinetic term is chiral. In d-D this generates an (inverse) propagator ∝ /p(see e.g. [Bilal2008]).⇒ We need an actual d-D kinetic term: iΨi /∂Ψi

⇒ Equivalent to introducing a left-handed inert component to the fermions.(Inert because gets removed in interaction terms due to explicit presence of PR/L.)

2 How to promote in d-D the Ψ/GΨ interaction term: gSTRaijΨi /G

aPRΨj ?

While in 4D: γµPR = PLγµ = PLγµPR, it is not so in d-D: they dier by anevanescent term.

⇒ NO unique way of extending the model to d-dimensions!⇒ Use the interaction term that makes calculations the most simple:

gSTRaijΨiPL /G

aPRΨj (symmetric) .

This choice explicitly conveys the fact that we really started with RH spinors...



Nota about charge conjugation

While it is clear how to dene the charge conjugation operation in 4D with e.g. an

explicit construction: numerically C =(εαβ 0

0 εαβ

)∼ iγ0γ2 with the good

properties,In d-D we can dene a similar operation only by its action on the fermions such

that it turns fermions to their charge-conjugate and back: ΨC = CΨT, and its

action on Dirac 4-spinor bilinears:

(ΨC)C = Ψ , CT = −C ;

ΨC

i ΓΨCj = −ΨT

i C−1ΓCΨ

T

j = ΨjCΓTC−1Ψi = ηΓΨjΓΨi ,

with: ηΓ =

+1 for Γ = 1 , γ5 , γ

µγ5 ,

−1 for Γ = γµ , σµν , σµνγ5 .

(See e.g. Appendix A of [Tsai2011].)


Outline






6 Summary



The BRST symmetry [Becchi,Rouet,Stora1975,Tyutin1975]

There remains a residual symmetry even after xing the gauge: BRST symmetry(≈ weak or generalized version of the gauge symmetry).From innitesimal gauge transfo. of elds: ϕi → δαϕi linear in the (small) gaugeparameter α, replace αa → θca, with θ: Grassmann parameter, and ca:(anticommuting) ghost.⇒ BRST transformation of the elds: δBRSTϕ = θsϕ ≡ δαϕ|αa→θca .Some consequences:

Gauge-invariant terms in the action S0 are BRST-invariant, while the allowedgauge-xing terms are only BRST-invariant and have the form: sΨ[Φ].

Any BRST-invariant term built from the existing elds can be added to S0

without spoiling physical predictions made out of it.



BRST transformations of elds of R-Model

The d-dimensional BRST transformations on the elds are as follows:

sdGaµ = Dab

µ cb = ∂µc

a + gSfabcGbµc

c ,

sdΨi = sdΨRi = icagSTRaijΨRj , sdΨi = sdΨRi = +iΨRjc

agSTRaji ,

sdΦm = icagSθamnΦn ,

sdca = −1

2gSf

abccbcc ≡ igSc2 ,

sdca = Ba , sdB

a = 0 (ca, Ba) is a BRST doublet ,

with a similar form (noted s in what follows) in 4D.The BRST operator sd is nilpotent: sd(sdφ) = 0, similarly to its 4D counterpart.



Eective action Γ: Interpretation & notation (1/2)

Eective action: Generating functional for 1-particle irreducible (1PI) Green'sfunctions [Weinberg1996]:

Γ[Φ] =∑

n≥2

1

|n|!

∫ ( n∏

i=1

d4 xi φi(xi)

)Γφn···φ1(x1, . . . , xn)

(Fourier

transform

)=∑

n≥2

1

|n|!

∫ ( n∏

i=1

d4 pi(2π)4

φi(pi)

)Γφn···φ1(p1, . . . , pn)

Momentum conservation︷︸︸︷(2π)4δ4(

∑nj=1 pj) ,

Γφn···φ1are the 1PI Green's functions dened by:

iΓφn···φ1(x1, . . . , xn) =

iδnΓ[Φ]

δφn(xn) · · · δφ1(x1)

∣∣∣∣φi=0

= 〈Ω|T[φn(xn) · · ·φ1(x1)]|Ω〉 1PI

≡ 〈φn(xn) · · ·φ1(x1)〉 1PI ,

and iΓφn···φ1(p1, . . . , pn) ≡

⟨φn(pn) · · · φ1(p1)

⟩ 1PI

is dened similarly.



Eective action Γ: Interpretation & notation (2/2)

Γ[Φ] =∑

n≥2

−i|n|!

∫ ( n∏

i=1

d4 xi φi(xi)

)〈φn(xn) · · ·φ1(x1)〉 1PI

=∑

n≥2

−i|n|!

.

Notation:O · Γ[Φ] =

∫dx O(x) · Γ[Φ] .

Field-Operator insertion in Γ[Φ] [Piguet,Rouet1981]:(e.g. counterterm insertions in loop diagrams...)

O(x) · Γ[Φ] =∑

n≥2

−i|n|!

∫ ( n∏

i=1

d4 xi φi(xi)

)〈O(x)φn(xn) · · ·φ1(x1)〉 1PI

=∑

n≥2

−i|n|!

.



BRST invariance at any loop order?

Aim: Formulation for verifying/enforcing BRST invariance ∀ orders ofperturbation.

I Reformulate BRST sym. invariance using quantum eective action Γ (up toO(~n)) → Slavnov-Taylor Identities (STI). (∼ Ward IDs with gauge transfos.)

I Employ formalism similar to Batalin-Vilkovisky: [Batalin,Vilkovisky1977,'81,'84]

∀φ, introduce in S0 external sources (antields) Kφ coupling linearly to sdφ.(When setting these sources to adequate values, S-matrix remains unchanged.)

I BRST invariance of Γ translates into a condition, the Zinn-Justin equation:(Γ,Γ) ≡ S(Γ) = 0, where S(Γ) is the Slavnov-Taylor (ST) functional:

S(Γ) ≡∫

dx

(∑

Φ

TrδΓ

δKΦ(x)

δΓ

δΦ(x)+Ba(x)

δΓ

δca(x)

).

This functional can be dened in either 4 or d dimensions (noted S and Sd resp.),that will either act on the renormalized Γren or on the dim-regularized ΓDReg resp.



The completed R-Model dening action S0 in d-D

Our complete dening action in d dimensions, including the antields, reads:

S0 =

∫dd x (LYM + LΨ + LΦ + LYuk + Lgh + Lg-x + Lext) ,

with: LYM =−1

4F aµνF

a µν , LΦ =1

2(DµΦm)2 −

λmnopH

4!ΦmΦnΦoΦp ,

LΨ ⇒ iΨi /DijRΨj = iΨi /∂Ψi + gSTR

aijΨRiPL /G

aPRΨRj ,

LYuk = −(YR)mijΦmΨRC

i PRΨRj + h.c. ,

Lgh = ∂µca ·Dab µcb , Lg-x =ξ

2BaBa +Ba∂µGaµ ,

Lext = ρµasdGaµ + ζasdc

a + RisdΨRi + sdΨRiRi + YmsdΦm .

Quantum numbers (mass dimension, ghost number and (anti)commutativity):

Gaµ Ψi, Ψi Φm ca ca Ba ρµa ζa Ri, Ri Ym ∂µ s

mass dim. 1 3/2 1 0 2 2 3 4 5/2 3 1 0

ghost # 0 0 0 1 -1 0 -1 -2 -1 -1 0 1

comm. + + + + + +



BRST invariance of the R-model @ tree-level?

The model is BRST-invariant at tree-level in 4D due to gauge symmetry:

s4S(4D)0 = 0.

Is it still so in d-dimensions? ⇒ No! ∃ BRST breaking ∆ at tree-level:

sdS0 =

∫dd x

(gS2TR/L

aij

)ca

∂µ(Ψiγ

µΨj)−+ Ψi

↔

/∂ γ5Ψj

≡ ∆ .

(R/L: results for right-handed / left-handed fermions resp.)



Notation: Normal Products N [O(x)] [Zimmermann1973]

Introduced by Zimmermann. (See also [Lowenstein1971].)

For a eld-product operator O(x), a normal product N [O(x)]is dened as the nite part of O(x) , i.e. via the nite partof the time-ordered Green's functions of O(x):

〈N [O]∏i φi(xi)〉

1PI= Fin.

(〈O∏i φi(xi)〉

1PI).

[Piguet,Rouet1981]

They depend on the chosen renormalization scheme:

I In BPHZ renormalization (original): done by subtracting the rst terms of aTaylor expansion of loop integrands up to a given order (degree ofsubtraction). → ∃ dierent normal products associated to the choice of thedegree of subtraction. [Piguet,Rouet1981]

I In dimensional renormalization (DimRen): the normal products are denedwith respect to the ε-pole subtraction. [Collins1974]



BRST restoration, Renormalized action (1/2)

Restore BRST symmetry, i.e. remove the irrelevant anomalies, if possible.The equation for renormalized 4D action, SΓren = ∆breaking, is generalizable viathe Regularized Action Principle [Breitenlohner,Maison1977] for ΓDReg:

SdΓDReg = ∆ · ΓDReg + ∆ct · ΓDReg +

∫dd x

∑

Φ

Tr [δS

(n)ct

δKΦ(x)· ΓDReg]

δΓDReg

δΦ(x),

with the relation: ∆breaking = LIMd→4

(SdΓDReg) = SΓren.

(LIM is by taking d→ 4 and cancelling the remaining evanescent parts.)At one-loop O(~), we have, using Γren = LIM

d→4(ΓDReg):

∆(1)breaking = LIM

d→4∆ · ΓDReg|(1)

sing. + bdS(1)sct +N [∆] · Γren|(1) + b4S

(1)fct .

The nite counterterm action S(1)fct is computed so that b4S

(1)fct cancels the

irrelevant anomalies from N [∆] · Γren|(1). S(1)fct is contained in Γren.



BRST restoration, Renormalized action (2/2)

SΓren(1) = ∆

(1)breaking = LIM

d→4∆ · ΓDReg|(1)


(1)fct .

Procedure:

1 Evaluate bdS(1)sct .

2 Evaluate ∆ · ΓDReg|(1)sing. by computing 1-loop diagrams with insertion of ∆.

Check whether it cancels with bdS(1)sct .

3 Evaluate N [∆] · Γren|(1) using Bonneau Identities; symbolically:

N [O] · Γren =∑

Oi

cOiN [Oi] · Γren .

4 Keep aside / cancel the relevant anomalies from N [∆] · Γren|(1), and dene

S(1)fct such that b4S

(1)fct absorbs the non-zero terms (irrelevant anomalies) from

N [∆] · Γren|(1).


Outline






6 Summary



Calculation machinery

Calculations performed with Mathematica.

Model programmed using FeynRules [Christensen...2009,Alloul...2014] (exceptwith no BRST sources since unsupported). Manually patched forsupporting arbitrary SU(N) (N not limited to numerical values).

Loop diagrams (w/o BRST sources) generated using FeynArts [Hahn2000].Amplitudes evaluated using FeynCalc [Mertig...1990,Shtabovenko...2016];ε-expansion obtained using the FeynCalc's interfaceFeynHelpers [Shtabovenko2016] to Package-X [Patel2017] (for 1-loop only).WARNING! Using development version of FeynCalc that includes needed xes(versions up to 17th June 2019 are OK).

Diagrams with sources manually generated, then evaluated using FeynCalc asdescribed above.

Semi-automated (manually and computer) evaluation of group-structureinvariants, using notations similarly dened as those in Machacek & Vaughn[Machacek,Vaughn1983,'84,'85].



1-loop singular counterterm (SCT) action

Compute 1-loop diagrams (self-energies, vertices, & sources insertions) from S0.

From them, dene the 1-loop SCT action S(1)sct = −Γ(1)|BMHV

div .

S(1)sct =

~16π2ε

g2S

22C2(G)−S2(S)

6LgS − g2

S

2S2(F )

3LgS − g2

S

3 + ξ

4C2(G)LG

−(g2SξC2(F )+

Y2(F )

2

)LΨR

2− g2

S

ξC2(G)

2Lc

− ~16π2ε

g2S

S2(F )

3

∫dd x

1

2Gaµ∂

2Gaµ +2Y2(S)

3S0

ΦΦ

⇐ Evanescent!

+

~16π2ε

1

2

g2S(3− ξ)C2(S)LDΦ − Y2(S)LDΦ +

(Y2(F )− 3g2

SC2(F ))Lψφψ

−g2SξΛ

SmnopS

0Φ4 + Y nik(Y mkl )∗Y nljS

0

ΨRCi ΦmΨRj

+ h.c.

+(3g4SAmnop − 4Hmnop + Λ2

mnop)S0Φ4

.

Lϕ: invariants under the linear BRST transformation bd, i.e. bdLφ = 0 (except for

bdLc = ∆). Dened as sums/dierences of eld-counting operators applied on S0.(see Backups for details.)Hermès Bélusca-Maïto (PMF Zagreb) γ5 in DReg χ YM Theory w/ Scalars SM & Beyond, Corfu 2019 22 / 34


Evaluation of bdS(1)sct Cancellation with ∆ · ΓDReg|(1)

sing. (1/2)

SΓren(1) = ∆

(1)breaking = LIM

d→4∆ · ΓDReg|(1)


(1)fct .

Since all the bdLφ = 0 except for bdLc = ∆ the tree-level breaking, and in

addition: bd

Y nik(Y m

kl )∗Y nljS

0

ΨRCi ΦmΨRj

+ h.c.

= 0, we obtain:

bdS(1)sct =

~16π2ε

−g2

S

ξC2(G)

2∆− g2

S

S2(F )

3bd

∫dd x

1

2Gaµ∂

2Gaµ−2Y2(S)

3bdS0

ΦΦ

+~

16π2ε

1

2bd

(3g4SAmnop − 4Hmnop + Λ2

mnop)S0Φ4

,

with:

bd

∫dd x

1

2Gaµ∂

2Gaµ =

∫dd x (∂µca + gSf

abcGbµcc)∂2Gaµ ,

bdS0ΦΦ = bd

∫dd x

−1

2Φm∂

2Φm =

∫dd x igSθ

amnc

aΦm∂2Φn .

The term in blue should cancel too (WIP!).Hermès Bélusca-Maïto (PMF Zagreb) γ5 in DReg χ YM Theory w/ Scalars SM & Beyond, Corfu 2019 23 / 34


Evaluation of bdS(1)sct Cancellation with ∆ · ΓDReg|(1)

sing. (2/2)

On the other hand we calculate the non-zero terms contributing to ∆ · ΓDReg|(1)sing.

Examples of ∆ insertion (left: does not

contribute; right: contributes for

i[∆ · ΓDRegji,a

ψψc](1)div .

Φm Φn

Gbµ Gc

ν

∆

ca

∆

ca

Φm

Ψi Ψj

Terms from bdS(1)sct×

( ~16π2ε

)−1Terms from ∆ · ΓDReg|(1)

sing.×( ~

16π2ε

)−1

−g2SS2(F )

3 bd∫

dd x 12 G

aµ∂

2Gaµ =

−g2SS2(F )

3

∫dd x (∂µca)(∂2Gaµ)

−g3SS2(F )

3

∫dd x fabcca(∂2Gbµ)Gcµ

i[∆ · ΓDRegba,µGc ]

(1)div + i[∆ · ΓDReg

cba,νµGGc ]

(1)div ⇒

ScG = g2SS2(F )

3

∫dd x (∂µca)(∂2Gaµ)

+ScGG = g3SS2(F )

3

∫dd x fabcca(∂2Gbµ)Gcµ

− 2Y2(S)3 bdS0

ΦΦ i[∆ · ΓDRegnm,aΦΦc ]

(1)div ⇒ ScΦΦ = 2Y2(S)

3 bdS0ΦΦ

−g2SξC2(G)

2 ∆ i[∆ · ΓDRegji,a

ψψc](1)div ⇒ Scψψ = g2

SξGC2(G)

2 ∆



Bonneau Identities, graphical interpretation (1/2)

In DimRen, normal products N [O] of evanescent operators O of the form

O ≡ (gµν = gµν − gµν)Oµνρ··· are interpreted [Bonneau1980] as the dierencebetween two ways of performing a subtraction in this renormalization scheme.⇒ Zimmermann-like identities: Bonneau Identities. (see Amon's talk)

N [O] · Γren = −nmax=4∑

n=2

∑

J=j1,··· ,jn,0≤r≤δ(J)

∑

i1,··· ,ir/1≤ij≤n

(−i)r

r!

∂r

∂pµ1

i1· · · ∂pµrir

· (−i~)r.s.p.

⟨n∏

i=1

φji(pi)N [ qO](q = −∑pi)

⟩ 1PI∣∣∣∣∣∣pi=0g=0

×N

1

n!

1∏

k=n

∏

α/iα=k

∂µα

φjk

· Γren + similar with additional BV sources insertions .

r.s.p.: residue of simple pole in ν = 2ε = 4−d. Overline: 1PI minimally subtracted.g ∼ g/ν, where this ν is not submitted to Laurent ν-expansion for the r.s.p..

N [O] · Γren =∑

Oi

cOiN [Oi] · Γren .

Expands evanescent operators Od on a basis of quantum4D operators of the renormalized 4D theory. − ( a

p2 )↔(bp3

).



Bonneau Identities, graphical interpretation (2/2)

N [O] · Γren =∑

Oi

cOiN [Oi] · Γren

−( ν or

(−gµν)

)·

=∑

Γi

=∑

Oi

cOi



Evaluation of N [∆] · Γren|(1)

SΓren(1) = ∆

(1)breaking = LIM

d→4∆ · ΓDReg|(1)


(1)fct .

Using Bonneau Identities. Some considered diagrams (non-exhaustive...):

etc.



Putting results together Finite counter-terms (FCT) (1/3)

SΓren(1) = ∆

(1)breaking = LIM

d→4∆ · ΓDReg|(1)

sing. + bdS(1)sct +N [∆] · Γren|(1)+b4S

(1)fct .

Find S(1)fct such that b4S

(1)fct = −N [∆] · Γren|(1) UP TO relevant anomalies that

cannot be absorbed (if so, BRST symmetry broken and model not renormalizable).First, the anomalous terms that cannot be transformed out:

~g2S

16π2

S2(F )

3

∫d4 x gSd

abcR εµνρσca(∂ρG

bµ)(∂σG

cν)

+ ~g2S

16π2

1

18

∫d4 x g2

SDabcdR εµνρσca∂σ(GbµGcνG

dρ) ,

with: dabcR = Tr[TRaTRb, TRc] ,

DabcdR = (−i)3! Tr[TRaTR

[bTRcTR

d]] =12(dabeR fecd+daceR fedb+dadeR febc) .

Cancel the unwanted gauge anomalies due to the RH fermions by e.g. introducing

left-handed fermions under suitable group representations; see e.g. the SM.




SΓren(1) = ∆

(1)breaking = LIM

d→4∆ · ΓDReg|(1)

sing. + bdS(1)sct +N [∆] · Γren|(1)+b4S

(1)fct .

Now the nite counterterms S(1)fct such that b4S

(1)fct = −N [∆] · Γren|(1):

S(1)fct = ~

g2S

16π2

∫d4 x

(−S2(F )

3

1

2Gaµ∂

2Gaµ + SGG + C2(F )(1 +ξ − 1

6)SΨΨ

−ξC2(G)

4(SRsΨ + SsΨR)

)+

~16π2

Y2(S)

3SΦΦ

+ · · · (Work In Progress!) + ~

∑

ϕ

CξϕLϕ + CξΨRLΨR + CξcLc

.

They are of order ~1.

Terms in red have arbitrary coes Cϕ because b4Lϕ = 0.




They won't of course eect in any case the RGEs we can calculate at 1-loop,but they are still there and don't appear to cancel even in the case we adaptmanually our model so as to cancel the unwanted gauge anomalies.

Therefore we expect that their presence will matter for higher-ordercalculations, e.g. for RGEs, due to 1+ - loop diagrams with insertion of nitecounterterms. Unless, they precisely cancel extra contributions that wouldappear at these higher-orders...



What about a L-Model?

How do the results modify for left-handed (LH) fermions? Two approaches:

1 Either note that PR ↔ PL, corresponding to the change γ5 ↔ −γ5, andrelated change εµνρσ ↔ −εµνρσ.

2 Or, view LH fermions in a left (L) representation of G, as being thecharge-conjugate of corresponding RH fermions that would belong to theconjugate representation of the left ones: PLΨL ≡ (PRΨR)C , andTL ↔ TR ≡ TL.

WARNING! We have not yet taken into account possible mixings between theseright-handed and left-handed fermions (in the Yukawa sector...)!


Outline






6 Summary



Towards an RGE Work In Progress! (1/2)

Suppose gauge anomalies are cancelled (e.g. through suitable eld contents).

I Problem: Cannot use straightforwardly the technique with bare ϕ's & g's, andthe Z renormalization factors in order to dene the βg and γϕ functions,because we have (non-physical) evanescent operators for which βO would haveto be dened...The Renormalization-Group Equations (RGEs) give the expansion of µ∂µΓren.

I Requirement: RGEs must be BRST-compatible, i.e.:µ∂µ(SΓren) = SΓren(µ∂µΓren) = 0 up to O(~n). It can be shown that a solutionis given by a linear combination of functionals ∂gΓren ≡ ∂Γren/∂g and NϕΓren

where Nϕ are linear combinations of eld-counting operators:

µ∂µΓren =

−

∑

g=gS ,Y,λH

βg g∂g +∑

ϕ=G,Φ,ΨR,c

γϕNϕ

· Γren .

The βg's and γϕ's coecients have the standard interpretation.



Towards an RGE Work In Progress! (2/2)

Sketch: From the Quantum Action Principle [Lowenstein1971,Piguet,Sorella1995],[Piguet,Rouet1981], express the dierential operators applied on Γren:

g∂gΓren = N [g∂g(S0 + Sct)] · Γren (for g = gS , Y, λH) ,

NϕΓren = N [Nϕ(S0 + Sct)] · Γren .

Cannot do the same for µ∂µΓren because neither S0 nor Sct depend on therenormalization scale µ, since the latter is not introduced at the Lagrangian /action level, but as a modication of the loop integration measure: µε

∫dd x .

⇒ Trick: use modied Bonneau-like IDs for µ∂µ · Γren (not shown here).Next step: re-express all the operators insertions into a basis of (independent)operator insertions, and do the same for g∂gΓren and NϕΓren:

µ∂µΓren =∑

Oi

ciOi =∑

Oi

(−∑

g

βg g × dg,i +∑

ϕ

γϕ × eϕ,i

)Oi ,

thus obtaining a system of equations for the βg's and the γϕ's to be solved.


Outline






6 Summary



Summary

Using the algorithm from [Martin,Sanchez-Ruiz1999], based on algebraicrenormalization techniques in DimReg with BMHV scheme and MS(bar)subtraction, we have revisited the renormalization of a Chiral Yang-Mills Theorywith real scalar elds, doing an explicit calculation at 1-loop. BRST symm. isrestored when possible, and gauge anomaly results are recovered as expected.

Once the d-D extension of the model is chosen, the γ5 matrix (and otherintrinsically 4D Lorentz objects) is treated unambiguously, thus making themethod clear of usage and explicitly consistent for any loop order.

Our current work in progress is to obtain RGEs using algebraic techniques andcompare the obtained results against the usual ones.

The scalars appear to just resize contributions of the gauge sector, orsupplement them with separate (possibly evanescent) terms, that do notactually break BRST since they can be reabsorbed into nite counterterms.

These results constitute rst steps for next-loop (2+ - loop) calculations.


Thank you!

Backups

The BRST bd invariants L used in S(1)sct (1/2)

L quantities, invariant under the linear BRST transformation bd, are dened:

LG = bd

∫dd x ρµaG

aµ =

(NG −Nc −NB −Nρ + 2ξ

∂

∂ξ

)S0

=

∫dd x

(2SGG + 3SG3 + 4SG4 + SΨGΨ + SΦGΦ + 2SΦGGΦ − ρµa(∂µca)

),

Lc = − bd∫

dd x ζaca = (Nc −Nζ)S0

=

∫dd x ρµasdG

aµ + ζasdc

a + RisdΨi + sdΨiRi + YmsdΦm ,

LΦ = bd

∫dd x YmΦm = (NΦ −NY)S0

=

∫dd x ((DµΦm)2 ≡ 2SΦΦ + 2SΦGΦ + 2SΦGGΦ) + 4λmnopSΦ4

mnop

+ ((YR)mijSΨRCi ΦmΨRj

+ h.c.) ,

LΨR = − bd∫

dd x (RiPRΨi + ΨiPLRi)−

∫dd x iΨi /∂Ψi = −(NR

Ψ +NLΨ−NR −NR)S0

−∫

dd x iΨi /∂Ψi = 2

∫dd x iΨi /∂PRΨi + SΨGΨ + ((YR)mijSΨR

Ci ΦmΨRj

+ h.c.) ,


The BRST bd invariants L used in S(1)sct (2/2)

In the previous calculations the eld-counting operators have been used; they aredened as:

Nϕ =

∫dd x ϕ(x)i

δ

δϕ(x)i, for ϕi = Gaµ,Φ

m, ca, ca, Ba, ρµa , ζa, R

i, Ri,Ym ,

NR/LΨ =

∫dd x (PR/LΨi(x))s

δ

δΨi(x)s, N

L/R

Ψ=

∫dd x (Ψi(x)PL/R)s

δ

δΨi(x)s.

Other bd invariants are:

the pure Yang-Mills term LgS and its equivalent 4-dimensional version LgS :

LgS =−1

4

∫dd x F aµνF

a µν = SGG + SG3 + SG4 ;

the Yukawa interaction: Lψφψ = (YR)mijSΨRCi ΦmΨRj

+ h.c. ,

the four-scalar interaction: LΦ4 = λmnopSΦ4mnop

.

Since LΦ =∫

dd x 2SΦΦ + 2SΦGΦ + 2SΦGGΦ + 4LΦ4 + Lψφψ is a bd invariant, it

follows that the combination LDΦ =∫

dd x (DµΦm)2 = LΦ − Lψφψ − 4LΦ4 isalso a bd invariant by itself.


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