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\begin{document}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Holomorphy}
\setcounter{equation}{0}
\setcounter{footnote}{0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Non-renormalization theorems}
\label{Chap7:holo:sec:nonrenorm}

 Consider 
\beq
W_{\rm tree} = {{m}\over{2}} \phi^2 + {\lambda\over{3}}\phi^3 
\eeq
$\phi$ is a chiral superfield;  scalar component
$\phi$, fermion component by $\psi$.

$R$-charge 
\beq
[R,Q_\alpha]=-Q_\alpha
\eeq
 $R[\psi] = R[\phi] -1$, $R[\theta] = 1$
 \newline
 Lagrangian in  toy model has Yukawa coupling
\beq
{\mathcal L} \supset \frac{\lambda}{3} \phi \psi \psi
\eeq
which must have zero $R$-charge, so
\beq
3R[\phi] -2=0
\eeq
therefore $R[W]=2$, or ${\mathcal L}_{\rm int} = \int d^2 \theta \, W$

\myhead{Toy Model}

\beq
\matrix{\quad& U(1)& \times & U(1)_R \cr
\phi & 1 && 1 \cr
m & - 2 && 0 \cr \lambda & -3 && -1 \cr}
\eeq
 treat the mass and coupling as background spurion
fields

integrate out
modes from $\Lambda$ down to $\mu$, then the symmetries and 
holomorphy of the effective superpotential restrict
it to be of the form
\beq
 W_{\rm eff}
= m \phi^2 \, h\left( \frac{\lambda \phi}{m} \right) 
= \sum_n a_n \lambda^n m^{1-n} \phi^{n+2}~,
\eeq
 weak coupling limit $\lambda \rightarrow 0$ restricts $n \ge 0$\newline 
the massless limit $m  \rightarrow 0$ restricts $n \le 1$ so


\beq
W_{\rm eff}= {{m}\over{2}} \phi^2 +  \frac{\lambda}{3} \phi^3 = W_{\rm tree}
\eeq
superpotential is not renormalized


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Wavefunction renormalization}
\label{Chap7:holo:sec:wavefunction}


\beq
{\mathcal L}_{\rm kin.} = Z \partial_\mu \phi^* \partial^\mu \phi +i Z \overline{\psi}
\overline{\sigma}^\mu \partial_\mu \psi
\eeq
$Z$ is a non-holomorphic function
\beq
Z=Z(m,\lambda,m^\dagger,\lambda^\dagger,\mu,\Lambda)
\eeq
If we integrate out modes down to $\mu > m$ at one-loop order
\beq
Z=  1 + c \lambda \lambda^\dagger \ln\left({{\Lambda^2}\over{\mu^2}}\right)
\eeq
where $c$ is a constant determined by the  perturbative calculation.
If we integrate out modes down to scales below $m$ we have
\beq
Z=  1+ 
  c \lambda \lambda^\dagger \ln\left({{\Lambda^2}\over{m m^\dagger}}\right)
\eeq
Wavefunction renormalization means that  couplings of canonically
normalized fields run
\newline  running mass and running
coupling are given by
\beq
{{m}\over{Z}}, {{\lambda}\over{Z^{{3}\over{2}}}}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Integrating out}
\label{Chap7:holo:sec:integratingout}

Consider a model with two different chiral superfields:
\beq
W = {{1}\over{2}} M \phi_H^2 +{{\lambda}\over{2}} \phi_H \phi^2
\label{Chap7:holo:3:superpot}
\eeq
three global $U(1)$ symmetries:
\beq
\begin{array}{cccc}
 & U(1)_A & U(1)_B & U(1)_R \\
 \phi_H & 1 & 0 & 1 \\
 \phi & 0 & 1 & {{1}\over{2}} \\
 M & -2 & 0 & 0 \\
 \lambda & -1 & -2 & 0 \\
 \end{array}
\eeq
where  $U(1)_A$ and $U(1)_B$ are spurious symmetries for $M$, $\lambda \ne 0$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Integrating out}
If we want to integrate out modes down to $\mu < M$, we must 
integrate out $\phi_H$.
An arbitrary term in the effective superpotential has the form 
\beq
\phi^j M^k \lambda^p 
\eeq
To preserve the symmetries we must have $j=4$, $k=-1$, and  $p=2$.
By comparing with tree-level perturbation theory we can determine
the coefficient:
\beq
W_{\rm eff} = -{{\lambda^2 \phi^4}\over{8 M}}
\label{Chap7:holo:3:supereff}
\eeq
algebraic equation of motion:
\beq
{{\partial W}\over{\phi_H}}=M \phi_H +{{\lambda}\over{2}}  \phi^2= 0
\eeq
solve this equation for $\phi_H$ and plug the result back into
the superpotential
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Another Example}
\beq
W = {{1}\over{2}} M \phi_H^2 +{{\lambda}\over{2}} \phi_H \phi^2 +{{y}\over{6}} \phi_H^3
\eeq
 $\phi_H$  equation of motion:
\beq
\phi_H=-\frac{M}{y}\left(1 \pm \sqrt{1-\frac{ \lambda y \phi^2}{ M^2} }\right)
\eeq
as $y \rightarrow 0$, the two vacua approach $\phi_H=- \lambda \phi /(2 M)$ (as in
previous example) and  $\phi_H=\infty$.
 Integrating out  $\phi_H$ yields
\beq
W_{\rm eff}= {{M^3}\over{3 y^2}}
\left(1-\frac{ 3 \lambda y \phi^2}{2 M^2}
\pm \left(1-\frac{ \lambda y \phi^2}{ M^2}\right) 
\sqrt{1-\frac{ \lambda y \phi^2}{ M^2}}\right)
\eeq
 singularities in $W_{\rm eff}$ indicate points in the parameter space
and the space of $\phi$ VEVs where $\phi_H$ becomes
massless and we should not have integrated it out

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Singularities}

The mass of $\phi_H$ can be found by calculating
\beq
\frac{\partial^2 W}{\partial \phi_H^2}= M+y \phi_H
\eeq
and substituting in the solution for$\phi_H$:
\beq
\frac{\partial^2 W}{\partial \phi_H^2}= \mp M  \sqrt{1-\frac{ \lambda y \phi^2}{ M^2} }
\eeq
Using  holomorphy   assign y  charges (-3,0,-1)
under $U(1)_A \times U(1)_B \times U(1)_R$ then
\beq
W_{\rm eff}=\frac{M^3}{y^2}  f\left(\frac{  \lambda y \phi^2}{ M^2}\right)
\eeq
for some function $f$, just as we found from explicitly integrating out $\phi_H$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The holomorphic gauge coupling}
\label{Chap7:holo:sec:gaugecoupling}

chiral superfield for
an $SU(N)$ gauge supermultiplet:
\beq
W^a_\alpha = -i\lambda^a_\alpha(y) +\theta_\alpha D^a(y) 
-(\sigma^{\mu \nu} \theta)_\alpha F^a_{\mu\nu}(y)
-(\theta \theta)\sigma^\mu D_\mu \lambda^{a\dagger}(y)~,
\eeq
 $a=1,\ldots,N^2-1$


\beq
\tau \equiv {{\theta_{\rm YM}}\over{2 \pi}} +{{4 \pi i}\over{g^2}}~,
\eeq
SUSY Yang--Mills action as a superpotential term
\beq
&&\frac{1}{16 \pi i} \int d^4 x \int d^2 \theta \, \tau \, W^a_\alpha
W^a_\alpha + h.c. =\\
&&\int d^4 x\left[ -\frac{1}{4 g^2} F^{a \mu \nu}F^a_{\mu\nu} 
-{{\theta_{\rm YM}}\over{32\pi^2}}
F^{a\mu\nu}{\wtilde F}^a_{\mu\nu} +{{i}\over{g^2}} \lambda^{a\dagger} 
\overline{\sigma}^\mu D_\mu \lambda^a +{{1}\over{2 g^2}}D^a D^a \right] \nonumber
\label{Chap7:holo:SYMaction}
\eeq
 $g$ only  in $\tau$ which is 
a holomorphic parameter, but  gauge fields are
not canonically normalized

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Running coupling}
one-loop running  $g$ is
given by the RG equation:
\beq
\mu \frac{d g}{d \mu} = - \frac{b}{16 \pi^2} g^3 
\eeq
where for an $SU(N)$ gauge theory with $F$ flavors and ${\mathcal N}=1$ 
SUSY, 
\beq
b=3N-F
\eeq
The solution for the running coupling is
\beq
\frac{1}{g^2(\mu)} = - \frac{b}{8 \pi^2} \ln \left( \frac{|\Lambda|}{\mu}
\right)
\eeq
where $|\Lambda|$ is the intrinsic scale of the non-Abelian gauge theory

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Holomorphic Intrinsic Scale}

\beq
\tau_{1-{\rm loop}} &=& {{\theta_{\rm YM}}\over{2 \pi}} +{{4 \pi i}\over{g^2(\mu)}}\\
&=& \frac{1}{2 \pi i} \ln \left[ \left(\frac{|\Lambda|}{\mu}\right)^b
 e^{i \theta_{\rm YM}} \right]
\eeq

\beq
\Lambda &\equiv& |\Lambda| e^{i \theta_{\rm YM}/b}\\
&=& \mu e^{2 \pi i \tau/b}
\label{Chap7:holo:holomorphicscale}
\eeq
\beq
\tau_{1-{\rm loop}} = \frac{b}{2 \pi i} \ln \left(\frac{\Lambda}{\mu}\right)
\label{Chap7:holo:taupert}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{CP Violating Term}

\beq
 F^{a\mu\nu}{\wtilde F}^a_{\mu\nu} = 4 \epsilon^{\mu \nu \rho \sigma}
\partial_\mu {\rm Tr} \left( A_\nu \partial_\rho A_\sigma + \frac{2}{3}
A_\nu A_\rho A_\sigma \right) 
\eeq
total derivative: no effect
in perturbation theory 
\newline nonperturbative
effects:  instantons have a nontrivial,
topological
winding number, $n$
\beq
\frac{\theta_{\rm YM}}{32\pi^2} \int d^4x \, 
F^{a\mu\nu}{\wtilde F}^a_{\mu\nu} = n\, \theta_{\rm YM}~.
\eeq

Since the path integral has the form
\beq
\int {\mathcal D} A^a {\mathcal D} \lambda^a {\mathcal D} D^a \, e^{i S}
\eeq
\beq
\theta_{\rm YM} \rightarrow \theta_{\rm YM} + 2 \pi
\label{Chap7:thetashift}
\eeq
is a symmetry of the theory 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Instanton Action}

The Euclidean action of an instanton configuration can be bounded
\beq
0 &\le& \int d^4x Tr\left( F_{\mu \nu} \pm {\wtilde F}_{\mu\nu}\right)^2
= \int d^4x Tr\left(2  F^2 \pm 2 F {\wtilde F}\right)
\eeq

\beq
 \int d^4x Tr  F^2 \ge |\int d^4x Tr F {\wtilde F} | = 16 \pi^2 |n|
\eeq
 one instanton effects are
suppressed by
\beq
e^{-S_{\rm int}} = e^{-({8 \pi^2}/{g^2(\mu)}) + i \theta_{\rm YM} } = 
\left(\frac{\Lambda}{\mu}\right)^b
\label{Chap7:one-instanton}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Effective Superpotential}


 integrate down to the scale $\mu$ 
\beq
W_{\rm eff}={{\tau(\Lambda;\mu)}\over{16\pi i}} W^a_\alpha W^a_\alpha
\eeq
physics  periodic in $\theta_{\rm YM}$ equivalent to 
\beq
\Lambda \rightarrow e^{2 \pi i/b} \Lambda~
\eeq
in general:
\beq
\tau(\Lambda;\mu)
 = {{b}\over{2 \pi i}}\ln\left( {{\Lambda}\over{\mu}}\right) +
f( \Lambda;\mu)~,
\eeq
where $f$  has Taylor series
representation in positive powers of $\Lambda$.  
\beq
\Lambda \rightarrow e^{2 \pi i/b} \Lambda
\eeq
in perturbative term  shifts $\theta_{\rm YM}$ by $2 \pi$,
$f$ must be invariant under this transformation, so the Taylor series
must be in positive powers of $\Lambda^b$

\myhead{Effective Superpotential}

in general, we can
write:
\beq
\tau(\Lambda;\mu) = 
{{b}\over{2 \pi i}}\ln\left( {{\Lambda}\over{\mu}}\right) 
+ \sum_{n=1}^\infty a_n
\left( {{\Lambda}\over{\mu}}\right)^{b n}.
\label{Chap7:holo:taunonpert}
\eeq
holomorphic gauge coupling only receives one-loop corrections and
nonperturbative $n$-instanton corrections,  no perturbative running beyond one-loop


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Symmetry of $SU(N)$ SUSY YM}
\label{Chap7:holo:sec:gaugino}



$U(1)_R$ symmetry is broken by instantons \newline
 anomaly index: gaugino $R$-charge times $T(Ad)$
 
Because of the anomaly, the
chiral rotation
\beq
\lambda^a \rightarrow e^{i\alpha} \lambda^a
\label{Chap7:holo:u1R}
\eeq
is equivalent to  shift 
\beq
\theta_{\rm YM}\rightarrow \theta_{\rm YM} - 
2N \alpha 
\eeq
 $2N$  because
the gaugino $\lambda^a$ is in the adjoint representation, $2N$ zero modes in instanton background
\newline chiral rotation is only a symmetry
when
\beq
\alpha={{k\pi}\over{N}}
\eeq
 $U(1)_R$  explicitly broken  to 
discrete $Z_{2N}$ subgroup

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Spurion Analysis}
Treat  $\tau$ as a spurion
chiral superfield, define  spurious
symmetry
\beq
\lambda^a \rightarrow e^{i\alpha} \lambda^a
~,\,\,\,
\tau \rightarrow \tau + \frac{N \alpha}{\pi}
\label{Chap7:holo:spuriousR}
\eeq


Assuming that SUSY YM has no massless particles, then holomorphy and symmetries
determine the effective superpotential to be:
\beq
W_{\rm eff}&=& a \mu^3 e^{{2\pi i \tau}/{N}}
\label{Chap7:holo:Weffgaugino}
\eeq
This is the unique form because under the spurious $U(1)_R$ rotation the superpotential 
(which has $R$-charge 2) transforms as
\beq
W_{\rm eff} \rightarrow e^{2 i\alpha} W_{\rm eff}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino condensation}
\label{Chap7:holo:sec:gaugino}


 treat $\tau$ as a background chiral superfield, the ${\mathcal F}$ component
of $\tau$ (${\mathcal F}_\tau$) acts as a source
for $\lambda^a \lambda^a$  
\newline gaugino condensate  given by
\beq
\langle \lambda^a \lambda^a \rangle &=& 16 \pi i {{\partial}\over{\partial F_\tau}}
\ln Z
= 16 \pi i {{\partial}\over{\partial F_\tau}} \int d^2 \theta W_{\rm eff}\nonumber\\
&=& 16 \pi i {{\partial}\over{\partial \tau}}  W_{\rm eff}
= 16 \pi i {{2\pi i}\over{N}} a \mu^3 e^{{2\pi i \tau}/{N}}
\eeq
Drop nonperturbative corrections to  running, plug in  $b=3N$:
\beq
\langle \lambda^a \lambda^a \rangle 
&=& -{{32 \pi^2}\over{N}}   a \Lambda^3 
\label{Chap7:holo:gauginocondensate}
\eeq
vacuum does not respect
the discrete $Z_N$ symmetry since
\beq
\langle \lambda^a \lambda^a \rangle \rightarrow e^{2 i \alpha}
\langle \lambda^a \lambda^a \rangle~
\eeq
only invariant for $k=0$ or $k=N$
\newline $Z_{2N}\rightarrow Z_2$, implies $N$
degenerate vacua
\newline $\theta_{\rm YM}\rightarrow \theta_{\rm YM} +2 \pi$
sweeps out $N$ different values for $\langle \lambda^a \lambda^a \rangle$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{NSVZ revisited}
\label{Chap7:holo:sec:NSVZ}

Three seemingly contradictory statements:
\begin{itemize}
\item{} the SUSY gauge coupling runs only at  one-loop
\beq
\beta(g) = - {{g^3}\over{16 \pi^2}} \left(3 T(Ad) -\sum_j T(r_j)\right)
\eeq
with matter chiral superfields $Q_j$ in representations $r_j$
\item{}  the ``exact" $\beta$ function is
\beq
\beta(g) = - {{g^3}\over{16 \pi^2}} 
{{\left(3 T(Ad) -\sum_j T(r_j)(1-\gamma_j)\right)}\over{1-T(Ad){{g^2}/{8\pi^2}} }}~
\eeq
\item{} one- and two-loop terms in  $\beta$ function are scheme independent
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Changing renormalization schemes}

\beq
g^\prime = g + a g^3 + {\mathcal O}(g^5)~
\eeq
If  $\beta$ function is given by
\beq
\beta(g) = b_1 g^3 +b_2 g^5 + {\mathcal O}(g^7)~
\eeq
then
\beq
\beta^\prime(g^\prime) = \beta(g) {{\partial g}\over{\partial g^\prime}}
= b_1 g^{\prime 3} + b_2 g^{\prime 5} + {\mathcal O}(g^{\prime 7})~
\eeq
dependence on $a$ only appears at higher order

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Holomorphic vs Canonical Coupling}
\beq
{\mathcal L}_h = {{1}\over{4}} \int d^2 \theta {{1}\over{g_h^2}} W^a(V_h) W^a(V_h)+h.c.,
\eeq

\beq
 {{1}\over{g_h^2}} &=& {{1}\over{g^2}} -i{{\theta_{\rm YM}}\over{8 \pi^2}} 
 ={{\tau}\over{4 \pi i}}~,\\
 V_h&=&(A^a_\mu,\lambda^a, D^a).
\eeq
canonical gauge coupling for canonically normalized fields:
\beq
{\mathcal L}_c = {{1}\over{4}} \int d^2 \theta
\left( {{1}\over{g_c^2}}-i{{\theta_{\rm YM}}\over{8 \pi^2}} \right) 
W^a(g_c V_c) W^a(g_c V_c)+h.c.
\eeq
not equivalent under  $V_h = g_c V_c$
 because of rescaling anomaly\newline  with matter fields $Q_j$, additional  rescaling anomaly from:
\beq
Q^\prime_j = Z_j(\mu,\mu^\prime)^{1/2} Q_j~
\eeq

 rescaling anomaly completely determined by the axial anomaly



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Rescaling Anomaly}
for fermions with a rescaling $Z=e^{2i\alpha}$.
We can rewrite the axial anomaly in a manifestly supersymmetric form using the path integral measure as
\beq
{\mathcal D}(e^{i\alpha}Q) {\mathcal D}(e^{-i\alpha}{\overline Q})&=&
{\mathcal D}Q {\mathcal D}{\overline Q} \\
&&\times \exp\left(
{{-i}\over{4}} \int d^2 \theta
\left( {{ T(r_j)}\over{8 \pi^2}} 2 i\alpha\right) 
W^a W^a+h.c.\right)~ \nonumber
\eeq
take $\alpha$ to be complex gives general case:
\beq
{\mathcal D}(Z_j^{{1}/{2}} Q_j) {\mathcal D}(Z_j^{{1}/{2}}  {\overline Q_j})&=&
{\mathcal D}Q_j {\mathcal D}{\overline Q_j} \\
&&\times \exp\left(
{{-i}\over{4}} \int d^2 \theta
\left( {{ T(r_j)}\over{8 \pi^2}} \ln Z_j \right) 
W^a W^a+h.c.\right)~ \nonumber
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Rescaling Anomaly}
 for the gauge fields (gauginos) taking $Z_\lambda=g_c^2$
\beq
&&{\mathcal D}(g_c V_c) \\=
&&{\mathcal D}V_c \times \exp\left(
{{-i}\over{4}} \int d^2 \theta
\left( {{2 T(Ad)}\over{8 \pi^2}} \ln(g_c) \right) 
W^a(g_c V_c) W^a(g_c V_c)+h.c.\right)~\nonumber 
\eeq
Thus, for pure SUSY Yang--Mills we have
\beq
Z&=&\int {\mathcal D}V_h \exp\left(
{{i}\over{4}} \int d^2 \theta  {{1}\over{g_h^2}}
W^a(V_h) W^a(V_h)+h.c.\right)\\
&=&\int {\mathcal D}V_c \exp\left(
{{i}\over{4}} \int d^2 \theta \left( {{1}\over{g_h^2}}-
{{2 T(Ad)}\over{8 \pi^2}} \ln(g_c) \right) 
W^a(g_c V_c) W^a(g_c V_c)+h.c.\right)~ \nonumber
\eeq
So
\beq
{{1}\over{g_c^2}}= {\rm Re}\left({{1}\over{g_h^2}}\right)-
{{2 T(Ad)}\over{8 \pi^2}} \ln(g_c) ~
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Rescaling Anomaly}

 including the matter fields:
\beq
{{1}\over{g_c^2}}= {\rm Re}\left({{1}\over{g_h^2}}\right)-
{{2 T(Ad)}\over{8 \pi^2}} \ln(g_c) -
\sum_j{{ T(r_j)}\over{8 \pi^2}} \ln(Z_j)
\eeq
Differentiating with respect to $\ln \mu$, this leads precisely to 
\beq
\beta(g) = - {{g^3}\over{16 \pi^2}} 
{{\left(3 T(Ad) -\sum_j T(r_j)(1-\gamma_j)\right)}\over{1-T(Ad){{g^2}/{8\pi^2}} }}~
\eeq
relation between the two couplings is logarithmic, one
cannot be expanded in a Taylor series around zero in the other 


\end{document}


