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Ward identity for special conformal transformation in d dimensions

I am reading CFT from the yellow book ( "Conformal Field Theory" by Francesco, Mathieu, Sénéchal ). In section 4.3.2, they calculate three Ward identities corresponding to (i) translation ...
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Ward Identities in Conformal Theory

For a 2D free boson model, $$ S=\frac{1}{2} g \int d^2 x \; \partial_\mu \varphi\;\partial^\mu \varphi $$ The energy-momentum tensor should be $$ T_{\mu \nu}=g\left(\partial_\mu \varphi \partial_\nu \...
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Charge Renormalization in Abelian Gauge Theory under General Gauge Fixing Conditions

In scalar QED or fermionic QED, the relationship between bare quantities (subscript "B") and renormalized quantities is given by $$ \begin{aligned} A^\mu_B &= \sqrt{Z_A} A^\mu\,, \quad \...
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$Z_1=Z_2$ and its relation to vertex renormalization in QED

I have been working on the full renormalization of scalar QED with self-interactions, following the steps of Schwartz’s treatment on spinor QED (Chap 19). I have 3 main questions regarding this: Need ...
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Renormalization in Background field gauge

The purpose of this question is not very straightforward to explain. So, I just state the question. If we use Background field gauge for renormalization, due to the QED-like Ward identities, the ...
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N=4 Supersymmetric Ward identity

(This question pertains to exercise 4.13 of Elvang and Huang's textbook (which used to be lecture notes). This is not for a class, just to learn some new tools for work). Consider the expansion of the ...
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Ward identity in scalar QED; gauge transformations & plane wave solutions for polarization

I am prepping for my QFT2 exam tomorrow, and in one of the mock exams I found the following question (and I'm not quite sure how to go about this). Given the following Lagrangian: $$ L = -\frac{1}{4}...
Sophie Schot's user avatar
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How should I calculate the commutator between the Belinfante stress tensor and the field operator?

As known, there is an ambiguity on the definition of the stress tensor (or energy-momentum tensor). The canonical stress tensor, defined as the Noether's current corresponding to space-time ...
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Confusion about the derivation of stress tensor OPE from Ward Identity

I apologize for any difficulty in expressing my review. Allow me to briefly summarize the material and then pose my question. Review In David Tong's string lecture note, he derives the OPE between ...
Steven Chang's user avatar
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Three-point function general form

The general matrix element of the electromagnetic current can be written as $$iM^ {\mu} = \left< p^\prime,s^\prime | j^\mu (x) | p,s\right>=\bar{u}(p^\prime,s^\prime)\left( R(q^2) \gamma^{\mu} + ...
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Derivation of the Conformal Ward Identity in Di Francesco et al

I am reading section 5.2.2. (titled The Conformal Ward Identity) from Conformal Field Theory by Di Francesco et al. The authors write \begin{align} \partial_\mu(\epsilon_\nu T^{\mu\nu}) &= \...
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Consequences for symmetries of the equations of motion in QFT

In general, if a Quantum Field Theory is described by a Lagrangian $\mathcal{L}$, the symmetries of $\mathcal{L}$ lead to classically conserved currents along the equations of motion and Ward ...
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Ward identity of correlation function

For local observables $\{O_i(x_i)\}^n_{i = 1}$, one defines the Ward identity as $$\partial_{\mu}\langle j^{\mu}(x)\prod^n_{i = 1}O(x_i)\rangle = \sum^n_{i = 1}\delta(x-x_i)\langle O_1(x_1)\cdots\...
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Confusion regarding time ordering in the Ward Identity derivation

This question does not follow from reading any text, but I was watching Shiraz Minwalla's CFT lectures on YouTube. At 51:36 into the lecture, he raises the question that, as an operator statement, we ...
QFTheorist's user avatar
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Contact terms in Schwinger-Dyson equation and Ward-identity

I am reading Weigand's notes for the derivation of Ward-identity. The Second last paragraph on page 133, says the following statement "The Schwinger-Dyson equation and the Ward-identity show ...
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Derivation of Ward identity

Can we derive Ward identity from Noether current of a Lagrangian density, under the assumption that path integral measure is invariant? Suppose that $\delta\psi(x) = \frac{d\hat{\psi}(x, \epsilon)}{d\...
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Meaning of a Feynman diagram in proof of Ward-Takahashi identity in chapter 7 of Peskin and Schroeder

I'm trying to understand what the external photon in this diagram (page 238 in P&S) corresponds to exactly. This diagram is supposed to be a contribution to the Fourier transform of a QED ...
Function's user avatar
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What are the conformal Ward identities associated with the correlation functions of the stress-energy tensor?

I found two papers on this matter, but am having trouble parsing the answer from either of them. https://arxiv.org/abs/1911.05359 https://arxiv.org/abs/2108.06767 For that matter, what even are the ...
Logan J. Fisher's user avatar
4 votes
2 answers
204 views

Propagator and Ward identity in the $R_\xi$ gauge

The full gauge propagator in the $R_\xi$ gauge is $$D_{\mu\nu} = \frac{i}{k^2+i\epsilon}\left(-g_{\mu\nu}+\frac{1-\xi}{k^2}k_\mu k_\nu\right).\tag{1}$$ Now if we take $\xi=0$, we get the Lorenz gauge, ...
Mohamed Ahmed's user avatar
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1 answer
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Different version of conformal Ward identity

In the book by Di Franceso, Mathieu, Senechal, equation (5.46) shows that (assuming $\bar\epsilon = 0$) $$ (*) \qquad \langle \delta_{\epsilon, 0} \mathcal{O}\rangle = - \oint_\infty \frac{dz}{2\pi i ...
user060606's user avatar
2 votes
1 answer
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Charge renormalization using Ward identity

In Mandl & Shaw's Quantum Field Theory (p 181), the Ward identity $$\frac{d\Sigma(p)}{dp_\mu} = \Lambda^\mu(p,p)\tag{9.60}$$ where $\Sigma(p)$ and $\Lambda^\mu (p^\prime,p)$ are respectively the ...
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Gauge invariance and Ward Identity?

I have to work on vacuum polarization and gauge contributions for a given problem. I have to compute and show that their sum is gauge invariant, which according to the exercise, is equivalent to ...
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Diagrammatic Ward Identity for the QED vertex

The QED Ward identity for the vertex reads \begin{equation} q^\mu\Gamma^\mu(p,p')=\Sigma(p)-\Sigma(p') \end{equation} with $q=p-p'$. In the limit $q\rightarrow 0$, \begin{equation} \Gamma^\mu(p,p)=\...
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Faddeev-Popov trick in QED Peskin and Schroeder

On page 297 of Peskin and Schroeder, the book obtains the propogator $$\tag{9.58} \tilde{D}_F^{\mu\nu}(k)=\frac{-i}{k^2+i\epsilon}\bigg(g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2}\bigg).$$ The book then ...
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Conformal Ward Identity (Di Francesco et al)

In the yellow pages (Conformal Field Theory, Di Francesco, Mathieu, Sénéchal), the authors derive the conformal Ward identity in the following way: They show that, for a conformal transformation, $$ \...
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How to use Ward identity to abbreviate the photon propagator into $\frac{-i\ g_{\mu\nu}}{q^2 (1- \Pi(q^2))}$?

How to derive abbreviated form (equation 7.75) from original form (equation 7.74) via Ward identity? (In Peskin's QFT Charpter 7 P246) I still can't see this result after read this paragraph many ...
a Fish in Dirac Sea's user avatar
1 vote
1 answer
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Ward Identity formulated by Tetrad

Let us consider a QFT on a curved Riemannian manifold and use the following definition for the stress tensor: \begin{eqnarray} \delta\langle\cdots\rangle_e=-\int e\delta e^a_\mu\langle T^\mu{}_a\cdots\...
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Ward Identity, Green Function with Current Insertion and Amputated Green Function

The QED Lagrangian (for one fermion) is: $$\mathcal{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} + \bar{\psi}\left(i \gamma^{\mu}\partial_{\mu} - m \right) \psi - q \bar{\psi} \gamma^{\mu}A_{\mu}\psi = \...
Aleph12345's user avatar
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Does the Ward-Takahashi identity work for a virtual photon?

In the reaction of the transformation of an electron pair into a muon pair is performed Ward–Takahashi identity. Why if there is no interaction with a real photon?
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Unphysical gauge boson polarizations in non-abelian gauge theory

I am learning P&S's chapter 16 , quantization of non-abelian gauge theory.(all my ref. formula in this post lie in P&S's book) I am puzzled for the logic of the unphysical polarization of ...
Daren's user avatar
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Do Ward identities imply that there is an (effective) Lagrangian invariant under the symmetry?

In usual perturbative QFT, if the UV Lagrangian is invariant under a symmetry $G$ and the regularization of the path integral does not break $G$, the Feynman rules are explicitly invariant under $G$. ...
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QED gauge invariance

On P & S page.297, in the second paragraph from bottom, the book discussed gauge invariance of Faddeev-Popov procedure, following a QED example. Where the photon propagator is: $$ \widetilde{D}_F^{...
Daren's user avatar
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Conformal invariance in 2d and correlation functions

It is well-known that 2d global conformal invariance constrains the 2, 3-point functions to some very simple form, and 4-point function must be $$ f(\eta, \bar \eta) \prod_{i < j}z_{ij}^{...} \bar ...
user31415926's user avatar
4 votes
1 answer
394 views

Ward-Takahashi Identity in QED

P&S write in Section 7.4 on page 238: We will prove the Ward-Takahashi identity order by order in $\alpha$……The identity is generally not true for individual Feynman diagrams; we must sum over ...
Daren's user avatar
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Conformal Ward identity minus sign error

I'm trying to track down what seems like a fairly crucial minus-sign error in Di Francesco et al's conformal field theory book. The minus sign has to do with the Ward identity for Lorentz ...
Zack's user avatar
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The presence of $\zeta^{\mu}(k)$ in the Ward-Takahashi identities in QED

On page 132 of Timo Weigand's QFT notes we introduce the Ward-Takahashi identity for QED, this is the statement that: $$k^{\mu}\mathcal M_{\mu}(k)=0 \tag{5.35},$$ with $$\mathcal M(k)=\zeta^{\mu}(k)\...
Charlie's user avatar
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1 answer
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Gauge-invariant vertex structure for $h\to\gamma\gamma$ via fermion loop

I am struggling (a bit) with the following diagram for scalar Higgs to two photons. $h\to\gamma\gamma$" /> If I put $q_\mu$ on-shell (or at the very least if I put both $q_\mu$ and $q'_\nu$ on-shell), ...
KilianM's user avatar
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1 answer
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On the Ward Identity in QED

I am reading P&S, particularly Chapter 5.5. The authors are trying to derive an expression for the Ward identity (not formally, but still). They claim that the amplitude describing a photon ...
schris38's user avatar
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2 votes
1 answer
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Why is $\mathcal{M}(k)$ given by this? (Ward Identity derivation in Peskin & Schroeder)

In page 160 of Peskin & Schroeder we are considering an amplitude $\mathcal{M}(k)$ with an external photon as given in equation (5.77) $$ \sum_{\epsilon}|\epsilon_\mu^*(k)\mathcal{M}^\mu(k)|^2=|\...
twisted manifold's user avatar
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Explicit check of Ward identity (Peskin & Schroeder p. 160)

I am trying to check explicitly that the (Compton) amplitude $$i\mathcal{M} = -ie^2\epsilon^*_\mu(k’)\epsilon_\nu(k)\bar u(p’)\left[\frac{\gamma^\mu \not k\gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k}+\...
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Symmetry implies Ward identity

I am thinking about symmetries and that their "quantum" consequences are Ward identities of the form $$<\beta|[Q,S]|\alpha>=0,$$ where $Q$ is the conserved charge associated with the ...
schris38's user avatar
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Is the 1PI self-energy of a massive photon transverse? EDIT: Upsetting consequences for the photon mass and renormalizability

Suppose we had the Lagrangian: $$\mathcal{L} = -\frac{1}{4} F^{\mu \nu}F_{\mu \nu} + \overline{\psi} (i \gamma^{\mu}\partial_{\mu} -m)\psi -e\overline{\psi} \gamma_{\mu} \psi A^{\mu} +\frac{1}{2} m_{\...
OutrageousKangaroo's user avatar
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1 answer
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Symmetric stress-energy tensor in CFT

I'm a bit confused reading about the stress-energy tensor and conformal Ward identities in Di Francesco. My question is in a similar spirit to this one from several years ago, but the question was not ...
Zack's user avatar
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2 votes
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On-shell propagator to an off-shell propagator

I am learning the Ward-Takahashi Identity part of Peskin and Schroeder's textbook of quantum field theory. In the prove process, it involves a diagram 7.66. Then it says that I can understand ...
David Shaw's user avatar
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PCAC - Ward Identity for non-conserved currents - Derivative and $T$-Order Commutation

I'm currently studying Goldberger-Treiman relation from the book by S. Coleman (Aspects of Symmetry, chapter 2) in which, working in the framework of a not better precised "weak interaction ...
Boreanaz's user avatar
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Why the matrix element grows with energy in this case?

Consider the QED with massive photons: $$ \mathcal{L} = g\bar{\psi}\gamma^{\mu}\psi A_{\mu}, \quad g = \text{const} $$ Because of the current conservation, the contribution of the longitudinal ...
Name YYY's user avatar
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Conformal Ward identities for local conformal algebra: error in textbook?

In Schottenloher's mathematically-oriented CFT textbook, "A Mathematical Introduction to Conformal Field Theory," Proposition 9.8 on page 160 states the conformal Ward identities for 2D CFTs ...
Daniel Ranard's user avatar
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1 answer
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Is gauge invariance necessary to have Ward identity hold for off-shell amplitudes?

In this other SE post: Is it really proper to say Ward identity is a consequence of gauge invariance? it is shown that the on-shell Ward identity is a consequence of global $U(1)$ symmetry for QED. ...
OutrageousKangaroo's user avatar
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A Question about Conformal Invariance in String Theory

I've been revisiting my lecture notes on string theory. There's one question about conformal invariance suddenly popped up in my head which made me very confused. Starting from Polyakov action $$S[X,g]...
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Do the Ward identities contain contact terms in Euclidean QFT?

In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to ...
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