Return to Answer

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact as in my answer was not enough without the explicit technical details of the Longhitano magic hat trick in your implicit hidden question, which I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine but esoteric reparameterization to the exponential Gürsey realization that has been the trusty side-knife of some of us. So I'll archive the explicit details here for possible utility to nitpickers.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant Higgs is $\sigma$, soon to be frozen to decoupling rigidity by taking its mass to infinity.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value of hypercharge, i.e. -1, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^- \end{pmatrix} , $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi ~,$$ hence $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi ~.$$

Now, the celebrated Higgs matrix is defined as a juxtaposition of these two left-doublets/columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$

It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard $\sigma$-model limit, rotate the definition of the three Goldstone variables a bit, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ and normalize $M$ to a unitary one, $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}\frac{1}{v}= \mathbb {1}\sqrt{1-\varpi^2/v^2} + i\frac{\vec{\varpi}}{v}\cdot \vec \tau ~ . $$

Finally, to ward off loss of mind, we change goldston variables to parallel ones in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} , $$ the chiral Goldstone boson matrix with the striking weak hypercharge property discussed.

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact (as in my answer above) was not enough, without the explicit technical details of the Longhitano magic-hat trick in your implicit hidden question, which I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine, but still esoteric, reparameterization to the exponential Gürsey realization that has been the trusty side-knife of many. So, I'll archive the explicit details here for possible utility to future nitpickers.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant physical Higgs is $\sigma$, soon to be frozen to decoupling rigidity by taking its mass to infinity, thus leaving only the goldstons behind.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value (-1) of weak hypercharge, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^- \end{pmatrix} , $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi ~,$$ hence $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi ~.$$

Now, the celebrated Higgs matrix is defined as a side-by-side juxtaposition of these two left-doublets serving as columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$

It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the core group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard linear $\sigma$-model limit to the non-linear one, orthogonally rotate the definition of the three (adjoint) Goldstone variables a bit, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ and normalize $M$ to a unitary matrix, $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}\frac{1}{v} \\ = 1\!\! 1 ~\sqrt{1-\varpi^2/v^2} + i\frac{\vec{\varpi}}{v}\cdot \vec \tau ~ . $$

Finally, to ward off loss of mind, change goldston variables to parallel ones, in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} ~, $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} ~, $$ the standard chiral Goldstone boson matrix with the originally striking weak hypercharge property in question.

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact as in my answer was not enough without the explicit technical details of the Longhitano magic hat trick in your implicit hidden question, which I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine but esoteric reparameterization to the exponential Gürsey realization that has been the trusty side-knife of some of us. So I'll archive the explicit details here for possible utility to nitpickers.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant Higgs is $\sigma$, soon to be frozen to decoupling rigidity by taking its mass to infinity.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value of hypercharge, i.e. -1, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^- \end{pmatrix} , $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi ~,$$ hence $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi ~.$$

Now, the celebrated Higgs matrix is defined as a juxtaposition of these two left-doublets/columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$

It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard $\sigma$-model limit, rotate the definition of the three Goldstone variables a bit, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ and normalize $M$ to a unitary one, $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}/v= \mathbb {1}\sqrt{1-\varpi^2/v^2} + i\vec{\varpi}\cdot \vec \tau /v . $$

Finally, to ward off loss of mind, we change goldston variables to parallel ones in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} , $$ the chiral Goldstone boson matrix with the striking weak hypercharge property discussed.

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact as in my answer was not enough without the explicit technical details of the Longhitano magic hat trick in your implicit hidden question, which I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine but esoteric reparameterization to the exponential Gürsey realization that has been the trusty side-knife of some of us. So I'll archive the explicit details here for possible utility to nitpickers.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant Higgs is $\sigma$, soon to be frozen to decoupling rigidity by taking its mass to infinity.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value of hypercharge, i.e. -1, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^- \end{pmatrix} , $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi ~,$$ hence $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi ~.$$

Now, the celebrated Higgs matrix is defined as a juxtaposition of these two left-doublets/columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$

It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard $\sigma$-model limit, rotate the definition of the three Goldstone variables a bit, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ and normalize $M$ to a unitary one, $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}\frac{1}{v}= \mathbb {1}\sqrt{1-\varpi^2/v^2} + i\frac{\vec{\varpi}}{v}\cdot \vec \tau ~ . $$

Finally, to ward off loss of mind, we change goldston variables to parallel ones in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} , $$ the chiral Goldstone boson matrix with the striking weak hypercharge property discussed.

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact as in my answer was not enough without the explicit technical details of the Longhitano magic hat trick in your implicit hidden question I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine but esoteric reparameterization to the exponential Gürsey realization that has been the trusty side-knife of some of us. So I'll archive the explicit details here for possible utility to those in need of them.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant Higgs is $\sigma$, soon to be sent to decoupling heaven by taking its mass to infinity.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value of hypercharge, i.e. -1, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^-, \end{pmatrix} $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi $$ hance $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi .$$

Now the celebrated Higgs matrix is defined as a juxtaposition of these two doublets/columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$ It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard $\sigma$-model limit, and rotate the definition of the three Goldstone variables and normalize $M$ to a unitary one, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ hence $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}/v= \mathbb {1}\sqrt{1-\varpi^2/v^2} + i\vec{\varpi}\cdot \vec \tau /v . $$

Finally, to ward off loss of mind, we change goldston variables to parallel ones in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} , $$ the chiral Goldstone boson matrix with the striking weak hypercharge property discussed.

Geeky footnote for the above answer

I realized from your conceptual comment request that a mere reassuring demonstration of fact as in my answer was not enough without the explicit technical details of the Longhitano magic hat trick in your implicit hidden question, which I understand as

How does the hypercharge transformation $e^{i\beta /2}$ on a complex Higgs doublet $\Phi$ morph into the $e^{-i\beta \tau_3/2}$ acting on the right of the Goldstone boson matrix picture?

Referring you to Longhitano's thesis paper of 1981, as I did, again glosses over the routine but esoteric reparameterization to the exponential Gürsey realization that has been the trusty side-knife of some of us. So I'll archive the explicit details here for possible utility to nitpickers.

Longhitano starts from the standard Higgs weak isodoublet and weak hypercharge 1 (as in WP) $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\equiv \frac{1}{\sqrt 2} \begin{pmatrix} \varphi_1-i\varphi_2 \\ \sigma +i\chi \end{pmatrix}. $$ The remnant Higgs is $\sigma$, soon to be frozen to decoupling rigidity by taking its mass to infinity.

The conjugate doublet is also a left isotriplet, but, naturally, with the opposite value of hypercharge, i.e. -1, $$ \tilde \Phi =i\tau_2 \Phi^*= \begin{pmatrix} \phi^{0~~*} \\ -\phi^- \end{pmatrix} , $$ so that $$ \Phi \mapsto e^{i(\beta +\vec{\alpha}\cdot \vec{\tau})/2} \Phi ~,$$ hence $$ \tilde \Phi \mapsto e^{i(-\beta +\vec{\alpha}\cdot \vec{\tau})/2}\tilde \Phi ~.$$

Now, the celebrated Higgs matrix is defined as a juxtaposition of these two left-doublets/columns, $$ M\equiv \sqrt{2}(\tilde\Phi, \Phi)= \sqrt {2} \begin{pmatrix} \phi^{0~~*} &\phi^+ \\ -\phi^- & \phi^0 \end{pmatrix}. $$

It is then evident that its transform is $$ \bbox[yellow]{ e^{i\vec{\alpha}\cdot \vec{\tau}/2} \sqrt{2}(\tilde\Phi e^{-i\beta/2}, \Phi e^{i\beta/2})= e^{i\vec{\alpha} \cdot \vec{\tau}/2}\sqrt {2} \begin{pmatrix} \phi^{0~~*}e^{-i\beta/2} &\phi^+e^{i\beta/2} \\ -\phi^- e^{-i\beta/2} & \phi^0 e^{i\beta/2} \end{pmatrix}= e^{i\vec{\alpha}\cdot \vec{\tau}/2} M e^{-i\beta \tau_3/2} }. $$ This is the group theory trick.

All one needs now is to send the mass of the Higgs to infinity, so $\sigma \to v\sqrt {1-\chi^2/v^2}$, the standard $\sigma$-model limit, rotate the definition of the three Goldstone variables a bit, $$ \Phi\to \frac{1}{2}\begin{pmatrix} \varpi_2+i\varpi_1 \\ v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}, $$ and normalize $M$ to a unitary one, $$ M/v\to U= \begin{pmatrix}v\sqrt{1-\varpi^2/v^2}+i\varpi_3& \varpi_2+i\varpi_1 \\ -\varpi_2+i\varpi_1& v\sqrt{1-\varpi^2/v^2}-i\varpi_3 \end{pmatrix}/v= \mathbb {1}\sqrt{1-\varpi^2/v^2} + i\vec{\varpi}\cdot \vec \tau /v . $$

Finally, to ward off loss of mind, we change goldston variables to parallel ones in the more elegant/sensible chiral model parameterization of a unitary matrix, $$ \vec \varpi /v\equiv \hat \pi \sin \frac{|\vec \pi|}{v} $$ so that $$ U= \mathbb {1} \cos \frac{|\vec \pi|}{v} + i\hat \pi \cdot \vec \tau \sin \frac{|\vec \pi|}{v}= e^{i\vec \pi \cdot \vec \tau/v} , $$ the chiral Goldstone boson matrix with the striking weak hypercharge property discussed.