Nambu's Child Comes of Age: Quantization of The...

Nambu’s Child Comes of Age:Quantization of The Effective String

Simeon HellermanKavli Institute for the Physics and Mathematics of the Universe

Tokyo University Institutes for Advanced Study

S.H. and Ian Swanson, Phys.Rev.Lett. 114 (2015) 11, 111601S.H., J. Maltz, S. Maeda, I. Swanson, JHEP 1409 (2014) 183

S.H., and Ian Swanson, In ProgressS.H., and Shunsuke Maeda, In Progress

Nambu and Science FrontierOsaka UniversityOsaka, Japan

November 17, 2015

Classical string model of the Regge spectrum

One of Nambu’s great contributions to science was the inventionof string theory.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg

Image credit http://courses.washington.edu/phys55x/ %Physics20557_lec11_files/image064.jpg


The dual resonance model was originally formulated to explainremarkable, robust patterns in hadronic spectral and scatteringdata.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



The theory was originally formulated in terms of an abstractscattering matrix.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



Nambu and Goto’s relativistic string provided a dynamical modelthat could in principle generate such an S-matrix.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



The key insight had to do with the poles in the S-matrix, which isto say, the spectrum of (almost stable) states in the dual resonancemodel.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



All hadronic states appear to lie in a series of towers of resonances.Each tower can be plotted on a graph of mass-squared versusangular momentum, as straight lines with a common, universalslope.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



m2 =J

α′, α′ =

12πTstring

Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



We know today that the string theory of strong interactions isJUST WRONG at distances ∼<

√α′, where QCD takes over.

However we can still treat string theory as a perfectly good

::::::::effective theory at scales >>

√α′.




For a string with large angular momentum, its::::::length is '

√Jα′ so

we should be able to use the effective theory of the stringworldsheet when J >> 1.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



This point of view predicts corrections to the Regge spectrumin the form

m2 =J

α′·[1 + O

(J−κ

) ], κ > 0 .




The subleading powers in J correspond to higher-dimensionoperators in the wordsheet action, and also quantm corrections.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



However, since the original proposal of the Nambu-Goto model, nosystematic quantization of the string as an effective theory wasdeveloped to the point where subleading corrections could becalculated... until recently.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



This talk will describe the calculation of the first subleadingcorrections.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



In particular, we shall show that the subleading correcions includean adjustable parameter corresponding to a renormalized quarkmass, and also an interesting universal term corresponding to theasymptotic intercept of the linear Regge trajectory.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



We will show the mass-squared has an expansionM2 = J

α′ + cq J+ 1

4 + c0 where cq is theory-dependent andc0 isuniversal and calculable.Image credit http://phys.columbia.edu/kabat/why_strings/Regge.jpg



I You might ask: Why does the Nambu-Goto action work atall, in any approximation?

I When the string is large, the short-distance structureshould become irrelevant, in the technical sense of therenormalization group.

I The dynamics should be described by the most relevantterms one can write in a local action for a string, invariantunder all the appropriate symmetries.

I The most relevant term invariant under the Poincarésymmetry of D-dimensional spacetime is the Nambu-Gotoaction:

ISNG = Tstring · Areaworldsheet ,

Tstring ≡1

2πα′.


I The Nambu-Goto action describes the spectrum witharbitrarily good precision when the string is large, with typicalsize sale "R".

I Less relevant terms in the action should contribute withpowers (perhaps including logarithms) of R/

√α′.

I An operator scaling as Length−p contributes to anyobservable at relative order R−(p+2) ("Relative" to the leadingNambu-goto contribution, that is).

I The coarse analysis of large-R corrections is easy – to learnthe power laws that appear rather than their coefficients, justclassify possible invariant operators up to some order ininverse length.


I For this talk we are exclusively interested in the firstsubleading correction to any given amplitude.

I The first question should be "Is the Nambu-Goto actionenough"?

I In certain situations, the answer is:::yes – assuming the theory

is::::::::::quantized

::::::::::::consistently.

I The leading:::::::::correction to the NG action – including the

::::::::::::::::::curvature-squared term – scale as |X |−2

I Therefore it contributes to M2meson at order J−1 at most.

I Therefore the:::::::::::asymptotic

:::::::Regge

::::::::::intercept – the order J0 term

in the::::::large-J expansion of M2

meson – is calculable anduniversal – i.e.

:::::::::::::independent of details of the

:::::::::::Lagrangian.


I To carry out the analysis, we must pick a gauge.I The two most commonly used gauges (for D not equal to the

critical dimension) are orthogonal gauge and static gauge.I For non-static problems, static gauge is inconvenient.I I’ll therefore describe effective string theory in conformal

gauge in a simplified framework embedded in the Polyakovformalism.

Covariant effective string theory simplified

I Let’s begin by considering the usual Polyakov action for thebosonic string, but with an arbitrary number D of embeddingcoordinates.: The Polyakov string is defined by the pathintegral

Z =

∫DMPolyakov

[g ] exp (−SPolyakov) ,

DMPolyakov[g ] ≡

D[g ]X D[g ]g

D[g ]Ω

SPolyakov =

∫d2σ

√|g••|LPolyakov ,

LPolyakov =1

4πα′gab ∂aX

µ ∂bXµ ,

The action SPolyakov is Weyl-invariant but the measureDMPolyakov

[g ] is not.


Z [g ] =∫DMPolyakov

[g ] exp (−SPolyakov) .

Under a Weyl transformation g•• → exp (+2ω) g••,

the individual factors of the integrand transform as:

SPolyakov → SPolyakov

DMPolyakov[g ] → exp

[(D − 26)Fanom[g , ω]

]DMPolyakov

[g ]

The form of the::::::::anomaly

:::::::::::functional F [g , ω] is determined uniquely to be

Fanom[g , ω] ≡ 124π

∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

by the Wess-Zumino consistency condition.




SPolyakov → SPolyakov

DMPolyakov[g ] → exp

[(D − 26)Fanom[g , ω]

]DMPolyakov

[g ]


∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

So the

:::::path

::::::::integral is not invariant unless D = 26:

Z [g ]→ exp[

(D − 26)F [g , ω]

]Z [g ]





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ exp

[(D − 26)F [g , ω]

]Z [g ]

In order to obtain a:::::::::::::::Weyl-invariant

:::::::::partition

:::::::::function,

we can augment the action by a term Sanom that transforms under

a:::::Weyl

:::::::::::::::transformation as

Sanom → Sanom + (D − 26)F [g , ω].



[g ] exp (−SPolyakov − Sanom) .


∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]






∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


In the "linear dilaton theory", one cancels this anomaly byassigning a nontrivial Weyl transformation to one of the scalarsXD−1 ≡ 1

|V | VµXµ:

XD−1 → XD−1 −√

26− D

6α′ω .

Weyl-invariance requires ∆L = 14π R(2) V · X .





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


From X we define a Liouville field φ,

XD−1 = −√

6α′

26− Dφ , φ = −

√26− D

6α′XD−1 ,

In terms of which the transformation of φ is

φ→ φ+ ω ,

and the anomaly action is

Sanom ≡ Sφ = +26− D

24π

∫d2σ

√|g |(gab ∂aφ∂bφ− φR(2)

)





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,

Sanom ≡ Sφ = +26−D24π

∫d2σ


).

This is a consistent string theory in D dimensions, but with only aD − 1-dimensional Poincaré symmetry.





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,


∫d2σ


).





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,


∫d2σ


).

However there are other choices for the anomaly actionpreserving a full D-dimensional Poincaré symmetry.Instead of modifying the Weyl transformation of one of the coordinates,we can construct a

::::::::::composite operator φ that transforms

as a Liouville field.





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,


∫d2σ


).

Any choice of φ transforming as a scalar under diffeomorphismsand a Liouville field under Weyl transformations will lead to ananomaly-free effective string theory by this prescription.We would also like to preserve D-dimensional Poincaré symmetry,so we want to make a Poincaré-invariant choice for φ.





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,


∫d2σ


).

The very simplest such choice of φ is φ ≡ −12 ln

(g•• ∂•X

µ∂•Xµ).

This choice of φ has all the correct properties.





∫ √|g | d2σ

g•• ∂•ω∂•ω + ωR(2)[g ]

Z [g ]→ Z [g ]


φ→ φ+ ω ,


∫d2σ


).

φ ≡ −12 ln

(g•• ∂•X

µ∂•Xµ).

This choice of φ is convenient and natural but not unique.



[g ] exp (−SPolyakov − Sφ) .

Sanom ≡Sφ = +26−D24π

∫d2σ


).

φ ≡ −12 ln

(g•• ∂•X

µ∂•Xµ).


I Now we can fix conformal gauge gab = δab. In the usualholomorphic coordinate w ≡ σ1 + iσ2 we haveδww = 1

2 , δww = 2 with all other components vanishing.

I In:::this

:::::::gauge we have

Lanom =β

2π(∂2X · ∂X )(∂X · ∂2X )

(∂X · ∂X )2, β ≡ 26− D

12

+(terms proportional to ∂∂X ) .

I Here we have abbreviated ∂ ≡ ∂w , ∂ ≡ ∂w .I People familiar with the

:::old covariant effective string

formalism of Polchinski and Strominger (1989) in orthogonalgauge will recognize this Lagrangian as their ad hoc anomalyterm.

Effective string theory for rotating strings

I For rotating strings we know the length R should scale as√α′J when J is large.

I So calculating the relative-order R−2 corrections for rotatingstrings corresponds to calculating the relative-order J−1

corrections.I The leading-order value of the mass-squared for an open

string in four dimensions is

m2leading Regge =

J

α′.

In fact, this relationship defines the (asymptotic) Regge slopeα′.

I Computing a relative order J−1 term would corrspond tocomputing the asymptotic Regge intercept on the "leadingtrajectory" – that is, the set of states of lowest mass for agiven angular momentum.


I Formally, the relative order J−1 correction to the dispersionrelation on the leading trajectory is particularly simple,because the lowest state with given Noether charges isautomatically Virasoro-primary, so the physical stateconditions are automatically satisfied, except the mass-shellcondition from L0.

I The correction to the mass-squared of the string state is givenby

∆M2∣∣ closedfirst−order

=2α′

∆Ews∣∣first−order ,

∆M2∣∣ openfirst−order

=1α′

∆Ews∣∣first−order ,

∆Ews∣∣first−order = 〈(P, J)|free Hfirst−order |(P, J)〉free ,


I This in turn is given by the Casimir energy −D−212 , plus the

classical value of the interaction Hamiltonian in the classicalrotating solution with the appropriate angular momenta.

I The classical value of the perturbing Hamiltonian is equal tothe negative of the classical value of the perturbingLagrangian. This follows from elementary manipulations inclassical mechanics and applies only to the lowest state of asystem with fixed Noether charges.

I No higher loops or even one-loop diagrams involvinginteraction vertices contribute at NLO in J. Each additionalinteraction vertex, and each additional quantum loop, issuppressed by

::at

::::::least

::::one additional power of J.


I Let’s see how this works, concretely, for open strings inconformal gauge.

I The solution for the lowest-lying state with angularmomentum J in a single plane is of the form

X 0 = 2α′ P0 σ0 ,

Z = −i√α′ J

(e iσ

++ e iσ

−),

with σ1 running from 0 to π. The classical solution satsifiesthe Neumann boundary condition at σ1 = 0, π.


I For this case, our analysis breaks down in its own terms.I The Lagrangian is singular near σ1 = 0, π in the classical

solution.I This is a non-integrable singularity. The integral diverges:

L PSrotating solution

= − β

2π2sin2(2σ1)

(1− cos(2σ1))2 .


I For the open string, this singularity is present because theboundary is moving at the speed of light and there is acurvature singularity in the

::::::::::Lorentzian

::::::::induced

:::::::metric.

I For the closed string, there is a singularity representing a foldin the string.

I In both cases, the integrated anomaly term diverges.I We will first consider a model calculation that avoids this

singularity.I This breakdown of the theory is a short-distance singularity, to

be removed by renormalization.I But first, let us consider a simpler case, where there is no such

singularity.

Closed strings with rotation in two planes

I Let us now perform a calculation in a simple case to illustratethe general idea of large-J universality at subleading order.

I The properties of rotations are different in higher dimensions.So we consider closed strings rotating in D ≥ 5, which neednot have folds: The Polchinski-Strominger denominator isnonvanishing everywhere.

I We consider closed strings in D ≥ 5, with nonzero classicalangular momenta J1,2 in two planes simultaneously.

I In terms of the SO(4) = SU(2)+ × SU(2)− subgroup of theSO(D − 1) little group, the total angular momenta areJ± ≡ 1

2(J1 ± J2) where we assume WLOG that J1 > J2 > 0.


I The classical solution is

X 0 = α′ P0 σ0 ,

Zi = −i√α′

2

(αZi−1e

iσ++ αZi

−1eiσ−),

Zi = i

√α′

2

(αZi

1 e−iσ+

+ αZi1 e−iσ

−),

I Here, the mode amplitudes are

αZ1−1 = αZ1

1 = αZ1−1 = αZ1

1 =√

J1 ,

aZ2−1 = αZ2

1 = −αZ2−1 = −αZ2

1 =√J2 .


I Evaluated in this rotating solution, the contribution of the PSanomaly term, evaluated in the rotating ground state, takesthe form

L PSrotating solution

= −βJ2−

2π2sin2(2σ1)

(J+ − J− cos(2σ1))2 .

I This Lagrangian density becomes singular at the endpointsσ1 = 0 and π, in the limit J+ = J−. This limit is imposedautomatically in D = 4, as the little group SO(D − 1) hasrank one, and J2 must vanish.

I But for generic biplanar angular momenta, this density issmooth.


I The resulting mass shift is

M2closed =

1α′

[2(J1 + J2)− D − 2

6

+26− D

12

((J1

J2

) 14

−(J2

J1

) 14)2]

+ O(J−1) .

I The contribution from the PS term is nonzero unless J1 = J2,or D = 26.

I When J2 is taken to zero, this diverges as a fold develops.I At present, we do not understand how to renormalize the

singular Hamiltonian at the fold.

Renormalization of boundary singularities

I Since we don’t understand that, let us return to our originalfocus on strings with boundaries.

I Our approach is to regulate and renormalize the boundarysingularities in the standard way.

I This works, because all UV-divergences are local terms.


I The classical solution is

X 0 = 2α′P0σ0

Z1 = i

√α′

2αZ1

1

(e−iσ

++ e−iσ

−),

Z2 = i

√α′

2αZ2

22

(e−2iσ+

+ e−2iσ−),

Z1 = −i√α′

2αZ1−1

(e iσ

++ e iσ

−),

Z2 = −i√α′

2αZ2−2

2

(e2iσ+

+ e2iσ−),


I Here,

αZ11 =

√2J1 αZ1

−1 =√

2J1

αZ22 = 2

√J2 αZ2

−2 = 2√J2 .


I Remember that we can modify our choice for the compositeLiouville field φ.

I We would like to do so so that it our choice is smooth nearthe boundary.

I Such a choice is

φ ≡ −14ln(X 2

11 − ε4 α′ X22) ,

X22 ≡ X22 −X12X21

X11, Xpq ≡ ∂p+X · ∂

q−X

I Near the boundary, this behaves as X22 ' −X22, which isnonzero and smooth.


I Now, modulo terms that do not contribute, the density of thePS term is

LPS, reg ≡β

2πX12X21

X 211 + ε4 α′ X22

.

I Note that wherever and whenever X11 6= 0, we haveLPS, reg → LPS. The short-distance modification is irrelevant,whenever the leading-order action is nonzero. Theshort-distance modification kicks in only at boundaries andfolds.

I The integral is

∆M2open =

1ε

26− D

24α′(J1 + 8J2)1/4 + (finite) .


I The short-distance singularity can be cancelled by a local termat the boundary, of the form

Oquark ≡ (X22)+ 14 = (−X22)+ 1

4

I This may seem like a peculiar operator, but in fact allboundary operators for open strings with Neumann boundariesare nonsingular operators Xpq dressed with powers of X22.

I This is true in any model where one obtains thefour-dimensional effective string theory starting out with amicroscopically well-defined microscopic string theory andintegrating out degrees of freedom.


I After renormalization, we find

M2open =

1α′

[J1 + 2J2 −

D − 224

+26− D

24

−4 +3 J1 + 4 J2

J121√J1 + 8J2

]+ O(J−1) .

I For angular momenta lying in a single plane (i.e., whenJ2 = 0), the mass-squared equals M2

open = (J1 − 1)/α′,independent of D. Of course, when D = 26, we obtainM2

open = (J1 + 2J2 − 1)/α′.I This is the case in which the bosonic string theory is

well-defined microscopically, and the singular PS anomalyterm is absent.


I It is worth emphasizing that we have fine-tuned the coefficientof the quark mass operator O(quark) so that there is no termof order J1/4 in the mass-squared formula.

I Generically we should expect a J1/4 term in the open-stringmass-squared, unless the mass of the quark at the endpoint islight compared to the scale of the string tension.


I In real QCD there will be additional degrees of freedom at theendpoints, carrying spin and flavor degrees of freeedom.

I The attachment of fermionic endpoints to the bosonic interiorof the worldsheet can be done in the effective framework. Thisis work in progress with Shunsuke Maeda and Ian Swanson.

I These degrees of freedom carry symmetries that constrain theallowed operators. In particular, chiral symmetry forbids quarkmasses, which are associated with the J+ 1

4 term in theboundary action.

I We expect that in the correct effective boundary CFT of thereal QCD string, the J+ 1

4 term in the action will be related byexact chiral symmetry to an amplitude for soft pion emission.

Structure of boundary operators

I Now one is led to ask an overwhelming question:I One might ask, why is the answer universal at all?I And why do the boundary operators appear containing these

strange:::::::::::::::quarter-integer powers X22?


I We will explain this in future work to appear with IanSwanson.

I This is one of the more::::::::::surprising

::::::::features of the effective

string theory with::::::::::Neumann

::::::::::boundary

:::::::::::conditions.

I We will demonstrate that::::any short-distance modification of

the theory such that the::::bulk of the worldsheet is an

::::::::ordinary

::::::::effective

:::::::string

:::::::theory with an

::::::::::::organization

:::of

::::::::::operators such

as we have described, with operators:::::::dressed with powers of

X11 – generically::::::::negative

:::::::::::::::quarter-integer ones.

I For such a theory, the:::::::::boundary operators always appear

dressed with powers of X22 – generically::::::::negative

:::::::integer ones.

I This is so for::::::::artificial

::::::::::::::short-distance

::::::::cutoffs preserving the

symmetries and also for::::real

:::::::::::::::short-distance

:::::::::effective

::::::::theories.


I The result is that all boundary operators are of the form(∏

pq Xpq)/X k22.

I We can quickly see that there are no::::::::marginal boundary

operators of vanishing X-scaling.I First, use the EOM to reduce all derivatives of X to the form∂p0X or ∂p0∂1X .

I Then use:::::::::Neumann boundary conditions to eliminate the

:::::latter.


I Now, all bilinear invariants of::X at the boundary are of the

form B(pq) ≡ ∂p0X · ∂

q0X .

I All all boundary operators are of the form (∏

pq B(pq))/Bk(22).

I Now consider only:::::::::marginal boundary operators.

I If the "undressed" operator (the numerator) has dimension∆ ≡

∑pq p + q, then the dressing is B−(∆−1)/4

(22) .


I Then in order to have positive or zero X -scaling, theundressed operator must have ∆ ≤ 5.

I The operators B11 and B12 vanish as independent operatorsbecause they are proportional to free-field

::::::stress

::::::::tensors and

::::first

:::::::::::derivatives thereof.

I The only marginal operator with ∆ = 5 is B(23)/B(22) which isa total derivative along the boundary.

I So after modding out by Virasoro descendants, the onlymarginal operator with nonnegative X -scaling is the quarkmass term, corresponding to ∆ = 4.

I The asymptotic intercept is universal, independent of anyshort-distance modification with the same light degrees offreedom and symmetries.

Conclusions

I We have found that the effective string theory framework ispredictive for large-J corrections to the spectrum of rotatingstrings.

I For closed strings in D = 4 and open strings in any dimension,the leading power corrections are ∆M2 ∝ J+ 1

4 with atheory-dependent coefficient. These terms are associated withlocalized terms at bounaries and folds.

I The (asymptotic) Regge intercept is is universal and

::::::::::calculable, modulo the

::::::quark

:::::mass term.

I By the way... So is:::::every

::::::other observable at NLO.

I Nambu’s child, the quantum mechanical theory of therelativistic effective string, has finally come of age.

I Thank you.

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