2050207 (7 pages) 2
© World Scientific Publishing Company 3
4
DOI: 10.1142/S0217732320502077 5
The Gervais Neveu gauge in presence of
6
a constant magnetic background
7
Igor Pesando∗and Federico Piola
8
Dipartimento di Fisica, Universit`a di Torino and I.N.F.N. – sezione di Torino 9
Via P. Giuria 1, I-10125 Torino, Italy 10 ∗ipesando@to.infn.it 11 Received 17 January 2020 12 Revised 5 May 2020 13 Accepted 21 May 2020 14 Published 15
We compare the gauge fixed Abelian Dirac-Born-Infeld action with string amplitudes 16
in presence of a constant magnetic background in order to find the mapping between 17
QFT polarizations and string polarizations, and to extract the gauge suggested by string 18
theory in presence of a constant magnetic background, which generalizes the Gervais– 19
Neveu gauge. 20
Keyword : String theory. 21
In this paper we aim to derive the generalization of the Gervais–Neveu gauge1 in
22
presence of a constant magnetic background and for DBI action. This is done by
23
comparing gauge fixed DBI with open string amplitudes with a constant magnetic
24
background.2 As in the previous case the way the string reproduces the QFT
am-25
plitudes is minimal, in fact many terms that arise from expanding the DBI around
26
a magnetic background are total derivatives, which are not reproduced by the
min-27
imal off shell extension of the string amplitudes. Nevertheless the string adds a
28
term, the first line of Eq. (14), which has to be reabsorbed into the gauge fixing
29
because it arises naturally from the expansion in momentum powers of the string
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amplitude. Our point of view is to consider the plain commutative QFT and not
31
its non commutative version.
32
We define Eµν = gµν+ 2πα0Fµν with µ, ν = 0, . . . D − 1 and g = (−, (+)D−1).
33
We define also its inverse as Eµν = Gµν−Θµν
2πα0 where Gµν is the open string metric.
34
The DBI Lagrangian density then reads
35
L = −TDe−Φ0p− det(g + 2πα0F ) = −TDe−Φ0p− det(E). (1)
36
We now switch on a constant magnetic background Fµν(0) and consider the
fluctu-1
ations around it as Fµν = Fµν(0)+ fµν with fµν = ∂µaν− ∂νaµ. Next we consider
2
the expansion of the DBI up to third order in f as L = L(0)+ L1+ L2+ L3+ O(f4)
3 where (2πα0= 1) 4 L(0)= −TDe−Φ0p− det(E0), L1= L(0)12tr(E −1 0 f ) = −L(0) 1 2tr(Θ(0)f ) = total der., (2) 5 and 6 L(2) = L(0) 1 8(tr(E −1 0 f )) 2−1 4tr(E −1 0 f E −1 0 f ) 7 = L(0) −1 4tr(G −1 (0)f G −1 (0)f ) + total der., (3) 8
where the terms containing Θ(0) ⊗ Θ(0) are total derivatives as required by
con-9
sistency with the propagator suggested by string theory. Finally the terms O(f3)
10 read 11 L(3) = L(0) − 1 48(tr(Θ(0)f )) 3+1 8tr(Θ(0)f )tr(G −1 (0)f G −1 (0)f ) 12 +1 8tr(Θ(0)f )tr(Θ(0)f Θ(0)f ) − 1 6tr(Θ(0)f Θ(0)f Θ(0)f ) 13 −1 2tr(Θ(0)f G −1 (0)f G −1 (0)f ) 14 = L(0) +1 8tr(Θ(0)f )tr(G −1 (0)f G −1 (0)f ) − 1 2tr(Θ(0)f G −1 (0)f G −1 (0)f ) 15 + total der., (4) 16
where all terms Θ(0)⊗ Θ(0)⊗ Θ(0) are total derivatives as required by consistency
17
with string theory. In order to compute amplitudes we need a gauge fixing action,
18
which we assume of the form Lgf = −12Fgf2 with Fgf, a composite scalar as
sug-19
gested by previous studies. As for L we can expand Fgf = Fgf (1)+ Fgf (2)+ · · · ,
20
where Fgf (1) is linear in aµ(x) and so on.
21
We now consider the dipole string in a constant magnetic field. We write the
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open string expansion with Euclidean worldsheet coordinate u = eτE+iσ ∈ H as
23 Xµ(u, ¯u) = xµ0− i√2α0hE−T (0) ln(u) + E −1 (0)ln(¯u) iµν pν 24 + i1 2 √ 2α0X n6=0 1 n h E(0)−Tg u−n+ E(0)−1g ¯u−ni µ ν ανn. (5) 25
The commutation relations read 1 [xµ0, xν 0] = iΘ µν (0), [xµ0, pν] = iδµν, [αµ n, ανm] = nG µν (0)δm+n,0 (6) 2
and the vacuum is defined as usual as pµ|0i = αµn|0i = 0 for n > 0. Then we have
3
the relation among radial ordering, normal ordering, and Green function given by
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R[Xµ(u, ¯u)Xν(v, ¯v)] =: Xµ(u, ¯u)Xν(v, ¯v) : +Gµν(u, ¯u; v, ¯v) with
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Gµν(u, ¯u; v, ¯v) = −α0g−1ln(u − v) + g−1ln(¯u − ¯v) + E(0)−TE(0)g−1ln(u − ¯v)
6
+ E(0)−1E(0)T g−1ln(¯u − v) + D
µν
. (7)
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The energy–momentum tensor is T (u) = −α10 : ∂uXTg∂uX : from which we read
8
the Virasoro generators
9 L0= α0p ◦ p + α−1◦ α1+ · · · , L1= √ 2α0p ◦ α 1+ · · · , (8) 10 with a ◦ b = aµG
µνbν = aµGµνbν according to the fact that a and b are naturally
11
covariant or controvariant. In particular p ◦ p = pµGµνpν.
12
Finally the vertex describing the fluctuations of the gauge field aµ(x) is
13 Va(x; k, ζ) = i r 1 2α0 : ζµ∂xX µ(x, x)eikνXν(x,x) :, (9) 14
where it is worth noticing that ∂xX(x, x) = 2G(0)−1E(0)∂uX|u=x. The state associated
15
16 to the previous vertex is V
a(0; k, ζ)|0i = ζµαµ−1|ki where the physical conditions
17
are k ◦ k = k ◦ ζ(k) = 0.
18
Following the approach of Ref. 4 we can derive the propagator string theory and
19
then the gauge fixed kinetic term, that is, the 2 vertex is
20 V2= (2π)DδD(k1+ k2) −1 2 k1◦ k1ζ1◦ ζ2, (10) 21 with ζi= ζ(ki). 22
As a final step, before starting to compare with the DBI action, we compute the
23
three photons Green function. We start with the basic on shell correlator, which
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reads (we use Polchinski’s book notation and hence hhk|pµ = kµhhk|)
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A(on shell)123 = A(on shell)(k1, ζ1; k2, ζ2; k3, ζ3)
26 = hhk1, ζ1|V (x = 1; k2, ζ2)|k3, ζ3i 27 =√2α0e−i2 P 1≤i<j≤3ki∧kj[−ζ 1◦ ζ2k2◦ ζ3+ ζ2◦ ζ3k2◦ ζ1 28 + ζ3◦ ζ1k3◦ ζ2− 2α0ζ1◦ k2ζ2◦ k3ζ3◦ k2](2π)Dδ(k1+ k2+ k3), 29 (11) 30
with a ∧ b = aµΘµν(0)bν. The S-matrix element is then 1 S123(k1, ζµ1; k2, ζµ2; k3, ζµ3) = C0(A (on shell) 123 + A (on shell) 132 ) = 0, (12) 2
where C0 is the coupling constant. In order to compare with the DBI action and
3
get the gauge fixing implied by string theory, we need an off shell definition on
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the previous S matrix element. Given the off shell A(off shell)123 we can compute the
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3-point vertex as V3 = A
(off shell)
123 + A
(off shell)
132 and the part of the action that is
6 cubic in ζ as S3 = R Q3i=1 d Dk i (2π)D 1
3!V3(k1, k2, k3), which we can compare with the
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gauge fixed DBI at the same order.
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The off shell A(off shell)123 can be obtained in at least two different ways. One
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approach is to take (13) off shell simply by not imposing the physical constraints.
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This however cannot be done immediately in such a naive way since we want a
3-11
point vertex V3and such a vertex must be totally symmetric in the exchange of the
12
photons. Using the naive off shell extension of A(on shell)123 has not such a property but
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we can fix the problem by requiring the off shell amplitude to be cyclical symmetric.
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The result is then
15 A(off shell)123 =√2α0e−i 3! P 1≤i≤3ki∧ki+1[+ζ 1◦ ζ2k2◦ ζ3+ ζ2◦ ζ3k3◦ ζ1 16 + ζ3◦ ζ1k1◦ ζ2− 2α0ζ1◦ k2ζ2◦ k3ζ3◦ k1](2π)Dδ(k1+ k2+ k3). (13) 17
with k4= k1, which is the on shell equivalent to the amplitude (13).
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The other way that gives the same result as above is to consider the CSV
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vertex,5 which is by construction cyclically symmetric. In Ref. 6, it was shown
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that it cannot be taken as the only vertex of OSFT since the 4 pts would not be
21
cyclically symmetric. It is anyhow believed that adding higher point vertexes in
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order to satisfy the A∞structure, a well-defined OSFT can be obtained, therefore,
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it is a good off shell starting point.
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Since we are considering the commutative version of the DBI we have to extract
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from the previous amplitude the terms that can be compared with the L3 ∼ f3∼
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k3ζ3. Actually there are two classes of terms L3 ∼ Θ(0)f3 + Θ3(0)f3. If we look
27
to the string we realize that from the expansion of e−i2
P
1≤i≤3ki∧ki+1 we get A ∼ 28
kζ3+ Θ(0)k3ζ3+ Θ2(0)k5ζ3+ · · · . Terms like Θ(0)k3ζ3 are those that are needed
29
but also kζ3 terms cannot be neglected since they arise from the first order of the
30
expansion of the exponential. Since these kζ3 terms are not in L they must be and
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are in Lgf. On the other side L(3) contains also terms Θ3(0)f3 and these cannot
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come from string amplitudes and in fact they are total derivatives as previously
33
noted. The limit of the string 3 vertex we consider is then
34 V3= √ 2α0C 0(−ζ1◦ ζ2k3◦ ζ3+ 2 cyclic permutations) + −3i 2 √ 2α0C 0 35 × (k2∧ k3ζ1◦ ζ2k2◦ ζ3+ 5 permutations) (2π)Dδ(k1+ k2+ k3), (14) 36
Author, Eqs. (11)
and (13) have the
same label.
Eq. (13) correctly
mentioned here?
Author,
Eq. (13) correctly
mentioned here?
where the first line was originally proportional to (+ζ1◦ ζ2k2◦ ζ3+ 5 permutations)
1
and in the second line we used k1∧k2= k2∧k3= k3∧k1as follows from momentum
2
conservation to write the last term.
3
We now proceed to the comparison between the gauge fixed action suggested by
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string theory and the DBI with gauge fixing term so determining the gauge fixing
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Lgforder by order in power of aµ. We express the QFT polarization µ(k)/(2πα0) =
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R dDxe−ikνxνa
µ(x) (where the factor (2πα0) is inserted in order to reabsorb the
7
same factor in the DBI) as a function µ= µ(ζ) of string polarization ζµ and the
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gauge fixing Lagrangian as a function of µ.
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We start with the first non trivial order, i.e. a2 ∼ 2. It is easier to compare
10
the action than the vertex since in this way we have not issues associated with the
11
symmetrization of the vertex. From R dDx(L
2−12Fgf (1)2 ) we get 12 S2= Z 2 Y i=1 dDk i (2π)D(2π) DδD(k 1+ k2) L(0) −1 2 [k1◦ k2(k1) ◦ (k2) 13 − k2◦ (k1)k1◦ (k2)] − 1 2 ˜ Fgf (1)(k1) ˜Fgf (1)(k2) , (15) 14
therefore to this order
15 µ(k) = 1 p−L(0) ζµ(k) + O(ζ2), ˜ Fgf (1)(k) =p−L(0)k ◦ (k). (16) 16
Let us now examine the action S3 of order f3. Contributions to it come from
17
L(3), higher order field redefinition µ(2) ∼ ζ2 and Fgf (2). More explicitly setting
18 aµ(2)(x) = R Q3 i=2 dDki (2π)Dµ(2)(k2, k3)ei(k2+k3)νx ν
and similarly for Fgf (2) we can
19 write 20 S3= Z 3 Y i=1 dDk i (2π)D ( ˜ L(3)(k1, k2, k3) + L(0) 2 2k1◦ k1(1)(k1) ◦ (2)(k2, k3) 21 −1 22 ˜Fgf (1)(k1) ˜Fgf (2)(k2, k3) ) (2π)Dδ(k1+ k2+ k3), (17) 22
where the factors 2 in the second and third line come from the squares. The explicit
23 computation gives 24 L(3)(k1, k2, k3) = −L(0) 2 i3 3![2(k2∧ k31◦ 2k2◦ 3+ 5 permutations) 25 + k1◦ k11◦ (−k22∧ 3+ 22k2∧ 3+ permutations) 26 + k1◦ 1(+2∧ 3k1◦ k2− 2k1∧ 2k2◦ 3+ permutations)] 27 × (2π)Dδ(k 1+ k2+ k3), (18) 28
where the first line is the main stringy contribution, the second line is proportional
1
to the gauge fixed kinetic term and the third line is proportional to ˜Fgf,1(k).
Sub-2
stituting (18) into (17) and comparing with the stringy result (14) we can fix the
3 amplitude normalization to be 4 C0= r 2 α0 1 p−L(0) , (19) 5
the field redefinition to the second order
6 µ(2)(k2, k3) = i 2(−kµ2(1)(k2) ∧ (1)(k3) + 2µ(1)(k2)k2∧ (1)(k3)) 7 = i 2 p−L(0) 2(−kµ2ζ(k2) ∧ ζ(k3) + 2ζµ(k2)k2∧ ζ(k3)), (20) 8 which implies 9 aµ(x) = 1 p−L(0) ˆ aµ+ 1 2 p−L(0) 2Θ αβ (0)(−∂µˆaαaˆβ+ 2∂αaˆµaˆβ), (21) 10 where ˆaµ(x) = R dDk (2π)Dζµ(k)eikνx ν
is the gauge field fluctuation with string
polar-11
ization. Finally the second-order gauge fixing term reads
12 ˜ Fgf (2)(k2, k3) = − i 2p−L0(−(k2) ∧ (k3)k2◦ (k2+ k3) 13 + 2(k2+ k3) ∧ (k2)k2◦ (k3)) + (k2) ◦ (k3), (22) 14 which implies 15 Fgf = −iGµν∂µaν+ Gµνaµaν− i 2p−L0G µνΘαβ (0)(∂µ∂νaαaβ+ ∂νaα∂µaβ 16 − 2∂µ∂αaβaν− 2∂µaβ∂αaν), (23) 17
where the first line reduces to the Gervais–Neveu gauge when we switch the
mag-18
netic background off.
19
In Ref. 7 the gauge chosen by open bosonic string field theory without any
back-20
ground was determined. The starting point was the full OSFT, which includes all
21
ghosts necessary for the BV formalism. After a series of field redefinitions necessary
22
to decouple the ghosts from the matter fields the result was that the gauge chosen
23
was the plain Lorenz gauge. This is at variance from the results obtained in first
24
quantization where a certain degree of ambiguity is present in the choice of the way
25
of going off shell. It is also plausible that the reason is that the field redefinition
26
involved are the cause and it would be interesting understanding the true reason.
27
Independently of that the gauge suggested by the first quantized version of the
28
string is more efficient than the Lorenz gauge since the terms surviving the gauge
29
fixed action are fewer.
References
1
1. J. L. Gervais and A. Neveu, Nucl. Phys. B 46, 381 (1972), doi:10.1016/0550-3213(72)
2
90071-5.
3
2. M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144, 425 (1934), doi:10.1098/rspa.
4
1934.0059; E. Fradkin and A. A. Tseytlin, Phys. Lett. B 163, 123 (1985), doi:10.1016/
5
0370-2693(85)90205-9; A. Abouelsaood, C. G. Callan, Jr., C. Nappi and S. Yost, Nucl.
6
Phys. B 280, 599 (1987), doi:10.1016/0550-3213(87)90164-7.
7
3. E. Coletti, I. Sigalov and W. Taylor, JHEP 0309, 050 (2003), doi:10.1088/1126-6708/
8
2003/09/050.
9
4. P. C. Argyres and C. R. Nappi, Nucl. Phys. B 330, 151 (1990), doi:10.1016/
10
0550-3213(90)90305-W; I. Pesando, Nucl. Phys. B 918, 129 (2017), doi:10.1016/j.
11
nuclphysb.2017.02.007.
12
5. S. Sciuto, Lett. Nuovo Cimento 2, 411 (1969); A. Della Selva and S. Saito, Lett. Nuovo
13
Cimento 4, 689 (1970); L. Caneschi and A. Schwimmer, Lett. Nuovo Cimento 3S1,
14
213 (1970) [Lett. Nuovo Cimento 3 (1970) 213], doi:10.1007/BF02755850; L. Caneschi,
15
A. Schwimmer and G. Veneziano, Phys. Lett. B 30, 351 (1969), doi:10.1016/
16
0370-2693(69)90503-6.
17
6. A. LeClair, M. E. Peskin and C. Preitschopf, Nucl. Phys. B 317, 411 (1989),
18
doi:10.1016/0550-3213(89)90075-8.
19
7. H. Feng and W. Siegel, Phys. Rev. D 75, 046006 (2007), doi:10.1103/PhysRevD.75.
20
046006.
